Radio Aggregation Scheduling

Size: px

Start display at page:

Download "Radio Aggregation Scheduling"

William Baldwin
5 years ago
Views:

1 Radio Aggregation Scheduling ALGOSENSORS 2015 Rajiv Gandhi, Magnús M. Halldórsson, Christian Konrad, Guy Kortsarz, Hoon Oh

2 Aggregation Scheduling in Radio Networks Goal: Convergecast, all nodes send data item to sink Christian Konrad Radio Aggregation Scheduling 2 / 1

3 Aggregation Scheduling in Radio Networks Goal: Spanning Tree Christian Konrad Radio Aggregation Scheduling 2 / 1

4 Aggregation Scheduling in Radio Networks Goal: Conflict-free schedule of edge Christian Konrad Radio Aggregation Scheduling 2 / 1

5 Aggregation Scheduling in Radio Networks Difficulty: Limited Transmission range Christian Konrad Radio Aggregation Scheduling 2 / 1

6 Aggregation Scheduling in Radio Networks Difficulty: Interference Christian Konrad Radio Aggregation Scheduling 2 / 1

7 Aggregation Scheduling in Radio Networks Difficulty: Transmission radii may vary Christian Konrad Radio Aggregation Scheduling 2 / 1

8 Aggregation Scheduling in Radio Networks Difficulty: Transmission radii may be different from interference radii Christian Konrad Radio Aggregation Scheduling 2 / 1

9 Aggregation Scheduling in Radio Networks Difficulty: Obstacles Christian Konrad Radio Aggregation Scheduling 2 / 1

10 Radio Aggregation Scheduling Problem Definition: Radio Aggregation Scheduling (RAS) Given: Graph G = (V, E) and sink node s V Find: Directed matchings M 1, M 2,..., M t E so that: 1 i M i induce an in-arborescence directed towards s, 2 The M i are conflict-free (RAS-legal matching), 3 t minimal. Matching Induced matching RAS-legal matching sender receiver unused vertex transmission unused Christian Konrad Radio Aggregation Scheduling 3 / 1

11 Broadcast: Reversing the Slots Convergecast Christian Konrad Radio Aggregation Scheduling 4 / 1

12 Broadcast: Reversing the Slots Broadcast, by reversing the slots Christian Konrad Radio Aggregation Scheduling 4 / 1

13 Broadcast in Radio-unicast Model Brodcast in the Radio-unicast Model Given: Graph G = (V, E) and source node s V Each round, RAS-legal matching between informed & uninformed nodes 1 One-to-one communication (one sender to one receiver) 2 Interference constraint: Successful reception at receiver if exactly one neighbor transmits Relation to other Models Telephone model: One-to-one comm., no interference constraint Christian Konrad Radio Aggregation Scheduling 5 / 1

14 Broadcast in Radio-unicast Model Brodcast in the Radio-unicast Model Given: Graph G = (V, E) and source node s V Each round, RAS-legal matching between informed & uninformed nodes 1 One-to-one communication (one sender to one receiver) 2 Interference constraint: Successful reception at receiver if exactly one neighbor transmits Relation to other Models Telephone model: One-to-one comm., no interference constraint Christian Konrad Radio Aggregation Scheduling 5 / 1

15 Broadcast in Radio-unicast Model Brodcast in the Radio-unicast Model Given: Graph G = (V, E) and source node s V Each round, RAS-legal matching between informed & uninformed nodes 1 One-to-one communication (one sender to one receiver) 2 Interference constraint: Successful reception at receiver if exactly one neighbor transmits Relation to other Models Telephone model: One-to-one comm., no interference constraint Christian Konrad Radio Aggregation Scheduling 5 / 1

16 Broadcast in Radio-unicast Model Brodcast in the Radio-unicast Model Given: Graph G = (V, E) and source node s V Each round, RAS-legal matching between informed & uninformed nodes 1 One-to-one communication (one sender to one receiver) 2 Interference constraint: Successful reception at receiver if exactly one neighbor transmits Relation to other Models Radio model: One-to-many comm., interference constraint holds Christian Konrad Radio Aggregation Scheduling 5 / 1

17 Broadcast in Radio-unicast Model Brodcast in the Radio-unicast Model Given: Graph G = (V, E) and source node s V Each round, RAS-legal matching between informed & uninformed nodes 1 One-to-one communication (one sender to one receiver) 2 Interference constraint: Successful reception at receiver if exactly one neighbor transmits Relation to other Models Radio model: One-to-many comm., interference constraint holds Christian Konrad Radio Aggregation Scheduling 5 / 1

18 Broadcast in Radio-unicast Model Brodcast in the Radio-unicast Model Given: Graph G = (V, E) and source node s V Each round, RAS-legal matching between informed & uninformed nodes 1 One-to-one communication (one sender to one receiver) 2 Interference constraint: Successful reception at receiver if exactly one neighbor transmits Relation to other Models Radio model: One-to-many comm., interference constraint holds Christian Konrad Radio Aggregation Scheduling 5 / 1

19 Broadcast in Radio-unicast Model Brodcast in the Radio-unicast Model Given: Graph G = (V, E) and source node s V Each round, RAS-legal matching between informed & uninformed nodes 1 One-to-one communication (one sender to one receiver) 2 Interference constraint: Successful reception at receiver if exactly one neighbor transmits Relation to other Models Radio-unicast: One-to-one comm., interference constraint holds Christian Konrad Radio Aggregation Scheduling 5 / 1

20 Broadcast in Radio-unicast Model Brodcast in the Radio-unicast Model Given: Graph G = (V, E) and source node s V Each round, RAS-legal matching between informed & uninformed nodes 1 One-to-one communication (one sender to one receiver) 2 Interference constraint: Successful reception at receiver if exactly one neighbor transmits Relation to other Models Radio-unicast: One-to-one comm., interference constraint holds Christian Konrad Radio Aggregation Scheduling 5 / 1

21 Broadcast in Radio-unicast Model Brodcast in the Radio-unicast Model Given: Graph G = (V, E) and source node s V Each round, RAS-legal matching between informed & uninformed nodes 1 One-to-one communication (one sender to one receiver) 2 Interference constraint: Successful reception at receiver if exactly one neighbor transmits Relation to other Models Radio-unicast: One-to-one comm., interference constraint holds Christian Konrad Radio Aggregation Scheduling 5 / 1

22 Known Results Known Results: Converge-cast schedule in Θ(diam + ω(g)) on unit-disc graph If interference radius larger than transmission radius in unit disc graph: O(1)-approximation 2-approximation on unit interval graphs [Wan et al., MobiHoc 2009], [Xu et al., FOWANC 2009], [Chen et al., MSN 2005] [An et al., I. J. Comput. Appl. 2011], [Guo et al., J. of Combin. Opt. 2014] Our Objectives Systematic study of RAS, starting with general graphs Approximation algorithms for geometrically defined graph classes Christian Konrad Radio Aggregation Scheduling 6 / 1

23 Our Results General Graphs 1. It is NP-hard to approximate RAS within factors n 1 ɛ or dn, where d is the average degree 2. Polynomial-time O( dn)-approximation algorithm Interval Graphs 3. Polynomial-time O(log n)-approximation algorithm Christian Konrad Radio Aggregation Scheduling 7 / 1

24 Algorithm for General Graphs Christian Konrad Radio Aggregation Scheduling 8 / 1

25 Algorithm for General Graphs Simulating the Radio Model A round in the radio model can be simulated in (max degree) rounds in the radio-unicast model Theorem [Kowalski, Pelc, Dist. Comp. 2007] Broadcast in the radio model can be done in O(diam + log 2 (n)) rounds. Corollary Broadcast in the radio-unicast model can be done in O( (diam + log 2 (n))) rounds. Lower Bound on OPT diam is a trivial LB. Hence: Õ( )-approximation Christian Konrad Radio Aggregation Scheduling 9 / 1

26 Algorithm for General Graphs G = (V, E), d : average degree, informed node s, OPT known 1 L V : nodes of degree at least dn ( L dn) 2 Inform L sequentially along shortest paths from s in O(diam dn) rounds 3 Inform dn OPT centers adjacent nodes to L in O( dn OPT ) rounds 4 Inform remaining nodes by simulating radio broadcast algorithm in O( dn(diam + log 2 (n)) rounds Christian Konrad Radio Aggregation Scheduling 10 / 1

27 Algorithm for General Graphs G = (V, E), d : average degree, informed node s, OPT known 1 L V : nodes of degree at least dn ( L dn) 2 Inform L sequentially along shortest paths from s in O(diam dn) rounds 3 Inform dn OPT centers adjacent nodes to L in O( dn OPT ) rounds 4 Inform remaining nodes by simulating radio broadcast algorithm in O( dn(diam + log 2 (n)) rounds Christian Konrad Radio Aggregation Scheduling 10 / 1

28 Algorithm for General Graphs G = (V, E), d : average degree, informed node s, OPT known 1 L V : nodes of degree at least dn ( L dn) 2 Inform L sequentially along shortest paths from s in O(diam dn) rounds 3 Inform dn OPT centers adjacent nodes to L in O( dn OPT ) rounds 4 Inform remaining nodes by simulating radio broadcast algorithm in O( dn(diam + log 2 (n)) rounds Christian Konrad Radio Aggregation Scheduling 10 / 1

29 Algorithm for General Graphs G = (V, E), d : average degree, informed node s, OPT known 1 L V : nodes of degree at least dn ( L dn) 2 Inform L sequentially along shortest paths from s in O(diam dn) rounds 3 Inform dn OPT centers adjacent nodes to L in O( dn OPT ) rounds 4 Inform remaining nodes by simulating radio broadcast algorithm in O( dn(diam + log 2 (n)) rounds Christian Konrad Radio Aggregation Scheduling 10 / 1

30 Algorithm for General Graphs G = (V, E), d : average degree, informed node s, OPT known 1 L V : nodes of degree at least dn ( L dn) 2 Inform L sequentially along shortest paths from s in O(diam dn) rounds 3 Inform dn OPT centers adjacent nodes to L in O( dn OPT ) rounds 4 Inform remaining nodes by simulating radio broadcast algorithm in O( dn(diam + log 2 (n)) rounds Theorem Õ( dn)-approximation for radio-unicast broadcast Christian Konrad Radio Aggregation Scheduling 10 / 1

31 Approximation Hardness for General Graphs Christian Konrad Radio Aggregation Scheduling 11 / 1

32 Connection IS/Coloring and RAS Hardness of IS/Coloring: [Feige, Kilian, J. Comput. Syst. Sci. 1998] Deciding whether a graph has chromatic number χ(g) n ɛ or χ(g) n 1 ɛ is NP-hard. Connection IS/Coloring and RAS Christian Konrad Radio Aggregation Scheduling 12 / 1

33 Connection IS/Coloring and RAS Hardness of IS/Coloring: [Feige, Kilian, J. Comput. Syst. Sci. 1998] Deciding whether a graph has chromatic number χ(g) n ɛ or χ(g) n 1 ɛ is NP-hard. Connection IS/Coloring and RAS Large IS in G implies large RAS-legal matching in B(G) Christian Konrad Radio Aggregation Scheduling 12 / 1

34 Connection IS/Coloring and RAS Hardness of IS/Coloring: [Feige, Kilian, J. Comput. Syst. Sci. 1998] Deciding whether a graph has chromatic number χ(g) n ɛ or χ(g) n 1 ɛ is NP-hard. Connection IS/Coloring and RAS c-coloring in G implies RAS-legal matching cover of size c in B(G) Christian Konrad Radio Aggregation Scheduling 12 / 1

35 Connection IS/Coloring and RAS Hardness of IS/Coloring: [Feige, Kilian, J. Comput. Syst. Sci. 1998] Deciding whether a graph has chromatic number χ(g) n ɛ or χ(g) n 1 ɛ is NP-hard. Connection IS/Coloring and RAS Converse is also true: RAS-legal matching cover of size c in B(G) implies c-coloring in G Christian Konrad Radio Aggregation Scheduling 12 / 1

36 LB Construction Lower Bound Construction Binary +B(G) Christian Konrad Radio Aggregation Scheduling 13 / 1

37 LB Construction Lower Bound Construction One bipartition of B(G) can be informed in O(log n) rounds Christian Konrad Radio Aggregation Scheduling 13 / 1

38 LB Construction Lower Bound Construction OPT = O(log n) + size of RAS-legal matching cover OPT small induced RAS-legal matching cover small in B(G) coloring with few colors in G Christian Konrad Radio Aggregation Scheduling 13 / 1

39 LB Construction Lower Bound Construction Theorem It is NP-hard to approximate RAS within factor n 1 ɛ, for any ɛ > 0. Christian Konrad Radio Aggregation Scheduling 13 / 1

40 Interval Graphs Christian Konrad Radio Aggregation Scheduling 14 / 1

41 Interval Graphs Unit Interval Graphs [Guo et al., J. of Combin. Opt. 2014] Inform a diameter path (dominating set) Each color class of a coloring can be informed in O(1) rounds Runtime: O(diam + χ(g)), diam and χ(g) are LBs O(1)-approx. Interval Graphs Difficulty: claws Splitting into O(log n) length classes Informed length class informs other length class in O(OPT ) rounds Theorem There is a polynomial-time algorithm for RAS on interval graphs with approximation ratio O(log n). Christian Konrad Radio Aggregation Scheduling 15 / 1

42 Interval Graphs Unit Interval Graphs [Guo et al., J. of Combin. Opt. 2014] Inform a diameter path (dominating set) Each color class of a coloring can be informed in O(1) rounds Runtime: O(diam + χ(g)), diam and χ(g) are LBs O(1)-approx. Interval Graphs Difficulty: claws Splitting into O(log n) length classes Informed length class informs other length class in O(OPT ) rounds Theorem There is a polynomial-time algorithm for RAS on interval graphs with approximation ratio O(log n). Christian Konrad Radio Aggregation Scheduling 15 / 1

43 Interval Graphs Unit Interval Graphs [Guo et al., J. of Combin. Opt. 2014] Inform a diameter path (dominating set) Each color class of a coloring can be informed in O(1) rounds Runtime: O(diam + χ(g)), diam and χ(g) are LBs O(1)-approx. Interval Graphs Difficulty: claws Splitting into O(log n) length classes Informed length class informs other length class in O(OPT ) rounds Theorem There is a polynomial-time algorithm for RAS on interval graphs with approximation ratio O(log n). Christian Konrad Radio Aggregation Scheduling 15 / 1

44 Interval Graphs Unit Interval Graphs [Guo et al., J. of Combin. Opt. 2014] Inform a diameter path (dominating set) Each color class of a coloring can be informed in O(1) rounds Runtime: O(diam + χ(g)), diam and χ(g) are LBs O(1)-approx. Interval Graphs Difficulty: claws Splitting into O(log n) length classes Informed length class informs other length class in O(OPT ) rounds Theorem There is a polynomial-time algorithm for RAS on interval graphs with approximation ratio O(log n). Christian Konrad Radio Aggregation Scheduling 15 / 1

45 Interval Graphs Unit Interval Graphs [Guo et al., J. of Combin. Opt. 2014] Inform a diameter path (dominating set) Each color class of a coloring can be informed in O(1) rounds Runtime: O(diam + χ(g)), diam and χ(g) are LBs O(1)-approx. Interval Graphs Difficulty: claws Splitting into O(log n) length classes Informed length class informs other length class in O(OPT ) rounds Theorem There is a polynomial-time algorithm for RAS on interval graphs with approximation ratio O(log n). Christian Konrad Radio Aggregation Scheduling 15 / 1

46 Conclusion Summary Õ( dn)-approximation algorithm for RAS on general graphs n 1 ɛ -approximation hardness on general graphs O(log n)-approximation algorithm for RAS on interval graphs Open Questions O(1)-approximation on interval graphs? Is there a const/poly-log approximation on unit disc graphs? Disc Graphs? Christian Konrad Radio Aggregation Scheduling 16 / 1

47 Thank you. Christian Konrad Radio Aggregation Scheduling 17 / 1

Rumors Across Radio, Wireless, and Telephone

Rumors Across Radio, Wireless, and Telephone Jennifer Iglesias Carnegie Mellon University Pittsburgh, USA jiglesia@andrew.cmu.edu R. Ravi Carnegie Mellon University Pittsburgh, USA ravi@andrew.cmu.edu