ABSTRACT. Kiran Kumar Somasundaram, Doctor of Philosophy, 2010

Transcription

1 ABSTRACT Title of dissertation: TOPOLOGY CONTROL ALGORITHMS FOR RULE-BASED ROUTING Kiran Kumar Somasundaram, Doctor of Pilosopy, 2010 Dissertation directed by: Professor Jon S. Baras Department of Electrical and Computer Engineering In tis dissertation, we introduce a new topology control problem for rulebased link-state routing in autonomous networks. In tis context, topology control is a mecanism to reduce te broadcast storm problem associated wit link-state broadcasts. We focus on a class of topology control mecanisms called local-pruning mecanisms. Topology control by local pruning is an interesting multi-agent grap optimization problem, were every agent/router/station as access to only its local neigborood information. Every agent selects a subset of its incident link-state information for broadcast. Tis constitutes te pruned link-state information (pruned grap) for routing. Te objective for every agent is to select a minimal subset of te local link-state information wile guaranteeing tat te pruned grap preserves desired pats for routing. In topology control for rule-based link-state routing, te pruned link-state information must preserve desired pats tat satisfy te rules of routing. Te nontriviality in tese problems arises from te fact tat te pruning agents ave access

2 to only teir local link-state information. Consequently, rules of routing must ave some property, wic allows specifying te global properties of te routes from te local properties of te grap. In tis dissertation, we illustrate tat rules described as algebraic pat problem in idempotent semirings ave tese necessary properties. Te primary contribution of tis dissertation is identifying a policy for pruning, wic depends only on te local neigborood, but guarantees tat required global routing pats are preserved in te pruned grap. We sow tat for tis local policy to ensure loop-free pruning, it is sufficient to ave wat is called an inflatory arc composition property. To prove te sufficiency, we prove a version of Bellman s optimality principle tat extends to pat-sets and minimal elements of partially ordered sets. As a motivating example, we present a stable pat topology control mecanism, wic ensures tat te stable pats for routing are preserved after pruning. We sow, using oter examples, tat te generic pruning works for many oter rules of routing tat are suitably described using idempotent semirings.

3 TOPOLOGY CONTROL ALGORITHMS FOR RULE-BASED ROUTING by Kiran Kumar Somasundaram Dissertation submitted to te Faculty of te Graduate Scool of te University of Maryland, College Park in partial fulfillment of te requirements for te degree of Doctor of Pilosopy 2010 Advisory Committee: Professor Jon S. Baras, Cair/Advisor Professor Armand Makowski Professor Ricard J. La Professor Steve Marcus Professor S. Ragavan

4 c Copyrigt by Kiran Somasundaram 2010

5 Acknowledgments It is my pleasure to record my gratitude towards my advisor Prof. Jon S. Baras for is support and encouragement during my graduate studies at te University of Maryland. His diverse interests and expertise gave me te freedom to work on problems of my interest. I tank im for urging me to discover underlying principles tat connect disparate areas of researc, wic I believe as bougt clarity to my tinking and researc. I am also grateful to my dissertation committee members Prof. A. Makowski, Prof. R. La, Prof. S. Marcus and Prof. S. Ragavan for agreeing to serve in my committee. I must also tank Prof. S. Marcus for referring to related work on algebraic dynamic programming, wic elped improve te dissertation manuscript. I want to tank te administrative staff of te ISR and te ECE department, and in particular, Kim Edwards for elping me in several official and administrative matters. I am grateful to all my colleagues and friends wo ave offered elp in various ways. I like to specially tank, my good friend, Kaustub Jain for te several constructive conversations tat we ad concerning tis dissertation topic. I also like to tank Ranjit Kumaresan for introducing me to sortest pat problems, wic is pivotal to tis dissertation. My profound sense of gratitude is to my family: my moter, my fater and my broter. Teir constant support and confidence in me made graduate studies really enjoyable. I am greatly indebted to my beloved Gowri, for witout er love, support and encouragement, tis dissertation migt not ave been possible. ii

6 My researc work was supported by te Army Researc Office award number W911NF Tis support is acknowledged wit gratitude. iii

7 Table of Contents List of Figures List of Abbreviations vi viii 1 Introduction Motivation for Topology Control in Link State Routing Survey of Topology Control Algoritms Topology Control Mecanism of OLSR Oter Topology Control Algoritms Degrees of Local Information Contributions of te Dissertation Organization of te Disseration Stable Pat Topology Control Overview Stable Pat Topology Control Stability Metrics Limitations of Existing Topology Control Mecanisms for Topology Compression Notations and Definitions Graps and Neigboroods Local and Global Views Pat Stability Local Pruning Expressing Global Constraints Local Pruning Conditions Positivity Assumption and Sufficiency for Strong Pruning Degrees of local neigborood Salient Features of Stable Pat Topology Control Optimal Pruning as a Local Set-Cover Problem Stable Pat Topology Control Algoritm Computing ζ Greedy Approximation Algoritm to Solve Set-Cover Problem Simulation Setup Scenario Illustrating CDS Limitations Static Grid Topology Random Waypoint Mobility Scenario Battlefield Scenario Summary of Simulation Results Summary iv

8 3 Algebraic Pat Problems Overview Rule-Based Routing Graps and Metrics for Rule-Based Routing Composing Arc Metrics Rules for Pat Selection Distribution of Order Idempotent Semiring Algebraic Pat Problem for Routing Rules Semirings Idempotent and Selective Semirings - Pat Selection Summary Topology Control for Rule-Based Routing Overview Notations and Definitions Graps, Metrics and Views Local Pruning for Rule-Based Routing Strict Inflatory Condition, Sufficiency and Loop-Freedom Application of Generalized Pruning Hop-Count Lexicograpic Extension to Loop-Free SPTC Topology Control for Pareto Optimal Trusting Routing Summary Conclusions and Future Work Future Work A Grap Teory and Problems in Combinatorics 97 A.1 Elementary Grap Teory A.2 Classical Sortest Pat Problem A.3 Connected Dominating Sets A.4 Set Cover Problem B Semiring Algebra 102 B.1 Order teory B.1.1 Orders in Multi-criteria Optimization B.2 Canonically Ordered Monoids B.3 Semirings, Rings and Diods B.4 Semiring Algebraic Pat Problems Bibliograpy 115 v

9 List of Figures 1.1 Components of Link State Broadcast Mecanism Local view of te ost Recipe for Local Pruning Algoritms Example Local View wit ETX metric for eac link indicated Local View G local : Illustrates te local view of for a neigborood size k = 2. Te N 2 neigborood ball is sown by a dotted ellipse. Host discovers all te nodes inside tis ball. It also discovers te communication adjacencies and te link quality metrics for all te edges indicated by solid lines; it correspond to all te edges contained in te N 2 neigborood ball, oter tan te edges between te boundary nodes, i.e., te edges between te nodes in N 2. Te edges tat cannot discover in te neigbor discovery pase is indicated by dased 2.3 lines. Te boundary set of G local is N k = {j 7, j 8, j 9, j 10, j 11, j 12, j 13 }.. 24 Gateway for te pat p in te local view G local. Te N k ball is indicated as a dotted ellipse. Te gateway node γp is te node u t Broadcast View G broadcast. Te edges indicated by solid lines correspond to E broadcast, wile te edges indicated by dased lines are tose tat are pruned during te topology compression Global view of corresponding to te local view of Fig. 2.2 and te broadcast view of Fig Te edges indicated by te solid lines are visible in te global view, wile tose indicated by dased lines are neiter broadcast nor visible in te local view. Te global view is te grap union of te local and broadcast views; it also includes te weigts/labels for te edges in te global view, wic are te link quality metrics Illustrates te relation between a globally optimal pat p and its subpat p local. p local is te sub-pat from to te first occurrence of γp locally optimal pats in G local. Te boundary vertices are j 1 and j 2, and te -locally optimal pat sets are P local j 1 = {(, i1, i 2, j 1 ), (, i 3, j 1 )}, and P local j 2 = {(, i3, j 2 ), (, i 4, j 2 )} Viterbi decoding example: Te figure illustrates te online trajectory pruning procedure in Viterbi decoding. Te start state, s 1, and terminal state, s 3, are indicated by double-lined circles. Te figure sows all locally optimal trajectories wit solid lines. As examples, it sows two locally suboptimal trajectories (s 1, s 1, s 1 ) and (s 1, s 3, s 3 ) Example linear grap illustrating te local pruning condition Loopy pruning: Sows an edge-weigted grap, wit an instance of te strong pruning policy, Ω pruned Π Ω S (G local ), satisfied at every V. Te edges cosen for broadcast, E broadcast, are indicated by solid lines and tose tat are pruned away are indicated by dased lines vi

10 2.11 Pat construction procedure: sows a globally optimal pat p from u 1 to. p = p u1 local.p u1 not local is a concatenation of p u1 local from u 1 to γ u 1 p and p u1 not local, te remnant pat. q is u 1 -locally optimal pat from u 1 to γ u 1 p and u 2 = q (2) node topology to illustrate te limitation of CDS constructions Topology selection process at manet 0 for OLSR-ETX node grid network Carried load vs. Offered load for 100-node grid sown wit 95% confidence intervals Carried load vs. Offered load of Random Waypoint sown wit 95 % confidence intervals Battlefield Scenario Carried load vs. Offered load for te longest connection of Battlefield Scenario sown wit 95 % confidence intervals Spoon network to illustrate te distribution of order. Te network sows two pats from k to j, wit weigts b and c. Tere is an arc from i to k wit link weigt a Local View of : Te edges and arcs indicated in solid lines are in G local wile te tose indicated in dased lines are not Gateway for pat-set P = {p 1, p 2, p 3, p 4 } 2 P j G in local view G local. Te N k ball is indicated as a dotted ellipse. Since pats p 2 and p 3 intersect at te boundary N k, teir gateway vertices are identical, i.e, γp 2 = γ p3. Te gateway-set for P is Γ P = {γ p 1, γp 2, γp 4 }. Te -local sub-pat-set list is P local (γp 1 ) = {p local 1 }, P local (γp 2 ) = {p local 2, p local 3 }, and P local (γp 3 ) = {p local 3 } Broadcast view G broadcast : Te arcs in solid lines are broadcast and tey constitute te broadcast view, wile tose in dased lines are pruned away, i.e., not broadcast Global view of corresponding te local view of Fig. 4.1 and te broadcast view of Fig Te global view is te grap union of te local and global views; it also includes te arc labels. Again, tose arcs indicated in solid lines are a part of G global, wile tat are indicated in dased lines are not A.1 Dominating Sets of G B.1 Example network for sortest pat computation B.2 Example network for bi-objective sortest pat computation B.3 Transition diagram of a finite automaton. Te initial state is 1 and te final states, marked by double circles, are 3 and vii

11 List of Abbreviations Sets R Z R + Z + ˆR + Ẑ + Set of reals Set of integers Set of non-negative reals Set of non-negative integers Te set R + { } Te set Z + { } Graps G(V, E) Undirected grap wit vertex set V and edge set E G(V, A) Directed grap wit vertex set V and arc set A G G G is a subgrap of G p G p is a pat in grap G Pij G Set of pats between vertices i, j V 2 P ij G Set of all sub-sets of Pij G p (k) k 1, k t vertex in te sequence of vertices constituting pat p k 1, k t vertex in te sequence of vertices of pat p reversed Ω G i In an undirected grap G, te set of incident edges to i V ; in a directed grap G, te set of incident in-arc for i V d c (i, j) Minimal op-count distance between i, j V N k k-op neigborood of V N k Boundary of N k G local Local view of V γp Gateway for a pat p in G local Pruned incident edge set of Ω pruned G broadcast G global w p Broadcast view Global view, wic is G local G broadcast Scalar weigt of te pat p for scalar arc weigts Vector weigt of te pat p for vector arc weigts w P For P Pij G, w P = {w p, p P } x G ij Optimal metric from i V to j V in G Pij G P G ij x local ij P local ij P local ij For vector weigt, non-dominated solution Optimal pat-set from i V to j V in G Efficient pat-set from i V to j V in G Optimal metric from i V to j V in G local For vector weigt, non-dominated solution Optimal pat-set from i V to j V in G local Efficient pat-set from i V to j V in G local viii

12 Rules π R Π R π Ω R Π Ω R l Min Oters TC SPTC NDC STIDC TDC A global property at V for a rule R Subgraps tat ave te property π R A local property at V for a rule R Subsets of arcs tat ave te property π Ω R Componentwise arc composition operator Vector arc composition operator Pat selection operator Non-dominance function Topology Control Stable Pat Topology Control Neigbor Discovery Component Selector of Topology Information to Disseminate Component Topology Dissemination Component ix

13 Capter 1 Introduction 1.1 Motivation for Topology Control in Link State Routing Wireless multi-op networks, suc as Wireless Sensor Networks (WSNs) and Mobile Ad Hoc Networks (MANETs) [45] are, typically, constructed on te fly, wit no pre-existing infrastructure. For tese ad oc networks, mobility of te nodes and teir ad-oc association makes te routing control mecanism critical to te functioning of te network [45]. A successful routing protocol for tese networks is one tat provides a mecanism to deliver data packets to any destination of te network even under dynamic topologies [60]. Routing protocols in mobile multi-op networks are broadly classified as reactive, proactive and ybrid [45]. In reactive protocols, a source requests for a route to a particular destination if it as data to send. On te oter and, in proactive routing protocols, every source maintains at-least one route to every destination of interest, by periodic updates. Hybrid protocols adopt mecanisms from bot reactive and proactive protocols. Many routing protocols, proposed for MANETs, use network broadcasting as means to establis routes [30, 6, 12, 24]. Network broadcasting, referred to as broadcasting in tis dissertation, is a process by wic a packet sent from one station reaces all te stations in te network. In tis dissertation, we study te broadcasting associated wit proactive link- 1

14 state protocols. Link state protocols, suc as OSPF [39] and OLSR [11], broadcast link-state information necessary for routing: every station broadcasts its local linkstate, and consequently, every station is aware of te global link-state of te entire network. Subsequently, unicast traffic is routed by op-by-op routing to any destination station [39, 11]. In tis dissertation, we investigate te former aspect of link-state routing protocols: broadcasting of link-state information. Te motivation to study tis arises from a problem associated wit link-state broadcast in mobile networks: in mobile multi-op networks, link-states are very dynamic, and consequently, a large number of routing control packets, corresponding to every link-state cange, is broadcast in te network. Tis control overead trottles te limited resources of a multi-op wireless network. Tis problem is referred to as te broadcast storm problem [60]. A class of algoritms called Local Pruning or Neigbor Knowledge Topology Control (TC) algoritms [60, 61] reduces te broadcast storm by controlled flooding. Controlled flooding, as opposed to naive flooding, constructs an overlay-flooding network to broadcast a reduced topology information. In tese algoritms, controlled flooding becomes feasible due to some local neigborood information tat te stations learn in te neigbor discovery process: in te neigbor discovery pase for many link-state routing mecanisms, te stations discover only teir immediate neigboring stations; owever, in local pruning algoritms, te stations discover not only teir neigbors, but also te neigbors of teir neigbors and oter associated neigborood metrics. Tis additional information is ten used to reduce te broadcast storm by two means: 2

15 1. Topology Compression - Reducing te link-state information tat is broadcast by selecting only a subset of te topology information tat is sufficient for routing. 2. Efficient Broadcast - Reducing redundant broadcasts by selecting a subset of stations (flooding overlay network) tat ensures network-wide broadcast. In tis dissertation, our focus is on te former: we develop algoritms for topology compression, i.e., pruning to preserve a sufficient topology for routing. In particular, we develop algoritms tat preserve pats tat satisfy Quality of Service (QoS) requirements. 1.2 Survey of Topology Control Algoritms In [31], we introduced a component-based arcitecture for link-state routing protocols. Fig. 1.1 illustrates te primary components tat are necessary for topology broadcast. Link-state routing protocols ave a neigbor discovery component tat enables eac station to discover its local neigborood information. Tis local neigborood information is fed into two components, Selector of Topology Information to Disseminate Component (STIDC) and Topology Dissemination Component (TDC). Te functionality of te STIDC is to perform topology compression (Section 1.1) and tat of te TDC is to perform efficient broadcast (Section 1.1). Te functionality of te tree components are explained in greater detail in Capters 2 and 4. In tis dissertation, since we investigate only te topology compression aspect 3

16 of link-state broadcasting, we will assume tat tere exists a TDC tat is capable of disseminating te compressed topology. We separate te concerns between te STIDC and te TDC by making te functionalities of te two components ortogonal/independent. In oter words, te algoritms tat we develop for topology compression will be independent of te underlying broadcast mecanism. Neigbor Discovery Component (NDC) Selector of Topology Informa9on to Disseminate Component (STIDC) Topology Compression Topology Dissemina9on Component (TDC) Efficient Flooding Figure 1.1: Components of Link State Broadcast Mecanism Before we present a survey of different TC algoritms, we introduce te topology selection and te broadcast mecanism of OLSR [12, 11], in detail, because it forms te foundation for te local pruning TC algoritms tat we introduce in tis dissertation Topology Control Mecanism of OLSR In Optimized Link State Routing (OLSR) protocol [12, 11], every ost station, in te network, discovers its local neigborood by periodic HELLO messages [10]. Since every ost locally broadcasts to its neigbors te set of neigbors tat it can 4

17 ear, every ost discovers its one-op neigbors and all te neigbors for eac of tese one-op neigbors (called two-op neigbors). Fig. 1.2 sows te neigborood tat te ost discovers from te HELLO messages. Te neigbor discovery protocol [10] is designed to discover only symmetric neigbors (tat can ear eac oter), and consequently, all te links discovered are undirected. In te example of Fig. 1.2, {i 1, i 2,, i 5 } are te one-op neigbors, and {j 1, j 2,, j 7 } are te twoop neigbors of. Note tat te neigbor discovery protocol at does not expose te links between two-op neigbors, i.e., cannot discover te links between j m and j n, 1 m, n 7. Tis local grap discovered at te ost is called te local view of. Te notion of a local view is defined more formally in Capters 2 and 4. j 7 j 6 i 1 i 2 j 1 i 3 i 3 j 5 i 5 j 2 i 4 j 3 j 4 Figure 1.2: Local view of te ost Te original version of OLSR [12] treats te topology-pruning problem as Connected Dominating Set (CDS) construction given only te local view [61]. Te grap-teoretic concept of CDS is explained in Appendix A.3. Every ost solves a 5

18 local set-cover problem in te local view to find te minimum set of one-op neigbors tat cover all two-op neigbors. For instance, in te example local view sown in Fig. 1.2, te ost selects from te set of one-op neigbors, {i 1, i 2,..., i 5 }, a minimal subset tat cover all two-op neigbors, {j 1, j 2,..., j 7 }: all te two-op neigbors must be reacable from tis selected subset of one-op neigbors in te local view. Tis distinguised subset, of one-op neigbors, is called Multi-Point Relay (MPR) set in OLSR. Te pruning problem to compute te MPR set is sown to be NP-ard and a greedy euristic is used to obtain te MPR set [46]. For tis example of Fig. 1.2, te MPR set is {i 1, i 3, i 4 }. Ten te ost broadcasts links {(, i 1 ), (, i 3 ), (, i 4 )} across te entire network. Similarly, every station in te network selects and broadcasts a subset of its incident links. Te selection of a subset of te incident links corresponds to te topology compression functionality of te STIDC. It is sown in [29] tat tis reduced topology preserves te sortest op-count pats for routing. Te MPR set serves anoter functionality: OLSR uses te multipoint relay stations to form an overlay-flooding network to acieve network-wide broadcast of tis compressed topology. Since te MPRs form a CDS [61], flooding via te MPRs guarantees network-wide broadcast [29]. Tis reduced flooding corresponds to te efficient broadcasting functionality of te TDC. Interestingly, in OLSR bot STIDC and TDC functionalities are served by te MPR sets! However, in general, te two functionalities could be acieved by different mecanisms. 6

19 1.2.2 Oter Topology Control Algoritms Tere are several local pruning TC algoritms proposed in te literature. We present a brief summary of tese algoritms. Scalable Broadcast Algoritm (SBA) proposed in [43], uses local neigborood information, in particular, te connectivity of its neigbors to scedule local re-broadcasts. Te dominant pruning algoritm proposed in [1] identifies a dominant set of broadcast relays using a greedy set-cover algoritm. Te multi-point relaying algoritm (explained in Subsection 1.2.1) proposed in [46] is a more optimized version of dominant pruning. Te ad oc broadcast protocol proposed in [44] is a more efficient broadcast mecanism tat uses te MPR construction. For a summary of te different local pruning algoritms proposed for MANET routing, see [60]. In [61], te autors sow most of te pruning algoritms construct a CDS using a local dominance/priority function. Tis work encompasses a large class pruning algoritms tat includes [43, 1, 46, 44, 28, 20, 8, 54, 27, 59]. Appendix A.3 illustrates wy a CDS is used for constructing an overlayflooding network in multi-op wireless networks. A CDS is a covering property of te vertices and does not translate trivially to a property of pats. Consequently, te CDS constructions in literature do not offer any guarantees on te QoS of te pat preserved (tose preserved subsequent to pruning). 7

20 1.3 Degrees of Local Information In pure link-state mecanisms, suc as OSPF [39], te link-state broadcast makes visible te entire weigted communication grap. Tis raw information creates a complete global view of te network tat is more tan sufficient for routing. In tis dissertation, as introduced in Section 1.2, we investigate protocols tat ave local neigborood information (made visible by te NDC). Tis additional information provides an opportunity to reduce te amount of link-state broadcast. Te recipe in tese local pruning algoritms is illustrated in Fig Note tat te gains in topology compression are essentially due to te local link-state discovery using te NDC. Local Link State Discovery (NDC) Select a sufficient (for rou9ng) topology by local pruning (STIDC) Broadcast tis sufficient topology Figure 1.3: Recipe for Local Pruning Algoritms At a superficial level, te above recipe suggests tat tese optimized linkstate routing protocols break-down te link-state discovery mecanism into two stages: local discovery by NDC and global discovery by TDC. Wen viewed at tis superficial level, it migt appear tat tere is no reduction in te overead because 8

21 te cost of performing neigbor discovery migt balance or even outweig te savings from reduced link-state broadcast. However, in several protocols suc as OLSR [11], te neigbor discovery mecanism [10] offers te local link-state information for free. Any modification to tese protocols to discover more local link-state information will always incur an additional cost and must be considered in designing te protocol. Tere is also anoter advantage in discovering te local link-state. In mobile multi-op wireless networks, te local link-state is likely to be very dynamic, e.g., te link stability metrics can cange drastically in a fading wireless cannel. Usually suc igly dynamic links are not preferred for routing and consequently, tey can be discarded from broadcast. Consequently, te link-state information broadcast rate is reduced, i.e., observing te local link-state enables to create a more stable routing grap. Tis observation is used in te Stable Pat TC algoritm described in Capter 2. However, tere is always a trade-off between te cost in learning te local link-state information and savings in te link-state broadcast. Tis is will caracterized formally in Capter Contributions of te Dissertation In tis dissertation, we formulate te TC problem as a multi-agent grap pruning problem. Te agents are te stations/routers of te network, and tese agents ave access to only teir local neigborood information. Te agents, using tis constrained view of te network, select a minimal local topology to construct a pruned grap for routing. However, tere is a constraint on te pruned grap 9

22 tat it must preserve te globally desirable pats for routing described by a rule for routing. We identify a class of rules tat can be described as an algebraic pat problem in idempotent semirings allows specifying global properties of pats from local properties of te grap. We sow tat tis class of rules satisfy, wat we call, te distribution of order property tat enables local pruning; in essence, tis distribution of order property is Bellman s optimality principle extended to patsets and minimal elements of partially ordered sets. Tis becomes necessary wen you ave rules defined over multiple metrics vector metrics. Te primary contribution of tis dissertation is identifying a policy for pruning, wic depends only on te local neigborood, but guarantees tat te pruned grap preserves desirable pats described by generic rules of routing. We sow tat for tis local policy to ensure loop-free pruning, it is sufficient to ave wat is called an inflatory arc composition property. Tus, te two primary contributions of tis dissertation are te following: 1. Extension of algebraic routing to vector metrics, were notions of optimum and optimality are replaced by tat of non-dominance and efficiency, respectively. We present a version of Bellman s optimality principle extended to multiple vectors, minimal elements and efficient pat-sets. 2. Identification of sufficient local pruning conditions for rule described using idempotent semirings strict inflatory arc composition tat guarantee loopfree pruning. 10

23 1.5 Organization of te Disseration Tis dissertation is organized into tree primary parts (tree capters): 1. We introduce te stable pat topology control problem. Tis serves as a motivating example for te generic topology control algoritms tat will follow in te subsequent capters. In tis part, we develop te macinery tat is necessary to formally specify te topology control problem. Te results in tis part make extensive use of Bellman s optimality principle: Bellman s optimality principle yields a tool by te global QoS of a pat can be described by te QoS of a local pat (visible in te local neigborood). In tis part, we develop sufficient conditions of local pruning tat guarantee to preserve global QoS stable pats for link-state routing. We also formally caracterize te benefit and cost in aving additional local neigborood information. Finally, using simulations, we present quantitative comparisons between our topology control algoritm and tose in prior art. 2. We introduce rule-based routing. We illustrate tat rules of compositions tat yields pat weigts and rules of pat selections can be effectively described using idempotent semiring algebras. Tis part is intended to serve as a tutorial to algebraic routing. Te essence of tis part is to introduce te notion of distribution of order and to formally define it for pat-sets. 3. We develop te generic topology control (link-state selection) pruning conditions for rule-based routing. We extend te pruning conditions of te stable 11

24 pat topology control problem, of te first part, to a generic rule-based routing framework. Te local policies tat we develop illustrate te need for te distribution of order property, wic is a tool to translate local properties to global properties of directed labeled graps. In doing so, we sow a manifestation of te Bellman s optimality principle tat extends to pat-sets and minimal elements of partially ordered sets. We sow a sufficient condition, called strict inflatory arc composition, wic guarantees loop-free pruning. 12

25 Capter 2 Stable Pat Topology Control 2.1 Overview In tis capter, we introduce te stable pat topology control problem. Tis serves as a motivating example for te generic topology control algoritms tat will follow in te subsequent capters. In tis capter, we develop te macinery tat is necessary to formulate te topology control problem. Altoug, te notions are tailored for te stable pat topology control, tey are in a form from wic tey can be easily extended to te generic rule-based routing. Te results in tis capter make extensive use of Bellman s optimality principle: Bellman s optimality principle yields a tool by te global QoS of a pat can be described by te QoS of a local pat (visible in te local neigborood). Te main results in tis capter are in developing sufficient conditions of local pruning tat guarantee to preserve global stable pats for linkstate routing. We also formally caracterize te benefit and cost in aving additional local neigborood information. Finally, using simulations, we present quantitative comparisons between our topology control algoritm and tose in prior art. Tis capter is organized as follows. We first introduce te notion of stable pat routing in Section 2.2. Next, in Section 2.3, we introduce te matematical 13

26 notations, definitions and terminology necessary to describe te topology control problem. In Section 2.4, we introduce te notion of local pruning and policies for local pruning; we establis a sufficient condition for loop-free local pruning. Finally, in Section 2.5, we present an approximation algoritm to solve te stable pat topology control problem and substantiate its performance wit simulation results. 2.2 Stable Pat Topology Control One important metric for routing in MANETs is pat longevity or pat stability [9]. In tis dissertation, we refer to it as pat stability. Altoug pat stability as been studied for many reactive distance vector scemes [9, 49], tere is little work tat addresses topology control for stable pats in link state routing. In tis capter, we introduce a new topology control algoritm: Stable Pat Topology Control (SPTC), is a mecanism to prune te initial topology (to reduce te broadcast storm) wile guaranteeing tat te stable pats for (unicast-)routing from every ost to any target station are preserved in te pruned topology. Topology control for stable pats as a two-fold advantage: 1. Tese long-lived pats are ceaper to maintain because tey are less likely to cange. 2. It offers te iger layer traffic long-lived pats and consequently yields improved traffic carrying performance. Note tat our goal is not to engineer new metrics, but to develop TC algoritms 14

27 tat can use te metrics tat ave been proposed in literature. We will, next, introduce some popular stability metrics Stability Metrics Majority of routing protocols proposed for wireless multi-op networks, bot reactive and proactive, are mecanisms tat use op-count as te metric for routing [45]: traffic is routed along te minimum op-count pat from source to destination. However, wireless links in a multi-op network are vulnerable to frequent breakage due to mobility and cannel erasures [55, 21, 3]. Hence, scemes based merely on op-count, wic are inerently insensitive to te dynamic stability of te pats, ave sown poor performance [15]. Tis limitation as inspired a number of protocols tat use link stability as a metric for routing. Peraps, te earliest MANET protocol to use link stability metric for routing is te Associativity Based Routing (ABR) sceme [58], wic uses an associativity tresold used to predict te stability of a neigboring station. It assumes neigbors tat remain associated beyond tis tresold are less likely to move away and ence form stable links. Signal Stability based Adaptive routing (SSA) [18] is anoter link stability based routing protocol tat uses signal strengt and location information from neigboring stations to estimate te stability of te links. Routelifetime Assessment Based Routing (RABR) [2] is an extension to SSA tat uses tresolding of link ages to coose routes. Mobility prediction was suggested in [55] to improve unicast and multicast routing protocols for MANETs. Tis sceme uses 15

28 GPS location information to estimate te residual lifetime for links. In [21], te autors present a simulation study of te empirical distribution of link lifetimes for various mobility models [7]. From tese empirical distributions, tey also derive a metod to compute te residual lifetime distribution for tese different mobility models. Te study reveals tat tere are strict tresolds beyond wic te residual lifetimes exibit a positive correlation wit te link age. Anoter statistical caracterization of link lifetimes is presented in [9]. Tese simulation results sow tat longer lifetime pats tend to ave longer lengt (in op-count), suggesting tat tere is a tradeoff between pat stability and pat lengt (delay). Te Stability and Hop-count based Algoritm for Route Computing (SHARC) [53] identifies tis tradeoff and combines a link stability metric and te op count metric to find sort pats (in terms of op-count) tat also ave good stability (lexicograpic ordering). Anoter simulation study, presented in [48], sows tat pat life is inversely related to te maximum velocity and te op-count, and is directly related to te transmission range. Te autors observe tat under ig mobility patterns, te pat durations can be approximated using exponential distributions. In [25], Han et. al. use Palm calculus to sow tat under certain conditions, te pat durations converge to exponential distributions as te number of op-count increases. Te wireless mes networking community as also been actively developing several stability metrics for routing. Since te backbone routers of a wireless mes network are stationary, routing using link stability metrics, rater tan mere opcount, is more feasible compared wit te MANET case, were te network topology 16

29 is more dynamic [33]. To te best of our knowledge, te first metric proposed for wireless mes networks is te Expected Transmission Count (ETX) metric in [14, 13]. Te ETX metric for a link is te expected number of attempts for a packet to successfully reac te oter end of te link, in an Automatic Repeat request (ARQ) sceme. Te autors of [14] design te ETX metric for MAC wit ARQ. Tus, te ETX metric accounts for link stability bot in te forward and te reverse direction of te link. Te ETX metric of a link is calculated using te forward and reverse delivery ratios of te link. Te forward delivery ratio, df, is te measured probability tat a data packet successfully arrives at te recipient, and te reverse delivery ratio, dr, is te probability tat te ACK packet is successfully received. Te probability tat a transmission is successfully received and acknowledged is df dr. A sender will retransmit a packet tat is not successfully acknowledged. Since eac attempt to transmit a packet can be considered to be Bernoulli trial, te expected number of transmissions is given by ET X = 1 df dr Te ETX metric of a pat is te sum of te link ETX metrics along te pat; te sum gives te expected number of transmissions to deliver a packet successfully from te source vertex to te terminal vertex of te pat. Altoug, it is not included in te RFC [12], te ETX metric as been incorporated in popular OLSR implementations [62, 47]. In [42], te autors argue tat te ETX metric, being additive, suffers from 17

30 route oscillations. Instead, tey propose, a multiplicative metric, Minimum Loss (ML) metric tat computes te loss probability of a pat. Anoter problem wit te ETX metric computation is tat te data and control packets are typically larger tan te probe packets used to compute te metric, and consequently, te computed loss probabilities need not be equal to te loss probabilities of te data and control packets. Tis problem is identified and a new metric called Expected Transmission Time (ETT), wic computes te expected transmission time instead of te count, is proposed in [17]. ETT adapts te ETX for different PHY transmission rates and packet sizes. Tey also propose te Weigted Cumulative ETT (WCETT) metric tat modifies ETT to also consider intra-flow interference. Tis metric is composed of bot end-to-end delay and cannel diversity; a tunable parameter is used to combine bot components. In wireless networks, te link stability is usually igly dynamic, and consequently, several of te metrics proposed, if used crudely, can cause significant control overead or route oscillations [33]. In [33], te autors propose two metrics: modified ETX (metx) and Effective Number of Transmissions (ENT) tat consider te variance of te link-stability wile computing te metrics. Anoter metric tat considers link-quality variation is iaware [56]. Tis metric uses te signal to noise ratio and signal to interference and noise ratio to continuously reproduce neigboring interference variations onto routing metrics. A number of oter link stability metrics ave also been proposed for MANETs; tey include [58, 49, 38, 9, 63, 55, 53, 3, 18, 2]. 18

31 2.2.2 Limitations of Existing Topology Control Mecanisms for Topology Compression Algoritms tat make use of link stability metrics, in most cases, are modifications to reactive distance vector protocols suc as Dynamic Source Routing (DSR) [30] and Ad Hoc On-demand Distance Vector (AODV) [6]; tese include Link Quality Source Routing (LQSR) [17], Multi-Radio LQSR (ML-LQSR) [17], SrcRR [13] and oters. Tere are few proactive link-state routing protocols tat incorporate tese link stability metrics for topology control for controlled flooding. Most of tese are variants of OLSR s [12] pruning metods (Subsection 1.2.1). In [4, 40, 36], te autors propose modifications to OLSR s MPR selection algoritm to incorporate link stability metrics: te MPR selection is posed as a weigted set-cover algoritm [32]. Te implementations of te OLSR protocols in [62, 47] provide provisions to compute te ETX link metric. Tey also provide options to use te ETX metric in calculating te MPR set (Subsection 1.2.1). Let ET X(u, v) be te symmetric ETX metric for te link (u, v). In te topology control algoritm implemented in [62], te ost computes te ETX metric of te best two-op pat to reac a two-op neigbor j by min l ET X(, i l ) + ET X(i l, j), (2.1) were i l s are te one-op neigbors of. Te ost ten selects a minimal set of its one-op neigbors MPR set suc tat all te two-op neigbors are reacable from via te best two-op pat, using tese selected MPRs. In essence, tis is anoter set-cover problem were all te two-op neigbors are covered by a subset 19

32 of one-op neigbors, using te computed ETX weigts (Equation (2.1)). i i 1 1 j Figure 2.1: Example Local View wit ETX metric for eac link indicated However, tese set-cover metods offer no proof guarantees for te stability of te pruned pats, i.e., te stable pats for routing need not be preserved by tese pruning metods. To illustrate tis, consider a weigted grap sown in Figure 2.1. Te symmetric ETX metrics for te links are indicated in te figure. Te ost as two one-op neigbors i 1 and i 2 and one two-op neigbor j. In tis example, te link (, i 1 ) is unstable (ETX = 4), wile all oter links are stable. For tis topology, te set-cover metod of te implementations in [47, 62] as only one feasible (two-op) pat (, i 1, j) to reac j, wic as an ETX cost 5. Note tat in te implementation, all feasible pats from, te ost, to reac j, te two-op neigbor, is of te form (, i, j), were i is an one-op neigbor (Equation (2.1)). However, if we relax te artificial constraint tat two-op neigbors need to be reaced from te ost using strictly one one-op neigbor, tere exists an alternative better pat (, i 2, i 1, j) of ETX cost 3. Clearly, te set-cover pruning metods (of OLSR and its variants) will not preserve tis stable pat. Note tat tis example is not a mere pedagogical example. In wireless radio networks, any one-op neigbor tat covers several two-op neigbors is, typically, far from te ost. Consequently, te link between te ost and tis one-op neigbor 20

33 is unstable. However, te set-cover formulation of [47, 62] will always coose tis unstable neigbor as its MPR. In te fortcoming sections, we will formulate and solve a distributed pruning (topology control) problem tat can provably preserve all te stable pats to every destination station in te pruned topology. Our TC algoritm is not specific to te ETX metric, wic as been discussed in tis subsection, but can be applied to all te stability metrics discussed in Subsection Notations and Definitions In tis section, we will introduce some terminology and protocols tat are relevant for te TC problem. See Appendix A.1 for definitions from elementary grap teory. In tis capter, we will deal wit only edge-labeled undirected graps Graps and Neigboroods Let G(V, E) denote te communication grap, were V is te vertex set of stations and E is te undirected edge set (communication adjacency between te vertices). For (u, v) E, tere is an associated symmetric link stability metric a uv = a vu 0. Tus, G is an undirected edge-weigted grap. A subgrap of G, denoted by G G, is a grap G (V, E ) suc tat V V and E E (restricted to V V ). Te set of pats in any subgrap G from i V to j V is denoted by P G ij. For any pat p = (v 1, v 2,, v n ), p (k) = v k, for 1 k n, i.e., p (k) denotes te k t vertex of te sequence of vertices in p. For 21

34 n k 1, p (k) denotes te k t vertex of te reversed pat, i.e., p (k) = v n k+1, n k 1. For any vertex i V, te set of edges incident to i in G is denoted by Ω G i. We introduce te notion of op-based neigboroods, wic is defined for te grap G. Te op-count c(p) of a pat p is te number of edges in p. Te minimal op-count distance between a pair of vertices (i, j) in G is d c (i, j) = min c(p). If p Pij G j is not reacable from i in G, i.e., P G ij =, ten d c (i, j) =. We define te k-op neigborood for a ost/vertex V by N k = {j V : d c (, j) k}. Here, k is called te size of te neigborood. Te boundary set for te neigborood N k is given by N k = N k \N k 1, were N 0 = {}, and N k =, k < 0. Let N k = N k \{} denote te exclusive neigborood, wic is te neigborood excluding te ost Local and Global Views In [52], we extended te notion of local and global views introduced in [61] to encompass edge-weigted graps. We summarize tese extensions in te subsections. Link state routing protocols for MANETs, suc as [12, 11, 47], ave a local neigbor discovery pase/protocol [10], were every ost/node discovers its local neigborood, using periodic HELLO messages. Tis mecanism is called eartbeat neigbor discovery. 22

35 In OLSR [12], te HELLO message of a ost consists of all its symmetric neigbors: symmetric neigbors are tose tat can ear eac oter. Apart from tis adjacency information, te HELLO message also contains link quality information to eac of tese symmetric neigbors [11]. For instance, in te implementation of [47], te link quality information is te ETX link metric (Subsection 2.2.1) from te ost to its symmetric neigbor. Te HELLO message from every ost is broadcast locally to all its neigbors. Subsequently, every ost not only discovers its communication adjacencies and link qualities to all its neigbors, but also te communication adjacencies and link qualities of te neigbor to its neigbors. Formally, every ost V, discovers (, i) and a i, i N 1, and also (i, j) and a ij, i N 1, j N 1 i. We generalize tis functionality of te [10] protocol, to formalize te neigbor discovery pase. We assume tat te HELLO message from eac ost contains bot te communication adjacency and te link quality information for all its (k 1)-op neigbors (k 2). Consequently, every ost V discovers te communication adjacency and te link quality information in its k-op neigborood: a special labeled subgrap G local, wic is a subgrap of G, tat contains only te vertices in N k and all te edges between tem, except tose between any two vertices of te boundary set, i.e., te vertex set is N k, te edge set is {(u, v) E : u, v N k and {u, v} N k }, and te labels a uv for every edge in tis edge set. We call tis labeled subgrap, G local, te local view for te ost. Fig. 2.2 illustrates te notion of local view using an example. We next introduce te notion of gateways for pats restricted to a local view. 23

36 j 23 j 24 j 15 j 16 j 14 N 2 -ball j 7 j 1 j2 j 22 j 8 j 17 j 13 j 6 j 3 j 21 j 12 j 5 j 9 j 4 j 20 j 11 j 10 j 19 j 18 Figure 2.2: Local View G local : Illustrates te local view of for a neigborood size k = 2. Te N 2 neigborood ball is sown by a dotted ellipse. Host discovers all te nodes inside tis ball. It also discovers te communication adjacencies and te link quality metrics for all te edges indicated by solid lines; it correspond to all te edges contained in te N 2 neigborood ball, oter tan te edges between te boundary nodes, i.e., te edges between te nodes in N 2. Te edges tat cannot discover in te neigbor discovery pase is indicated by dased lines. Te boundary set of G local is N k = {j 7, j 8, j 9, j 10, j 11, j 12, j 13 }. 24

37 For any pat p = ( = u 1, u 2,..., u n = j) Pj G, te gateway of p in Glocal, denoted by γ p, is te first vertex of p tat is in te boundary set N k, i.e., γ p = u t if and only if u t N k and u s N k, 1 s < t. If te pat p never intersects N k, i.e., u s N k, 1 s n, ten γ p is not defined. Fig. 2.3 illustrates te notion of a gateway. j pat p N k ball u t γ p gateway vertex Figure 2.3: Gateway for te pat p in te local view G local. Te N k as a dotted ellipse. Te gateway node γ p is te node u t. ball is indicated In local pruning algoritms, te ost V, wic as discovered its local view G local, cooses a subset of its incident edges, wic we call te pruned incident edge set of. Te set of edges incident to in G local is denoted by Ω Glocal. However, since all te edges from te ost to its one-op neigbors are contained in G local, Ω Glocal = Ω G. Te pruned incident edge set is denoted by Ωpruned 2 ΩG, were 2 Ω G is te power set of Ω G, wic is te set of all subsets of ΩG. Te local pruning for compressed link-state selection, essentially, boils to computing tis Ω pruned, given 25

38 G local. In naive link-state flooding, te Ω G is broadcast network-wide. However, in compressed link-state te pruned link-state Ω pruned Ω G is flooded, and tis reduces te topology broadcast storm; te objective is to minimize Ω pruned wile guaranteeing tat te pruned grap satisfies some global properties for routing. Every ost broadcasts, using te Topology Dissemination Component (TDC), its pruned incident edge set, Ω pruned, and te link quality metrics for tese edges. Te corresponding broadcast edge set is given by E broadcast = V Ω pruned, and tis induces a labeled subgrap G broadcast, wic we call te broadcast view. Te broadcast view is common to all te osts V. Fig. 2.4 illustrates te notion of a broadcast view. Once te broadcast view is available at te ost V, it as te final linkstate information to compute te pats for routing. Tis link-state information is a combination of te local view and te broadcast view, and we call tis, te global view: At every ost station V, te global view G global is te grap union G local G broadcast along wit te corresponding edge weigts, were G global and G broadcast are exposed by te NDC and TDC respectively. Note tat te global view, like te local view, is specific to a ost ; every ost sees a different global view. Fig. 2.5 sows te global view corresponding te local and broadcast views of Figure 2.2 and 2.4 respectively. Note tat te edge-labeled grap G global is te only information tat as to compute optimal routes to different destinations, wic subsequently yields te best next-op to forward te packets to any destination node. We will define te notion of optimality for stable routing in te following subsection. 26

39 j 23 j 24 j 15 j 16 j 14 j 7 j 22 j 1 j 2 j 8 j 17 j 13 j 6 j 3 j 21 j 12 j 5 j 9 j 4 j 20 j 11 j 10 j 19 j 18 Figure 2.4: Broadcast View G broadcast. Te edges indicated by solid lines correspond to E broadcast, wile te edges indicated by dased lines are tose tat are pruned during te topology compression. 27

40 j 23 j 24 j 15 j 16 j 14 j 7 j 22 j 1 j 2 j 8 j 17 j 13 j 6 j 3 j 21 j 12 j 5 j 9 j 4 j 20 j 11 j 10 j 19 j 18 Figure 2.5: Global view of corresponding to te local view of Fig. 2.2 and te broadcast view of Fig Te edges indicated by te solid lines are visible in te global view, wile tose indicated by dased lines are neiter broadcast nor visible in te local view. Te global view is te grap union of te local and broadcast views; it also includes te weigts/labels for te edges in te global view, wic are te link quality metrics. 28

41 2.3.3 Pat Stability For te stability metrics discussed in Subsection 2.2.1, te stability of pat p, denoted by w p (weigt of te pat), is computed by composing te link stability metrics a uv, (u, v) p. Most of te metrics from Subsection follow eiter additive or multiplicative compositions. Since a multiplicative composition can be transformed to an additive composition, i.e., using logaritms, we only consider additive compositions for pat stability: w p = a uv. (u,v) p Te optimal value of pat stability between a vertex pair (i, j) in G is x G ij = min w p p Pij G = min p P G ij (u,v) p a uv, (2.2) and te corresponding optimal pat set, wic is te set of pats tat acieve te optimal value of te pat stability from i to j in te subgrap G, is Pij G = {p P G ij : w p = x G ij }. In essence, computing te optimally stable pats corresponds to computing te sortest pats in G. For an introduction to te sortest pat problem, see Appendix A.2. From ereon, we will call tese sortest pats, optimal pats. Also in tis context, since sortest pats and stable pats are identical, we abbreviate bot by sp. For brevity of notation, te optimal pat stability and te corresponding optimal pat sets restricted to te local view, G local P Glocal ij = P local ij. 29, are denoted by x Glocal ij = x local ij and

42 Finally, note tat tis additive pat stability metric follows te Bellman s optimality principle: Lemma For any optimally stable pat p = (i = u 1, u 2,..., u n = j) Pij G, any sub-pat (u k, u k+1,..., u l ) P G u k u l for 1 k < l n. Proof Suppose tat te sub-pat (u k, u k+1,..., u l ) P G u k u l. Consider an alternative better pat (of lower weigt) (u k = v 1, v 2,,, v m = u l ) P G u k u l, i.e., w (v1,v 2,,,v m) < w (uk,u k+1,...,u l ). Ten te pat p = (i = u 1, u 2,, u k = v 1, v 2, v 3,, v m = u l, u l+1,, u n ) as a lower weigt tan p, i.e., w p < w p. Tis is a contradiction. Te above lemma yields te following results w.r.t. -locally optimal pats: Lemma For a globally optimal p = ( = u 1, u 2,..., u n = j) P G j, let p local = (u 1, u 2,..., u l = γ p ) be te sub-pat from to te first occurrence γ p, i.e., l = min{1 s n : u s = γp } (sown in Fig. 2.6). Ten p local P local γ, i.e., p p local is a -locally optimal pat. Proof From Lemma 2.3.1, w p local = x G γ p Since p local P local γ p P local γ p. Since P local γ P G, x local x G. p γp γp γp, wp local x local γ p. Tus, w p local = x local. γp Lemma Let p = ( = u 1, u 2,..., u n = j) P G j be a globally optimal pat in G, wit p local = (u 1, u 2,..., u l = γ p ) (te sub-pat from to te first occurrence γ p, see Fig. 2.6). Consider a -locally optimal pat p local = ( = v 1, v 2,, v m = γp ) P local γ. Ten te pat obtained by replacing p local wit p local p in p is also globally optimal, i.e., q = ( = v 1, v 2,, v m = u l = γ p, u l+1,, u n ) P G j. 30

43 p - local γ p pat p j N k ball Figure 2.6: Illustrates te relation between a globally optimal pat p and its subpat p local. p local is te sub-pat from to te first occurrence of γ p. Proof From Lemma 2.3.2, w p local = x G γ p = x local γ p = w p local. w q = w p local + w (u l,u l+1,,u n) = w p local + w (ul,u l+1,,u n) = w p. Note tat te principle illustrated in Lemma is te fundamental principle based on wic te teoretical results of te tis capter are developed. Tis principle is expressed, in greater generality, in an algebraic framework in Capter 3 and is used for te generalized topology control problem in Capter 4. In essence, Lemma is pivotal to tis dissertation. 31

44 2.4 Local Pruning Topology control by local pruning [60] can be viewed as a multi-agent grap optimization problem: Te objective of eac agent, V, is to make use of te local neigborood information, G local (Subsection 2.3.2), to select a subset of te topology information, Ω pruned, tat is ten broadcast to te network. Tis subset is cosen so tat te resulting pruned grap preserves some global properties of te original grap. Te non-triviality in tese problems is in translating te global properties/requirements to te local properties/requirements. Altoug tere ave been many local pruning algoritms [40, 11, 47, 60], tere are few algoritms tat guarantee global QoS given only local neigborood information. Notable of tem are te algoritms in [37, 61], wic provide constructions of global Connected Dominating Sets (CDS) tat satisfy some global constraints using only local neigborood information. However, as illustrated in Appendix A.3, te CDS constructions do not guarantee pat QoS. To te best of our knowledge, tere are no constructions tat guarantee global QoS of pats given only local neigborood information. In tis section, we present grap-pruning procedures tat preserve sp pats for link-state routing Expressing Global Constraints Te fundamental pruning problem for eac ost, V, is to construct a minimal pruned edge set, Ω pruned, suc tat G global preserves desired properties of G. Tis is an interesting multi-agent optimization problem were te objective 32

45 function (finding a minimal pruned edge set) for eac agent (ost) depends only on local neigborood information, G local. However, te agents (osts) togeter must satisfy a global constraint, i.e., te global view must preserve desired properties of G. Before we consider te optimization problem (of finding te minimal pruned edge set), we will matematically express te global constraint for stable pat topology control. Tis is non-trivial because te global constraint involves te global view, wile te osts ave access to strictly teir local view. We will introduce furter notation to express tis problem. For sp routing, we want G global to preserve stable routing pats (of G) from to every oter vertex j V. Tis can be expressed as a sortest pat property π S (G) for V : π S (G) : p P G j, j V. (2.3) Let Π S (G) denote all te subgraps of G for wic πs (G) olds, i.e., tose tat contain at-least one sp tree rooted at. Ten te desired stable pat preserving global constraint can be expressed as G global Π S (G). (2.4) Altoug we ave matematically expressed te global constraint for stable pat pruning, tis constraint cannot be directly imposed on te local view; we need local constraints tat will guarantee tat te global constraints are satisfied. Remember, local pruning conditions sould impose constraints on Ω pruned Ω G, given G local (Subsection 2.3.2). Next, we will formulate suc local constraints tat imply te desired global constraint (Equation (2.4)). 33

46 2.4.2 Local Pruning Conditions Te sp problem as an interesting structure tat arises from Bellman s optimality principle (Lemma 2.3.1). Lemma states tat for a global sortest pat, any sub-pat is also a sortest pat. Tis suggests tat locally suboptimal pats can be discarded/pruned-away in computing te sp solutions. Tis can be represented formally by constraints on Ω pruned Ω G. Given te local view G local, we can compute all te locally optimal pats (including multiplicities) from to every boundary node, j N k, using any sp algoritm [32]; tese pats are denoted by sets P local j, j N k. Note for any pat p P local j, p begins wit te edge (, ik ) were p (2) = i k is an one-op neigbor of, i.e., i k N 1. Consider te set of suc edges tat lie on any of te -locally optimal pats to any of te boundary nodes, j N k. We denote tem by te set Ω non dominated = j N k p P local {(, p (2) )}. j For example, consider te local view of sown in Fig. 2.7, wit a neigborood size k = 2. Te optimal pats to j 1 N 2 are (, i 1, i 2, j 1 ) and (, i 3, j 1 ), and tat to j 2 N 2 are (, i 3, j 2 ) and (, i 4, j 2 ). Te set Ω non dominated = {(, i 1 ), (, i 3 ), (, i 4 )} consists of all te edges wic are on all tese optimal pats. It does not contain te edge (, i 2 ) because none of te -locally optimal pats begin wit te edge (, i 2 ). We ave, now, introduced te required notations to define te local pruning conditions of interest. Given te local view, G local, te local property π Ω weak S (G local ) 34

47 i 1 i 2 i 3 i 4 j 1 j 2 Figure 2.7: -locally optimal pats in G local. Te boundary vertices are j 1 and j 2, and te -locally optimal pat sets are P local j 1 = {(, i1, i 2, j 1 ), (, i 3, j 1 )}, and P local j 2 = {(, i3, j 2 ), (, i 4, j 2 )}. is said to old for any Ω 2 ΩG if Ω non dominated Ω, and let Π Ω weak S (G local ) denote te set of all suc Ω s. Te local pruning condition, wic we are interested in, is given by Ω pruned Π Ω weak S (G local ). (2.5) We will prove tat if Ω pruned Π Ω weak S (G local ), ten G global Π S (G). Before we present te proof, we present te intuition beind tis local condition. As we alluded to, in te start of tis subsection, te primary structure for tis condition is Bellman s optimality principle. First, we sow an analogy between te procedure by wic Ω non dominated is constructed and te trajectory pruning procedure of online Viterbi decoding [26], 35

48 wic is a manifestation of Bellman s optimality principle at work. Viterbi decoder is an online maximum likeliood decoder for a convolutional encoder [26]: a convolutional encoder generates an output string of bits for an input string of bits (message from source); tis corresponds to a trajectory, a sequence of states, in te encoder. Te objective of te Viterbi decoder, at te receiver, is to detect tis input string, or equivalently, te state trajectory, from te output string tat is corrupted by noise. Te underlying principle beind Viterbi decoder is finite state space dynamic programming, wic is an algoritm tat makes use of Bellman s optimality principle. s 1 s 2 s 3 Star6ng state Terminal state Globally op6mal trajectory Locally op6mal trajectory Locally sub- op6mal trajectory s Decoding to- go k Figure 2.8: Viterbi decoding example: Te figure illustrates te online trajectory pruning procedure in Viterbi decoding. Te start state, s 1, and terminal state, s 3, are indicated by double-lined circles. Te figure sows all locally optimal trajectories wit solid lines. As examples, it sows two locally suboptimal trajectories (s 1, s 1, s 1 ) and (s 1, s 3, s 3 ). To better illustrate te algoritm, we present an example in Fig Te 36

49 figure sows, wat is called, a trellis diagram [26]. In tis example, te convolutional encoder can be in one of te 4 states {s 1, s 2, s 3, s 4 }. Te start state is s 1, wic is known at te receiver. Te trajectory corresponding to te maximum likeliood solution is te trajectory/pat (s 1, s 4, s 3, s 2, s 3, s 3 ). Te figure sows te online decoding stage k = 2. At k = 2, te decoder as an apriori information of only te first two bits of te corrupted output string, and consequently, can not make any decisions about te future, k > 2. However, witin k = 2, te decoder prunes away some locally suboptimal trajectories. Te figure sows all te locally optimal trajectories till k = 2 for eac of te states, i.e., tose sub-trajectories ave te maximum likeliood from s 1 to eac of s 1, s 2, s 3 and s 4 in two stages. Solid lines sow tese locally optimal sub-trajectories. As examples, we sow, wit dased lines, two locally suboptimal trajectories (s 1, s 1, s 1 ) and (s 1, s 3, s 3 ), i.e., likeliood(s 1, s 1, s 1 ) < likeliood(s 1, s 2, s 1 ), likeliood(s 1, s 3, s 3 ) < likeliood(s 1, s 4, s 3 ). Tese locally suboptimal trajectories can be pruned away, i.e., tey need not be stored to compute te desired globally optimal trajectory. However, te locally optimal trajectories too every state (s 1, s 2, s 3, s 4 ) at k = 2 need to be stored for te global computation because any of tese trajectories can be potentially a subtrajectory of te globally optimal trajectory. Te example of trajectory pruning in Viterbi decoding illustrates te necessity to preserve all te locally optimal pats, wen we ave only local information. Tis is te intuition for considering te set of pats j N k P local j, i.e. -locally optimal pats to every boundary node. 37

50 Figure 2.9: Example linear grap illustrating te local pruning condition Altoug te trajectory pruning procedure in Viterbi decoding explains te need for preserving all locally optimal pats, it does not explain te local condition on Ω pruned (Equation (2.5)). Tis local condition is a condition on te edges of Ω G. To understand te intuition beind tis local condition, consider a line grap sown in Fig. 2.9, were all edge-weigts are 1. Let te size of te neigborood be k = 2. In tis example, only te osts 2 and 3 are potentially responsible of selecting te edge ( 2, 3 ) (Subsection 2.3.2). Consider te pruning policy at 2. For 2, N k 2 = { 4 }, and ( 2, 3, 4 ) is te only pat from 2 to 4. And for 3, N k 3 = { 1, 5 }, and ( 3, 2, 1 ) is te only pat from 3 to 1. If ( 2, 3 ) Ω pruned 2 and ( 3, 2 ) Ω pruned 3, ten ( 2, 3 ) G broadcast. Since ( 2, 3 ) G local 5 and ( 2, 3 ) G broadcast, ( 2, 3 ) G global 5. Consequently, G global 5 does not contain te globally optimal pats from 5 to 1 and 2. Tis leads us to te following teorem. Teorem If Ω pruned Π Ω weak S (G local ), ten G global Π S (G). Proof If Ω pruned Π Ω weak S (G local ), we will sow tat for any j V, all globally optimal pats from to j are contained in G global. Tis implies G global Consider p = ( = u 1, u 2,, u n = j) P G j. Tere are two cases: Π S (G). Case 1: p is completely contained in G local. Ten p is contained in G global because G global = G local G b roadcast. 38

51 Case 2: p is not completely contained in G local. We will prove by construction tat p is preserved in G global. For some 1 < l < n, te sub-pat (u 1, u 2,, u l ) is completely contained in G local, and consequently, it is contained in G global (Case 1). We will sow tat te remnant pat (u l+1, u l+2,, u n ) is contained in G broadcast. For any l < m n, consider te local view at u m, G local u m. Consider te upstream of sub-pat p from u m to = u 1, q = (u m, u m 1,, u 1 ). By Lemma 2.3.1, q is globally optimal, i.e., q P G u mu 1. Consider te sub-pat of q from u m to te first occurrence of γ um q optimal, (u m, u m 1,, γ um q Ω pruned u m Π Ω weak S u m, (u m, u m 1,, γq um ). By Lemma 2.3.2, tis sub-pat is u m -locally ) P umγ um q (G local ), (u m, u m 1 ) Ω pruned u m.. Ten, (u m, u m 1 ) Ω non dominated u m. Since Tis argument olds for l < m n, and consequently, te sub-pat (u l+1, u l+2,, u n ) is preserved. Te proof sows tat entire set of te optimal pats from to any destination j V, Pj G, is preserved in G global if te local pruning condition of Equation (2.5) olds. Tis is more tan sufficient for op-by-op routing [51]: for op-by-op routing every ost,, must be able to compute a best next-op for every destination, j V. So we need not preserve te entire set P j G ; preserving a subset of Pj G would suffice. Te local pruning condition of Equation (2.5) requires tat all locally optimal pats to be preserved, and consequently, gives lesser freedom to prune. It is for tis reason tat we call it a weak pruning condition. We now present a stronger local pruning condition. Tis stronger condition requires to preserve at-least one locally optimal pat to every boundary node, rater 39

52 tan all te locally optimal pats. It is expressed by anoter local property denoted by π Ω S at V : given te local view, G local, te property π Ω S (G local ) is said to old for an edge set Ω 2 ΩG if for all j N k tere exists a pat p P local j suc tat (, p (2) ) Ω. Let Π Ω S π Ω S (G local ) olds. i.e., (G local ) denote te set of all suc Ω s for wic Π Ω S (G local ) = {Ω 2 ΩG : ( j N k {p P local j : (, p(2) ) Ω}) }. Te strong local pruning condition is given by Ω pruned Π Ω S (G local ). (2.6) Note tat Π Ω weak S (G local ) Π Ω S (G local ), and consequently, te strong pruning conditions gives greater freedom to prune. To illustrate tis stronger pruning condition, we revisit te example of Fig Remember, te locally optimal pats from to j 1 are (, i 1, i 2, j 1 ) and (, i 3, j 1 ) and tat from to j 2 are (, i 3, j 2 ) and (, i 4, j 2 ). Clearly, Ω non dominated = {(, i 1 ), (, i 3 ), (, i 4 )} Π Ω S (G local ) because it intersects wit all te locally optimal pats. Interestingly, smaller subsets {(, i 1 ), (, i 3 )}, {(, i 1 ), (, i 4 )}, {(, i 3 ))}, {(, i 3 ), (, i 4 )} Π Ω S (G local ) because tey intersect wit at-least one -locally optimal pat to j 1 and j 2. As te example illustrates, te stronger pruning condition provides a greater degree of pruning. However, tis local condition is not sufficient to ensure G global Π S (G), since it does not guarantee loop-freedom. Tis is a well-known problem in distributed routing protocols [45]: Loops typically occur in distributed grap algoritms wen tie-breaking mecanisms are not employed. Using an example, we illustrate tat 40

53 a similar problem is likely to occur in stable pat distributed pruning witout tiebreaking Figure 2.10: Loopy pruning: Sows an edge-weigted grap, wit an instance of te strong pruning policy, Ω pruned Π Ω S (G local ), satisfied at every V. Te edges cosen for broadcast, E broadcast, are indicated by solid lines and tose tat are pruned away are indicated by dased lines. Consider an example edge-weigted grap sown in Fig. 2.10, were te edgeweigts correspond to some link stability metric (Section 2.2.1). Consider any stable pat pruning policy at stations 1, 2,..., 5. Let te size of te neigborood be k = 2. Te neigborood boundary sets are N 2 1 = { 3 }, N 2 2 = { 4 }, N 2 3 = { 1 }, N 2 4 = { 2, 5 } and N 2 5 = { 4 }. If te pruning mecanisms at tese stations satisfy te strong local pruning condition (Equation (2.6)), ten 4 coses ( 4, 3 ) and 5 coses ( 5, 3 ). However, te pruning mecanisms at 1, 2 and 3 ave multiple optimal pats to coose from. For 1 to reac 3, tere are two optimal pats, ( 1, 2, 3 ) and ( 1, 5, 3 ). For 2 to reac 4, tere are two optimal pats, ( 2, 3, 4 ) and ( 2, 1, 5, 3, 4 ). For 3 to reac 1, tere are two optimal pats, ( 3, 2, 1 ) and ( 3, 5, 1 ). Te Fig illustrates one pruning policy tat 41

54 satisfies te necessary conditions: 1 cooses ( 1, 2 ) for pat ( 1, 2, 3 ), 2 cooses ( 2, 1 ) for pat ( 2, 1, 5, 3, 4 ), and 3 cooses ( 3, 5 ) for pat ( 3, 5, 1 ). Te pruned grap G broadcast, sown in te Figure 2.10, is ten disconnected! Clearly, te distributed pruning does not preserve te stable optimal pats in te different global views Positivity Assumption and Sufficiency for Strong Pruning Te example of Fig suggests a sufficient condition, wic we call te positivity condition: all te edge weigts are strictly positive, a uv > 0, (u, v) E. We will sow tat under te positivity assumptions, te strong pruning conditions (Equation (2.6)) become sufficient to ensure tat optimal pats from a ost to every destination vertex are preserved in its global view. Teorem Under te positivity assumption,a uv > 0, (u, v) E, for all V, if Ω pruned Π Ω S (G local ), ten G global Π S (G). Proof We need to sow tat G global as at-least one optimal pat to any vertex j V. We will prove by construction tat one optimal pat is preserved under pruning. Similar to te proof of Teorem 2.4.1, tere are two cases. Case 1: p P G j is contained in Glocal, i.e., p P local j. Ten p is contained in G global because G global = G local G broadcast. Case 2: No p Pj G is completely contained in Glocal. We will prove by construction tat a reverse pat p Pj G is contained in Gglobal. Consider te pruning at j = u 1. Let p P G u 1 be any globally optimal pat from u 1 to. 42

55 Consider te sub-pat of p from u 1 to te first occurrence γ u 1 p, p u 1 local. Denote te remnant of te pat from γ u 1 p by p u 1 not local. Since te strong pruning condition, Ω pruned u 1 Π Ω S u 1 (G local ), olds at u 1, tere exists a pat q P u1 γ u 1 p suc tat (u 1, q (2) ) Ω pruned u 1. Let q (2) = u 2, wic we call te pruned next-op of u 1 to reac. Tis is step is illustrated in Fig u 1 p u 1- local γ p u 1 p u 1- not- local u 2 q Figure 2.11: Pat construction procedure: sows a globally optimal pat p from u 1 to. p = p u 1 local.p u 1 not local is a concatenation of p u 1 local from u 1 to γ u 1 p p u 1 not local, te remnant pat. q is u 1 -locally optimal pat from u 1 to γ u 1 p and and u 2 = q (2). By Lemma 2.3.3, te concatenation of te q and p u 1 not local is globally optimal, i.e., q.p u 1 not local P G u 1. Ten w q.p u 1 not local = x G u 1 = a u 1 u 2 + x G u 2. Since 43

56 a u1 u 2 > 0, we ave x G u 1 > xg u 2. Repeating te above construction, we get a sequence of vertices u 1, u 2, u 3,, u r, were u s+1 is te pruned next-op of u s to reac. Ten x G u 1 > xg u 2 > xg u 3 > > x G u r. Tis implies tat tere can be no loops in pruning, i.e., u r u s, 1 s < r (if u r = u s, ten x G u = r xg u s, wic is a contradiction). Since tere are no loops (eac u r is unique) and V is finite, tere exists a u l in te construction suc tat u l N k. Tere exists a -locally optimal pat from to u l in G local, say ( = u n, u n 1,, u l ) P local j. We ave proved by construction tat ( = u n, u n 1,, u 1 = j) is preserved in G global Degrees of local neigborood As alluded to in Section 1.3, in Capter 1, we study te degree of pruning possible wit increasing degrees of local neigborood information. In te teory developed so-far, for local pruning, we ave considered te neigborood size, k, to be fixed. In tis subsection, we present results for te effectiveness of local pruning wit increasing k. To present tese results, we parameterize te local view wit te neigborood size; we denote by G local (k), te local view wit a neigborood size k. Similarly, we denote te -locally optimal pats and solutions by P local j (k) and x local j (k) as functions of k. We illustrated in Subsection tat te strong pruning condition allows for better pruning. Tis was a consequence of te relation Π Ω weak S (G local ) Π Ω S (G local ); we also sowed tat Π Ω S (G local ) contained smaller subsets of Ω G. 44

57 Tis suggests tat te inclusion relation induces an order, wic we call te degree of pruning δ. For any two pruning policies, on Ω G, π i and π j, ordered by an inclusion relation, Π i Π j, were Π i and Π j are te subsets of Ω G tat satisfy π i and π j respectively, te degree of pruning is related by δ(π i ) δ(π j ). In oter words, tere is greater degree of pruning for a policy tat is stronger. Remember, Ω Π Ω S (G local (k)) implies tat for every j N k, tere exists a p P local j (k) suc tat (, p(2) ) Ω. Te monotonicity of te degree of pruning wit te neigborood size is caracterized by te following results. Lemma Π Ω S (G local (k)) Π Ω S (G local (k + 1)), k 1. Proof Let Ω Π Ω S (G local (k)). To sow tat Π Ω S (G local (k)) Π Ω S (G local (k + 1)), we need to sow tat Ω Π Ω S (G local (k + 1)), i.e., for every j N k+1, tere exists a p P local j (k + 1) suc tat (, p(2) ) Ω. Consider a p P local j (k + 1), j N k+1. Te pat can be decomposed into p = p.p, were p = ( = u 1, u 2,, u l ) is contained in G local (k), and p is te remnant pat. By Lemma 2.3.1, p P local u l (k). Since Ω Π Ω S (G local (k)), tere exists a q P local u l (k) suc tat (, q (2) ) Ω. Consequently for every p P local j tat (, q (2) (k +1), j N k+1, tere exists an alternative (optimal) pat q.p suc ) Ω. Tis implies Ω ΠΩ S (G local (k + 1)) Teorem Π Ω S (G local (k)) Π Ω S (G local (k )), 1 k k. Proof Proof follows directly from Lemma Corollary δ(π Ω S (G local )(k)) δ(π Ω S (G local )(k )), 1 k k. 45

58 Note tat te greater degree of pruning wit increased degree of local neigborood information comes at te cost of more complex neigbor discovery protocols Salient Features of Stable Pat Topology Control Note all te results in te section are a consequence of Lemma and its derivatives. Tere are two underlying principles tat are made use of in deriving te results: 1. Distribution of sub-optimality: Bot in te weak pruning and strong pruning conditions, te means by wic te local description is translated to a global description is based pruning of suboptimal pats. Tus sub-optimality, in a loose sense, distributes across pats: a locally suboptimal cannot be globally optimal. 2. Positivity and Downstream: Te positivity assumption ensures tat te pruning mecanism constructs pat by going downstream, i.e., subsequent next ops of pruning always progress towards te ost. In tis dissertation, we formalize te above two observations. In te later capters, we will introduce generic rule-based routing scemes, wic are described using semiring algebras. We will sow tat te above two observations translate to 1. Distribution of order, and 2. Inflatory arc composition 46

59 in te generalized pat problem framework. In essence, tese two principles provide te framework on wic topology control algoritms for generic rule-based routing are developed in tis dissertation Optimal Pruning as a Local Set-Cover Problem Te topology control problem for stable pat routing boils down to selecting a minimal pruned edge-set suc tat local pruning conditions are satisfied: min Ω pruned Ω G subject to Ω pruned (2.7) Ω pruned Π Ω S (G local ). Listing out all feasible subsets Ω pruned Π Ω S (G local ), in general, is computationally intractable. We will sow tat tis problem can be reduced to a set-cover problem. To formulate tis set-cover problem, we introduce furter notation. Let ζ : N 1 2 N k denote te covering function: for i N 1 and ζ (i) = {j N k : p P local j suc tat i = p(2) }, i.e., ζ (i), i N 1 is te set of boundary nodes wic are reacable via locally optimal pats starting wit te edge (, i). Te corresponding inverse function ζ 1 : N k 2 N 1 is, for j N k, ζ 1 (j) = {i N 1 : j ζ (i)}. Tis function ζ can be computed efficiently using any sortest pat procedures [32] (see Section 2.5). Ten te set-cover problem is 47

60 min 2 N1 (2.8) subject to i ζ (i) = N k. Teorem For any minimizer of te problem in Equation (2.8), {(, i) : i } solves te minimal pruning problem of Equation (2.7). Proof Since i ζ (i) = N k, Ωpruned ) = {(, i) : i } Π Ω S (G local ). 2.5 Stable Pat Topology Control Algoritm In tis section, we present te Stable Pat Topology Control (SPTC) algoritm tat solves te set-cover problem (to an approximation) in Equation (2.8) introduced in Subsection Finally, we demonstrate te performance of te SPTC algoritm by using te ETX metric (see Subsection 2.2.1) Computing ζ Algoritm 1 computes te covering function ζ used in te set-cover formulation (Equation (2.8)). Te local view G local is input to te algoritm and it outputs ζ. Given G local, te function computeallpairspfloydwarsall, used in Algoritm 1, computes te all pair sortest pats (-locally optimal pats) in te exclusive neigborood N k, using te well-know Floyd-Warsall algoritm [32]. It returns a matrix SP N k tat yields te sortest (-locally optimal restricted to N k ) pat metrics. Te next step of Algoritm 1 is called vertex expansion at. 48

61 Te locally optimal pats to every boundary node, j N k from is computed and te one-op neigbor,i N 1 on tese optimal pats is stored in ζ 1 (j). Note tat eac element ζ 1 (j) is a set of one-op neigbors. Algoritm 1 Compute covering function ζ at V INPUT: G local //Compute all-pair-sortest pats in exclusive neigborood SP N k computeallpairspfloydwarsall(g local ); //Vertex expansion for all j N k do ζ 1 (j) arg min i N 1 a,i + SP N k (i, j); end for OUTPUT: ζ Greedy Approximation Algoritm to Solve Set-Cover Problem Given ζ, Algoritm 2 is a greedy algoritm tat approximately solves Equation (2.8). Te algoritm first extracts te essential cover elements and appends it to te set R greedy. A one-op neigbor i is an essential cover for a boundary node j if tere are no oter one-op neigbors wic lie on te locally optimal pats to j, i.e., ζ (i) = {j}. In te next step, te Algoritm 2 performs a recursive greedy step, were it selects tat one-op neigbor i, wic covers te most uncovered 49

62 boundary nodes. Te recursive procedure terminates wen all te boundary nodes are covered. Algoritm 2 Greedy Set-Cover Algoritm at INPUT: ζ, G, N 1, N k INIT: R greedy, U N k; // Find and append essential cover elements for all {j N k : ζ 1 (j) = 1 } do R greedy R greedy ζ 1 (j); U U\{j}; end for // Greedy selection wile U do i arg max i N 1 {j U : j ζ (i)} R greedy R greedy {i } U U\{j U : i ζ (j)} end wile Output: R greedy Algoritm 2 is adapted from a set-cover approximation algoritm illustrated in Capter 11 of [32]. Let d = max i N 1 ζ (i). Ten te following Lemma gives te approximation bounds for te greedy solution R greedy : 50

63 Lemma Let te optimal solution to Equation (2.8) be and R greedy be te output of Algoritm 2 at ost, ten R greedy H(d ), were H(N) = N n=1 1. n Tis lemma is proved in Capter 11 of [32]. We call te Algoritms 1 and 2 togeter as te SPTC algoritm Simulation Setup All simulations were carried out in OPNET Modeler 14.5 [41]. For te simulations, te mobile node model manet station advanced was cosen. Te parameters given in Table 2.1 were used in te simulations. We modified te default code for te OLSR model, wic is an OLSR version 1 [12] implementation. We made suitable modifications to te neigbor discovery mecanism, as per [47], to compute te ETX metric online. We also modified te MPR selection algoritms to implement te SPTC algoritm. For te simulations, we used k = 2, size of te neigborood. To study te performance of te SPTC algoritm, we compare it wit te OLSR implementation in [47], wic uses te ETX metric to select MPRs using setcover metods (illustrated in Subsection 2.2.2). We implemented bot te OLSR and te SPTC algoritm to use te ETX metric. We call tese two implementations, OLSR-ETX and SPTC-ETX respectively. In te simulations, we compared bot te data traffic carrying and Topology Control (TC) overead performance of SPTC-ETX and OLSR-ETX. For te data 51

64 Group Parameter Value Protocol b MAC and PHY Transmission Rate Transmit Power Receiver Sensitivity Error Correction Capabilities 11 Mbps 5 mw -95 dbm None Routing and TC Protocol HELLO message interval Neigbor old time TC message interval OLSR-ETX or SPTC-ETX 2 s 32 s 5 s ETX Computation ETX Memory Lengt ETX Memory Interval ETX Hello Timeout Expiry 32 s 2 s 2.5 s Traffic Type Packet lengt UDP CBR 1024 bits Table 2.1: Parameters for simulation 52

65 traffic carrying performance, we studied te carried load for various offered loads. We set up a UDP traffic generator tat sends Constant Bit Rate (CBR) traffic between pairs of stations. We ten swept across tis CBR rate to study te traffic performance wit OLSR-ETX and SPTC-ETX. In link state mecanisms suc as OLSR, te TC broadcast mecanism is proactive, and consequently not all TC messages broadcast correspond to topology canges. To study te overead due to topology canges, we measured te rate of reactive TC messages and te total number of actual topology canges. Reactive TC messages are tose tat are generated due to canges in te selected topology. Tis is a good estimate of te actual topology overead for te pruned network. We will compare tis topology control overead for SPTC-ETX against tat of OLSR-ETX Scenario Illustrating CDS Limitations Before we present te results for complicated topologies, we will study te performance of SPTC-ETX and OLSR-ETX for a simple topology sown in Figure Tis scenario corresponds to te example topology in Subsection tat illustrates te fundamental limitation of CDS constructions. Te topology is set up suc tat tere is a long-distance unstable wireless link between manet 0 and manet 1. All oter links are sort and ence more stable compared to (manet 0,manet 1). Consider OLSR-ETX s MPR selection process at node manet 0. link (manet 0,manet 1) is unstable, it goes ON and OFF frequently. Since te Wenever 53

66 m manet_2 manet_0 manet_1 manet_3 500 m Figure 2.12: 4 node topology to illustrate te limitation of CDS constructions tis link is ON, manet 3 N 2 manet 0. Te OLSR-ETX s set-cover construction, selects te unstable link (manet 0,manet 1) as Ω pruned manet 0 because it te only edge in te two-op pat to reac manet 3. Wen te link (manet 0,manet 1) is OFF, manet 1 N 2 manet 0, and consequently, OLSR-ETX cooses (manet 0,manet 2) as Ω pruned manet 0. Tus as te unstable link (manet 0,manet 1) goes ON and OFF, Ω pruned manet 0 oscillates between (manet 0,manet 1) and (manet 0,manet 2). Tis is illustrated in Figure 2.13, wic sows a realization of te topology selection process at manet 0 obtained by OPNET simulation. Fundamental limitation of te set-cover OLSR construction is tat it is not designed to exploit te local pat diversity. Tis limitation is overcome by te SPTC-ETX tat provides a more stable Ω pruned manet 0. From simulations we observed tat SPTC-ETX almost always cooses Ω pruned manet 0 = {(manet 0,manet 2)} and Ω pruned manet 2 = 54

67 {(manet 0,manet 1)}, tus preserving te stable pat (manet 0,manet 2,manet 1). For a simulation period of 1 our, we observed 96 topology canges for OLSR-ETX and 6 for SPTC-ETX. 1.5 Link (manet_0! manet_1) Link (manet_0! manet_2)!* cosen by manet_ Time in s Figure 2.13: Topology selection process at manet 0 for OLSR-ETX Static Grid Topology Te next topology tat we consider is a 100-node static grid topology sown in Figure Te network consists of many stable and unstable links. Tis topology suffers from te same problem explained in Subsection Te oneop neigbors tat are far off (in pysical distance) typically cover more two-op neigbors. However, by te nature of radio propagation, tese links are unstable. Te UDP CBR traffic is sent between 5 different random source-destination pairs. Te comparison of te traffic-carrying performance of SPTC-ETX and OLSR- 55

68 m 6000 m Figure 2.14: 100-node grid network 56

69 ETX is sown in Fig Te simulation results indicate tat SPTC-ETX as a saturation capacity of 86kbps, wile tat of OLSR-ETX is 75kbps. 9 x Carried load in bps OLSR ETX SPTC ETX Offered load in bps x 10 5 Figure 2.15: Carried load vs. Offered load for 100-node grid sown wit 95% confidence intervals Te average number of total topology canges (for many runs of te simulation) was and 8280 for OLSR-ETX and SPTC-ETX respectively. Te corresponding rate of reactive TC messages was 930bps and 681bps respectively. Tis implies tat te pruned subnetwork of SPTC-ETX is stable/long-lived compared to tat of OLSR-ETX Random Waypoint Mobility Scenario Random waypoint mobility pattern is a commonly used to study protocol performances in a mobile environment [7]. Te mobility parameters tat we used for te random waypoint mobility pattern are sown in Table 2.2. All statistics were 57

70 Parameter Value No of stations 25 Simulation Area Speed Pause time 3000m 3000m (5,20] m/s 0 s Table 2.2: Random waypoint mobility parameters collected once te simulations reaced stocastic stationarity. Again, UDP CBR traffic is sent between two different random source-destination pairs. Te sample mean of te carried load as a function of te offered load is sown in Fig We observe tat SPTC-ETX is capable of carrying 13% more load tan OLSR-ETX. Tis is because in OLSR, we observed tat significantly more traffic is routed troug unstable links. Te average number of topology canges was and in one our of simulation time for OLSR-ETX and SPTC-ETX respectively. Te corresponding rate of reactive TC messages was 8.3kpbs and 6kpbs respectively Battlefield Scenario Finally, we consider a battlefield scenario, introduced in [5], wit an initial topology as sown in Sub-figure 2.17a. It comprises of 3 platoons of stations: Platoon A consists of nodes 0 to 9, Platoon B consists of nodes 10 to 19, and Platoon C consists of nodes 20 to 29. Te tree platoons move in te trajectories sown in 58

71 7 x Carried load in bps OLSR ETX SPTC ETX Offered load in bps x 10 5 Figure 2.16: Carried load vs. Offered load of Random Waypoint sown wit 95 % confidence intervals Sub-figure 2.17b: Platoon B moves forward along te x direction, and Platoons A and C move forward and away from platoon B at speed of 1.5 m/s in te y direction. Ten te platoons move togeter back to te initial formation. To ensure better connectivity among te platoons, two supporting nodes 30 (to support connections between Platoon A and B) and 31 (to support connections between Platoon B and C) move alongside te platoons (in te x direction). Te simulations were carried out wit te parameters sown in Table 2.1. Tis yields a radio range of approximately 900m. Hence witin eac platoon, all te nodes are at most two-ops from eac oter. Wen te platoons are close togeter, te inter-platoon communication is stable witout using te supporting nodes 30 and 31. However, wen te platoons move away from eac oter, te direct inter-platoon connections become unstable and te supporting stations become necessary for delivering ig traffic. Again for 59

72 Type Source-Destination Offered Load (kbps) (1,3),(2,9),(4,6),(7,5),(20,29), Intra-Platoon (14,17),(16,11),(17,18),(19,12), 12 (21,22),(23,27),(23,28) (1,18) 2.4 Inter-Platoon (20,11),(20,0) 6 (10,1),(21,10) 12 Table 2.3: Traffic connections for Battlefield scenario SPTC, we cose te current age as link stability metric (tis is only a euristic). UDP traffic was sent between 17 source-destinations pairs. Table 2.3 sows te base traffic for te scenario. For te traffic analysis, we focus on te connection (20, 0) (from Platoon C to A) because tis is a long connection and would be potentially sensitive to pat stability. We scale te base traffic (offered load) of all connections (in Table 2.3) by te same factor and obtain te carried load vs. offered load performance for connection (20, 0) sown in Fig Again, we observe tat SPTC carries significantly more load tan OLSR for tis connection. Tis is because wen te platoons are maximally apart, we observe tat for connection (20, 0), SPTC-ETX routes significantly more traffic (about 1.5 times more) troug te supporting nodes 30 and 31 wen compared to OLSR-ETX s routing mecanism. We observe tat te carried load for te oter connections is also iger. Tus te 60

73 overall network trougput is improved. For example, wen te offered load to te network (all connections) was 2M bps, SPTC-ETX was able to carry 923kbps, wile OLSR-ETX is able to carry only 890kpbs. Figure 2.18 compares te traffic carrying performance for te long connection (20, 0 for SPTC-ETX and OLSR-ETX (a) Initial topology (b) Node trajectories Figure 2.17: Battlefield Scenario In te TC study, we observed tat te average number of topology canges was and 5360 canges in one our of simulation time for OLSR-ETX and SPTC- ETX respectively. Te corresponding rate of reactive TC messages was 884bps and 338bps respectively. 61