Maximum Life Routing Schedule in Multihop Wireless Networks | CS 547, Exams of Computer Science

Download Maximum Life Routing Schedule in Multihop Wireless Networks | CS 547 and more Exams Computer Science in PDF only on Docsity! Maximum-Life Routing Schedule in Multihop Wireless Networks (draft) 1 Introduction In this chapter, we study max-life routing schedule for various communication tasks including concurrent unicasts, data aggregation, broadcast and multicast in multihop wireless networks subject to the energy constraints. The communication topology of a multihop wireless network is represented by an arc- weighted digraph D = (V;A; c) where the weight c (a) of a link a 2 A is the minimum power requirement by a. All nodes can adjust their transmitting power. The power consumptions due to transmissions and receptions follow the same model introduced in the previous chapter. As in the previous chapter, we also ignore the power consumption due to reception in all routings. For the purpose of uni…ed treatment, an instance of a communication task given implicitly by a collection R of routes. A routing schedule subject an energy budget function b 2 RV+ is a set of pairs (Hi; xi) 2 RR+ for i = 1;


; k satisfying that kX i=1 pHi (u)xi  b (u) ;8u 2 V: The life (or length) of this schedule is de…ned to be Pk i=1 xi. The problemMax-Life Routing Sched- ule (MLRS) seeks a routing schedule of maximum life subject an energy budget function b 2 RV+. The problem MLRS can be formulated as the following linear program (LP): max P H2R xH s:t: P H2R xHpH (u)  b (u) ;8u 2 V ; xH  0;8H 2 R: (1) This LP has jV j constraints (excluding the trivial constraints xH  0;8H 2 R), and consequently there always exists an optimal solution using at most n routes. However, since the number of variables jRj is prohibitively large (exponential in the number of nodes), standard LP solvers are not practical for solving this packing LP. In this chapter, we present three algorithms for MLRS: Ellipsoid Algorithm (EA), Price-Directive Algorithm (PDA), and Flow-Based Algorithm (FBA). The …rst two algorithms are generic and can be applied to produce a routing schedule for any communication task. The last 1 EA PDA FBA Conc. Unicasts exact 1 + " exact Aggregation exact 1 + " exact Broadcast 2H (n 1) 1 (1 + ") (2H (n 1) 1) N/A Multicast O (k") O (k") N/A Table 1: Optimality or approximation bound of the three algorithms. one are designed speci…cally for concurrent unicasts and data aggregation. Table 1 summarizes the optimality or the approximation bound of these three algorithms. The following notations are used throughout this chapter. We use n to denote the number of nodes in V , i.e., jV j = n. For any positive integer k, H(k) denotes the k-th Harmonic number 1+ 12 +


+ 1 k . For any real function f 2 RS and any subset S0  S, f (S0) denotes P e2S0 f (e). 2 Min-Cost Routing In this section, we introduce a generalization of Min-Power Routing for a communication task. Given a price function y 2 RV+, the cost of a subgraph H of D with respect to y is de…ned to P u2V y (u) pH (u). Note that if y is an all-one vector, the cost of H w.r.t. to y is exactly p (H). Given an instance of a communication task given implicitly by a collection R of routes and a price function y 2 RV+, the problem Min-Cost Routing (MCR) seeks a route H 2 R of minimum cost with respect to y. By applying the algorithms developed in the previous chapter for MPR, we immediately have the following algorithmic results: 1. MCR for Concurrent Unicasts can be solved in polynomial time. 2. MCR for Aggregation can be solved in polynomial time. 3. There is a (2H (n 1) 1)-approximation algorithm for MCR for Broadcasting. 4. For any …xed " > 0, there is an O (k")-approximation algorithm for MCR for Multicasting. The two problems MLRS and MCR are intrinsically related to each other. We refer to the LP in equation (1) as the primal LP. The dual to this primal LP associates a price y(u) for each node u 2 V : min P u2V b (u) y (u) s:t: P u2V pH (u) y (u)  1;8H 2 R y (u)  0;8u 2 V (2) 2 the separating hyperplanes found by the above separation oracle while running the ellipsoid algorithm on S (L "). Then, R0 is of polynomial cardinality. Consider the restricted dual LP: min P u2V b (u) y (u) s:t: P u2H pH (u) y (u)  1;8H 2 R0 y (u)  0;8u 2 V Its value is also at least L. So, we solve the following restricted primal LP of polynomial size, which is the dual of the restricted dual LP: max P H2R0 xH s:t: P H2R0 xHpH (u)  b (u) ;8u 2 V tH  0;8H 2 R0 The optimal solution of this restricted LP has value at least L, which is a -approximation to the original primal LP. Theorem 3.1 immediately implies the following algorithmic results on MLRS: 1. MLRS for Concurrent Unicasts can be solved in polynomial time. 2. MLRS for Aggregation can be solved in polynomial time. 3. There is a (2H (n 1) 1)-approximation algorithm for MLRS for Broadcasting. 4. For any …xed " > 0, there is an O (k")-approximation algorithm for for MLRS for Multicasting. To conclude this section, we remark that the approximation algorithms presented in this section are of theoretical interest only. They characterize the approximation hardness of the optimization problems studied in this section. However, the ellipsoid method with the approximation separation oracles is practically quite infeasible [6]. In the coming sections, we will develop practically feasible approximation algorithms for those problems. 4 Price-Directive Algorithm In this section, we present an iterative algorithm called Price-Directive Algorithm (PDA) for a given communication task, which is adapted from the Garg-Köneman algorithm [4] for fractional packing problems. The outline of the algorithm is described in Table 2. The basic idea of this algorithm is that by setting the prices of the nodes with low residue energy relatively higher, the nodes with low residue energy are protected from getting drained of energy quickly while the nodes with high residue energy are enforced to contribute more energy. The algorithm utilizes a -approximation algorithm 5 A for MCR for the same communication task (if  = 1, the algorithm A is optimal). A constant parameter " 2 (0; 1) is also part of the input, and the output solution has an approximation bound of at most (1 + "). The algorithm maintains the following variables:  H: the set of chosen routes, and a variable;  xH for each H 2 H: the duration of H;  z 2 RV+: the energy consumption percentage vector de…ned by z (u) = P H2H xHpH (u) b (u) ;8u 2 V ;  : the maximum energy consumption percentage maxu2V z (u);  y 2 RV+: the price vector;  : the total energy cost P u2V b (u) y (u). Initially, H is empty and the price y (u) of each node u is the reverse of its energy budget b (u). Accordingly, both z and  are initialized to zero, and is initialized to n accordingly. Each iteration …rst computes a route H 2 R using an algorithm A on D together with the current price vector y. A node v is said to be a bottleneck if b (v) =pH (v) is the smallest among all nodes. Let v be the bottleneck node and set t = b (v) =pH (v). The duration xH of H is increased by t, and both z and  are updated accordingly. After that, the price vector y is reset properly and the variable is updated accordingly. The stopping rule is that 0 <   1+"" ln n . Finally, each xH for H 2 H is scaled down by a factor of  to obtain a feasible solution. The next theorem gives both the running time and approximation ratio of the algorithm PDA. Theorem 4.1 The algorithm PDA produces an (1 + ")-approximation in at most n l (1+") lnn (1+") ln(1+")" m iterations. Theorem 4.1 immediately implies the approximation bounds of PDA listed in Table 1. The rest of this section is devoted to the proof of the Theorem 4.1. We …rst introduce the following algebraic inequality. Lemma 4.2 For any " > 0 and 0  t  1, z  log1+" (1 + "t). 6 Price-Directive Algorithm (PDA): H ;;8u 2 V; z (u) 0; 0; 8u 2 V; y (u) 1b(u) ; n; repeat compute an H 2 R using A on (D; y) ; t minv2V b (v) =pH (v); if H 2 H then xH xH + t, else H H[ fHg, xH t; 8u 2 V , z (u) z (u) + tpH(u)b(u) ;  maxu2V z (u) ; 8u 2 V , y (u) y (u)  1 + "tpH(u)b(u)  ; P u2V b (u) y (u) ; until 0 <   1+"" ln n ; Output f(H;xH=) : H 2 Hg. Table 2: Price-Directive Algorithm Proof. Let f (t) = (1 + ")z (1 + "z) : Clearly f (0) = f (1) = 0. Since (1 + ")t is convex and 1 + "t is linear, f (t) is also convex. Therefore, for any t 2 [0; 1], f (t)  max ff (0) ; f (1)g = 0; which implies that (1 + ")t  1 + "t: Taking the logarithm with base 1 + ", the lemma follows. Let H0, z0, 0, y0 and 0 be the initial values of H, z, , y and respectively. For each j  1, let Hj , zj , j , yj and j be the values of H, z, , y and respectively at the end of the j-th iteration. In addition, for each j  1, let j = max u2V yj (u) b (u) : We …rst claim that j  log1+" j : Indeed, for each j  1 and each u 2 V , by Lemma 4.2, zj (u) zj1 (u)  log1+" (1 + " (zj (u) zj1 (u))) = log1+" yj (u) yj1 (u) ; which implies zj (u)  log1+" yj (u) y0 (u) = log1+" (yj (u) b (u)) : 7 s; t 2 V . A vector f 2 RA+ is called a ‡ow from s to t, or simply a s t ‡ow, if for each for each v 2 V n fs; tg, f out (v)
= f in (v)
This condition is called the ‡ow conservation law : the amount of ‡ow entering a vertex v 6= s; t should be equal to the amount of ‡ow leaving v. The value of a ‡ow f from s to t is, by de…nition: val (f) = f out (s)
f in (s)
: So the value is the net amount of ‡ow leaving s, which is also equal to the net amount of ‡ow entering t. Figure 1 is an example of a s t ‡ow of value 11. It’s well-known that a set of s t paths of total value L de…nes a s t ‡ow of value L, and conversely any s t ‡ow of value L can be decomposed into at most jAj s t paths of total value L and possibly some circuits (see, e.g., [2]). A s t ‡ow f is subject to an arc-capacity z 2 RA+ if f  z. The problem Maximum Flow seeks a s t ‡ow f of maximum value subject to an arc-capacity z 2 RA+. This problem can be solved in polynomial time by the well-known ‡ow augmentation algorithms. 0 5 5 66 4 5 2 4 0 0 t v4 v3 v2 v s 1 Figure 1: An an s t ‡ow of value 11. Suppose that we are given with k commodities with si; ti being the source and sink, respectively, for commodity i. We use Fi to denote the set of si–ti ‡ows. A k-‡ow is a sequence hf1; f2;


; fki with fi 2 Fi for each 1  i  k. The concurrency of a k-‡ow hf1; f2;


; fki is de…ned to be min1ik val (fi). A k-‡ow hf1; f2;


; fki is subject to an arc-capacity z 2 RA+ if Pk i=1 fi  z. The problem Maximum Concurrent Multi‡ow seeks a k-‡ow hf1; f2;


; fki of maximum concurrency subject to an arc-capacity z 2 RA+. This problem can be formulated as the LP below and thus can be solved in polynomial time. max L s:t: fi 2 Fi;81  i  k val (fi) = L;81  i  kPk i=1 fi  z 5.1 Concurrent Unicasts Suppose that we are given with k unicasts with si; ti being the source and sink, respectively, of the i-th unicast. We treat the k unicasts as k commodities. A k-‡ow hf1; f2;


; fki is subject to an energy 10 budget b 2 RV+ if X e2out(v) c (e) kX i=1 fi (e) !  b (v) for each v 2 V . Then MLRS for concurrent unicasts is equivalent to Maximum Concurrent Multi‡ow subject to the energy budget b. Our algorithm proceeds in two steps. The …rst step of the algorithm solves the following LP of polynomial size: max L s:t: fi 2 Fi;81  i  k val (fi) = L;81  i  kP e2out(v) c (e) Pk i=1 fi (e)   b (v) ;8v 2 V In the second step, we decompose each fi into at most jAj si–ti paths of total value L and discarding the rest circuits if there is any. In total, we obtain at most k jAj paths. 5.2 Data Aggregation Let D = (V;A) be a digraph and s be a node in V . A fractional s-arborescence packing in D subject to an arc-capacity function z 2 RA+ is a set of k pairs (Tj ; j) with each Tj being a spanning s-arborescence in D and j 2 R+ for each 1  j  k satisfying that for each e 2 A,X 1jk;e2Tj j  z (e) : The value of this packing is de…ned to be Pk j=1 j . A maximum-value fractional s-arborescence packing in D subject to z can be computed in polynomial time by the Gabow-Manu algorithm [3]. The packing output by the Gabow-Manu algorithm uses at most jAj spanning s-arborescences. In addition, the following min-max relation for fractional arborescence packing was observed by Wong [7] and Maculan [5] (see also Theorem 52.7 in [6]). Theorem 5.1 The value of a maximum-value fractional s-arborescence packing in D subject to z is equal to min u2V nfsg mflow (u; z) ; where mflow (u; z) is the maximum s u ‡ow in D subject to z, for each u 2 V n fsg. 11 The …rst step of the algorithm assigns an arc-capacity function z 2 RA+ maximizing minu2V nfsgmflow (u; z) subject to the energy constraints:X e2out(v) c (e) z (e)  b (v) ;8v 2 V Such capacity function can be obtained by solving the following concise LP: max L s:t: P e2out(v) c (e) z (e)  b (v) ;8v 2 V val (fu) = L;8u 2 V n fsg fu 2 Fu;8u 2 V n fsg fu  z;8u 2 V n fsg In the second step, we apply the Gabow-Manu algorithm to compute a maximum packing of at most jAj spanning inward s-arborescences. Clearly, the length of the such packing is at least L. On the other hand, L is clearly an lower bound on optimum. Thus, L is optimum and the length of the packing is exactly L. References [1] G. Calinescu, S. Kapoor, A. Olshevsky, and A. Zelikovsky, Network Lifetime and Power Assignment in Ad-Hoc Wireless Networks, Proc. of European Symposium on Algorithms, September 2003, LNCS 2832, pp. 114-126. [2] L.R. Ford, and D.R. Fulkerson, Flows in Networks. Princeton University Press, Princeton 1962. [3] H.N. Gabow and K.S. Manu, Packing algorithms for arborescences (and spanning trees) in capaci- tated graphs, Mathematical Programming 82 (1998) 83–109. [4] N. Garg, and J. Könemann, Faster and Simpler Algorithms for Multicommodity Flow and Other Fractional Packing Problems, in Proceedings of the 39th Annual Symposium on Foundations of Computer Science, Palo Alto, CA, November 1998, pp. 300–309. [5] N. Maculan, A new linear programming formulation for the shortest sdirected spanning tree problem, Journal of Combinatorics, Information & System Sciences 11 (1986) 53–56. [6] A. Schrijver, Combinatorial Optimization: Polyhedra and E¢ ciency. Algorithms and Combinatorics, Vol. 24, Springer, Berlin, 2003. [7] R.T. Wong, A dual ascent approach for Steiner tree problems on a directed graph, Mathematical Programming 28 (1984) 271–287. 12