A GRASP for Broadcast Scheduling in Ad-Hoc TDMA Networks

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1 A GRASP for Broadcast Scheduling in Ad-Hoc TDMA Networks Sergiy I. Butenko Dept. of Industrial Engineering, Texas A&M University College Station, TX 77843, USA and Clayton W. Commander and Panos M. Pardalos Dept. of Industrial and Systems Engineering, University of Florida Gainesville, FL , USA ABSTRACT The Broadcast Scheduling Problem (BSP) arises in the study of ad-hoc networks. The BSP is an NP-Complete problem in which a finite set of stations are to be scheduled in a time division multiple access (TDMA) frame. In a TDMA frame, time is divided into equal length slots in which transmissions occur. Unconstrained message transmission can result in a collision of messages, therefore the objective of the BSP is to provide a collision free broadcast schedule which minimizes the total frame length and maximizes the slot utilization within the frame. In this article, we present a GRASP (Greedy Randomized Adaptive Search Procedure) for the BSP. GRASP is a two-phase multi-start heuristic for hard combinatorial problems. A reactive heuristic was used to automatically balance GRASP parameters. A variant of the post-optimization enhancement procedure Path Relinking is also applied. We report experimental results given by our approach. Keywords: GRASP, Ad-Hoc Networks, Broadcast Scheduling Problem, Combinatorial Optimization, Path Relinking 1. INTRODUCTION Ad-hoc networks provide high speed communication between potentially mobile receivers by using a multi-hop protocol. In such networks, each station can act as both a client and a server. Applications of ad-hoc networks can be seen in military battlefield scenarios and mobile commerce [2]. Since every station in the network shares the same radio channel, precautions should be taken when messages are scheduled so as to prevent message collision [21]. There are two types of collision in ad-hoc networks. Direct collision is a result of two adjacent stations broadcasting during the same time slot. Hidden collision occurs when two nonneighboring stations transmit simultaneously to a station that can receive messages from both senders. A collision free schedule will minimize the overall delay of the system [3]. 2. PROBLEM FORMULATION Consider a graph G =(V, E) where the vertex set V = {1, 2,..., n} represents the stations in the network. Then we can model the network by letting the edge set E represent the set of transmission links between adjacent stations in the network. We say that stations i and j are one-hop neighbors iff there exists an undirected edge (i, j) E. If (i, j) E but there exists an intermediate node k V such that (i, k) E and (k, j) E, then stations i and j are referred to as two-hop neighboring stations. A hidden collision is a result of two-hop neighbors transmitting in the same slot [4]. Notice that an equivalent interpretation of V is that it represents the set of direct collisions. Let C be an N N symmetric binary matrix, where N = V. Then we can represent the set of one-hop neighbors in the incidence matrix C = {c ij} as follows: { 1, if (i, j) E and i j, c ij = (1) 0, otherwise. We assume that there are M time slots per TDMA frame, and that each slot length is equal to the amount of time required to transmit one packet of data. We assume also that packets are received in the same slot they are transmitted and packets are sent at the beginning of each slot. We now represent the broadcast schedule as a M N binary matrix S = {s mn} defined as follows: { 1, if station n is to broadcast in slot m, s mn = (2) 0, otherwise. In order to analyze the efficiency of a broadcast schedule, we need to calculate the percentage of the available slots being assigned in a transmission frame [4]. Let ρ n be the slot utilization for station n. Then, ρ n = = the number of slots assigned to station n (3) frame length M m=1 smn M. (4) Hence, the total slot utilization of the entire network, ρ, is given by ρ = = N n=1 ρn N M N m=1 n=1 smn NM (5). (6) With these tools, we can now represent the Broadcast Scheduling Problem as Minimize M and Maximize ρ

2 procedure GRASPwPR(input, MaxIter) 1 i 1 2 do while i < MaxIter 3 call GreedyRandomizedConstructor(S) 4 call LocalSearch(S) 5 if i > PRSize 6 call PathRelinking(S, S g) 7 else 8 add current solution to PRSet 9 fi 10 call UpdateSolution(S) 11 od elihw end GRASPwPR Figure 1. GRASP with Path Relinking Heuristic subject to: M s mn 1, n, (7) m=1 c ij + s mi + s mj 2, i, j, m, i j, (8) ciks mi + c kj s mj 1, i, j, k, m, i j, j k, k i. (9) Contraint (7) ensures that each station transmits at least once per frame. Constraint (8) ensures that one-hop neighbors do not transmit in the same slot. Finally, (9) prevents two-hop neighbors from transmitting simultaneously [21]. The BSP was shown to be NP-complete by the current authors in [5]. 3. GRASP FOR BROADCAST SCHEDULING Greedy Randomized Adaptive Search Procedure (GRASP) is a two-phase, multi-start metaheuristic for hard combinatorial problems first introduced by Feo & Resende [8]. In the first phase, known as the construction phase, an initial solution is created by means of an adaptive greedy function. Since a construction phase solution is not guaranteed to be locally optimal, phase two implements a local search to improve the initial solution. The best solution produced from all GRASP iterations is returned. GRASP is easily adaptable and has been successfully applied to such problems as broadcast scheduling [4], quadratic assignment [14, 15], and most recently to the uncapacitated facility location problem [18]. For a list of other GRASP applications, please the bibliography of Festa and Resende [9]. Construction Phase In [4], GRASP is applied to the BSP with satisfying results. Here, we will implement the same construction phase originally employed in [4]. Initially, the stations are sorted in descending order of the number of one-hop and two-hop neighbors. Then, the station with the most neighbors is assigned. After this greedy choice, the restricted candidate list (RCL) is created and consists of the best α% of stations which may simultaneously transmit with the greedy assigned station. Initially, we use α = 20. A station is then selected at random from the RCL and assigned in the current slot. A new RCL is created, and another station randomly selected and assigned. This process continues until RCL =, at which point the slot number is incremented and the process restarts with another greedy choice. The selection of the greedy choice is biased towards those stations which have not been previously assigned. procedure PathRelinking(Guide, Current) 1 slot 1; 2 do while slot σ M 3 if guide(slot) > current(slot) 4 current(slot) guide(slot); 5 fi 6 if all stations assigned at least once 7 EXIT; 8 fi 9 slot slot + 1; 10 od elihw end PathRelinking Figure 2. Pseudocode for the Path Relinking subroutine Local Search As with the construction phase, the local search used here is taken from [4]. Using the schedule produced in the construction phase, the slots are sorted in descending order of the number of bursts. The two slots with the fewest scheduled transmissions are combined, and the number of slots is now k = m 1. Call this modified schedule s m,n and define E(m i) to be the set of collisions in slot m i. By summing along all the slots, we have the function to minimize as f(s) = k i=1 E(m i). We apply the following local search for this minimization. A station causing a collision from the combined slot is randomly selected and every attempt is made to swap this station with another from the remaining k 1 slots. After each swap, f(s) is re-evaluated. If the new value of f(s) is less than the value before the swap was made, then this swap is considered successful and another colliding station from the combined slot is selected and the swap exchange procedure is repeated. However, if f(s) is not improved, the swap is undone and another is attempted. If after every attempt no successful swap is made, a new colliding station is randomly chosen and again the swap procedure is attempted. This process continues until either a successful swap is made, or until some iteration limit is reached. If f(s) = 0, then the frame length has been successfully decremented by one slot. Again, the two slots with the fewest broadcasts are combined, the value of k is once

3 (a) an α i parameter is selected from ℵ with some probability p i. Initially, since no choice is favored, p i is uniform for all i = {1, 2..., k}. As the iterations progress, certain values of α will produce better results than others. Therefore, we have another set A = {a 1, a 2,..., a n} where each element a i A stores the average solution value found using parameter α i ℵ. Next, the values of Λ = {λ 1, λ 2,..., λ n}, where λ i = f(s ) a i and s represents the current best solution are calculated. For the BSP, we define f(s ) to be the minimum frame length found. Lastly, the probabilities p i are updated such that p i = λ i nk=1 λ k. It turns out that reactive GRASP can significantly improve standard GRASP solutions while adding little additional computation time. Since more options are available, the program can tailor itself to good solutions by varying the size of RCL [11] (b) (c) Figure 3. (a) 15 station network. (b) 30 station network. (c) 40 station network. again decremented, and the process repeats. In the end if f(s) > 0 then no improved solution was found and the original construction phase solution is returned as best. Reactive GRASP Reactive GRASP (RG) [16] is helpful enhancement for the standard GRASP. The RG method automatically determines the value of α parameter. Recall that α determines the size of the Restricted Candidate List in the construction phase. Without Reactive GRASP, it is up to the practitioner to determine the best value of α through extensive brute-force testing. Reactive GRASP for the BSP was proposed by the authors in [6]. The following is a description of this implementation. Initially, a set of potential α values is formed, such as ℵ = {α 1, α 2,..., α k }. A standard convention is to initialize k = 10 and ℵ = {.1,.2,..., 1.0}. From each iteration, Path Relinking First introduced by Glover path relinking (PR) was used as an enhancement routine for tabu searches [10]. PR was first applied to GRASP by Laguna and Martí in [13]. Path relinking introduces a memory to the GRASP which usually results in improved solutions. In the standard GRASP, the iterative nature of the heuristic includes no mechanism for remembering traits about solutions generated in each iteration. Thus nothing can be recalled about why or why not a certain solution was more favorable than another. Path relinking allows GRASP to remember these traits and favor them in successive iterations. GRASP with PR was successfully applied to problems such as job shop scheduling [1] and quadratic assignment [15]. In [6], a variant of path relinking was proposed for BSP which we will now describe. Path relinking works by using a set of elite solutions as guides and examines point to point trajectories in search of an optimal solution. With the GRASP for the BSP, the set of elite solutions is stored in memory. Our elite set contains the ten best solutions up to the current iteration. After a normal GRASP iteration, an elite solution is chosen randomly to be the guide. There is then a slot-wise comparison between the guiding solution and the current solution. In each slot, if the guiding solution has more scheduled transmissions than the current solution, then that slot in the current solution is replaced by the slot from the guiding solution. If however the current slot has more bursts than the guiding solution, then it is kept and the next slot examined. At this point, a check is performed to determine if all stations have been assigned. If they have, then the procedure stops and the solution is returned. If all stations have not been assigned, then the slot-wise comparison continues and after each slot another all-broadcast check is performed. The process is allowed to continue for at most σ M slots, where σ M = min{m guide, M current}. If at this point, all stations have still not been scheduled, then the program exits returning the solution with σ M slots as the best. Pseudocode for the PR subroutine can be seen in Fig-

4 Stations LB Frame Length Channel Utilization G RG+PR SVC MFA G RG+PR SVC MFA Figure 4. Comparison of frame length and utilization all tested heuristics. The lower bounds (LB) were determined by calculating the clique number as described in [12]. ure 2. Here guide(i) refers to the set of vertices in the i th slot of the guiding solution. Note that unlike the standard path relinking schemes, our procedure does not necessarily relink the starting and guiding solutions, i.e., it aims to introduce certain attributes of the guiding solution into the current solution, while the generated path may not reach the guiding solution in the end. 4. COMPUTATIONAL RESULTS In [4], the standard GRASP was tested on three networks introduced by Wang & Ansari [20] which have become the de facto test cases for broadcast scheduling heuristics. These graphs are shown in Figure 3. The solutions reported improved the solutions reported in [20] and [21]. Reactive GRASP with path relinking was also tested on these same test cases in [6] with satisfying results (see Figure 4). Also, for an example comparison of time to solution between standard GRASP (G) and reactive GRASP with path relinking (RG+PR), see Figure 5. In [5] to provide a more thorough examination of BSP heuristics the authors tested the heuristics of [4, 19, 20, 21] on several randomly generated unit-disk graphs with 50, 75, and 100 vertices and varying densities ranging from units. For these cases, GRASP was the best performer with respect to frame length minimization. Unit-disc graphs are popular models in which each vertex has the same transmission range equal to the radius of a Instance GRASP RG+PR n Rad. m ρ m ρ Figure 6. The average frame length and utilization are given for the randomly generated networks consisting of 50, 75, and 100 stations with radii of 20, 30, 40, and 50 units. disk centered at the vertex. In this article, we compare the results of the standard GRASP to those of reactive GRASP with path relinking when tested on the unit-disk graphs described above. Comparative results between GRASP and RG+PR for the unit-disk graphs can be seen in Figure 6. The data in the figure are representative of the average frame lengths and utilizations of 5 random networks for each station-density combination (class). As we suspected, the RG+PR routine improves the solutions of the standard GRASP. In all cases, notice that RG+PR has average frame lengths at least as small as GRASP for each class and improves the average utilization for many classes where the average frame lengths are the same. 5. CONCLUDING REMARKS In this paper, we propose apply a metaheuristic based on GRASP to the problem of scheduling broadcasts in an ad-hoc TDMA network. We enhance the basic GRASP by using reactive GRASP with path relinking which finds improved solutions by automatically balancing parameters and performing post-optimization improvement techniques. When compared to other well-known heuristics, these techniques appear to robust, reliable, and maintain their integrity as problem size increases. REFERENCES Figure 5. Time to solution comparison for standard GRASP and Reactive GRASP with Path Relinking for 40 station example network. 1. R.M. Aiex, S. Binato, and M.G.C. Resende, Parallel GRASP with Path Relinking for Job Shop Scheduling, Parallel Computing, vol. 29, pp , S.I. Butenko, X. Cheng, C.A.S. Oliveira, and P.M. Pardalos, A New Algorithm for Connected Dominating Sets in Ad Hoc Networks, in S. Butenko, R. Murphey, and P. Pardalos, editors, Recent Developments in Cooperative Control and Optimization, pp 61-73, Kluwer Academic Publishers, 2003.

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