Analysis of Minimum-Energy Path-Preserving Graphs for Ad-hoc Wireless Networks

Analysis of Minimum-Energy Path-Preserving Graphs for Ad-hoc Wireless Networks Mahmuda Ahmed, Mehrab Shariar, Shobnom Zerin and Ashikur Rahman Department of Computer Science and Engineering Bangladesh University of Engineering and Technology Dhaka, Bangladesh Abstract We consider ad-hoc wireless networks and the topology control problem defined as minimizing the amount of power needed to maintain connectivity. The issue boils down to selecting the optimum transmission power level at each based on the position information of reachable s. Local decisions regarding the transmission power level induce a subgraph of the maximum powered graph G max in which edges represent direct reachability at maximum power. We propose an analysis for constructing minimum-energy path-preserving subgraphs of G max, i.e., ones minimizing the energy consumption between pairs. We also propose an algorithm for constructing subgraph of G max based on 1-hop neighbor information. By presenting experimental results we show the effectiveness of our proposed algorithm. keywords: Ad-hoc Network. I. INTRODUCTION Minimizing energy consumption is a major design goal in ad-hoc wireless network. If networks are designed preserving minimum energy paths then significant reduction in energy consumption can be achieved. We have to choose the minimum power level for each to minimize the energy consumption for the whole network. We use the same model as [1] which considers, an n-, multi-hop, ad-hoc, wireless network deployed on a two-dimensional plane. Suppose that each is capable of adjusting its transmission power up to a maximum denoted by P max. Such a network can be modeled as a graph G=(V,E), with the vertex set V representing the s, and the edge set defined as follows: E = { x, y x, y V V d(x, y) R max } (1) where d(x,y) is the distance between s x and y and R max is the maximum distance reachable by a transmission at the maximum power P max. The graph G defined this way is called the maximum powered network. By setting a reduced power level for each, messages can be transmitted with a minimum energy among all the paths in the network. This will lessen average degree and produce a subgraph of G, while maintaining global connectivity. We say that a graph G G is a minimum-energy pathpreserving graph or, alternatively, that it has the minimum energy property, if for any pair of s (u,v) that are connected in G, at least one of the (possibly multiple) minimum energy paths between u and v in G also belongs to G. Minimum-energy path-preserving graphs were first defined in [2]. Typically, many minimum-energy path-preserving graphs can be formed from the original graph G. It has been shown that the smallest of such subgraphs of G is the graph G min = (V,E min ),where(u, v) E min iff there is no path of length greater than 1 from u to v that costs less energy than the energy required for a direct transmission between u and v. Let G i = (V,E i ) be a subgraph of G = (V,E) such that (u, v) E i iff (u, v) E and there is no path of length i that requires less energy than the direct one-hop transmission between u and v. Then G min can be formally defined as follows: G min = G 2 G4... Gn 1 (2) It is easy to see that any subgraph G of G has the minimumenergy property iff G G min. Thereby, each of G i G min, for any i = 2, 3,..., n-1 is a minimum-energy path-preserving graph. G 2 is the first of the series G i, it is the most practically useful derivative of the maximum powered graph G. An algorithm for constructing one such graph, G 2, was presented in [2]. It works reasonably well in dense networks but its performance degrades considerably when the network density drops to medium or low. Construction of G 2 efficiently in sparse and moderately dense networks with some assistance of the Medium Access Control (MAC) layer was presented in [1]. To construct G 3 we need 2-hop information that means information from the immediate neighbors and the neighbors of those immediate neighbors. Number of links in G 2 and G 3 is considerably less compared to the number of links within the maximum powered network. Interestingly, number of links in G 3 does not reduce significantly compared to G 2 but G 3 requires 1 hop more information than G 2. On the other hand, if we can construct G 3 and G 2 in a distributed way, we can also construct G 2 G 3. G 2 G 3 is definitely more close to G min than G 2 and it has a great amount of savings in comparison to G 2. Construction of G 3 involves exchanging many messages, and thus causes obvious energy overheads. We have tried not to increase the overhead but improve the performance of G 2. Then we construct G 3 with the information collected from only the neighbors found within the region having radius R max. The resulting graph is Local Minimum Energy Communication Network () is nothing but G 2 with limited information. We present here a distributed algorithm to locally construct G 2. This approach requires no further

information than G 2 but produces a reasonable amount of savings. II. RELATED WORKS Recent research has lead to exposure of mainly three dimensions directed at power control algorithms for wireless mobile networks. The first class of solutions looks at the issue from the perspective of MAC layer. Monks et al. [3] propose a modification to the IEEE 82.11 RTS-CTS handshake whereby a overhearing a CTS packet uses the received power to estimate its distance to the sender. Also, Sing et al. [4] propose techniques for powering-off the transceivers when they are not active. The significance of the second approach lies in its dealing with network layer. Most of the schemes in this class use distributed variants of the Bellman-Ford algorithm with various flavors of cost metrics derived from the notion of power. Metrics taken as standard are as follows: energy consumed per packet, time to network partition, variance in power levels, and(power) cost per packet. The findings in the second approach can play a leading role in the third approach that separates routing from topology control. Ramanathan [5] describe two centralized heuristic algorithms to minimize the maximum transmission power while trying to maintain connectivity or bi-connectivity, called Local Information No Topology (LINT) and Local Information Link-State Topology (LILT). COMPOW proposed by Narayanaswamy [6], chooses the smallest common power level for each emphasizing mainly on 1) connectivity, 2)traffic carrying capacity maximization, 3)contention reduction in MAC layer, and 4) low power requirement. It is pre-assumed that distribution of s are homogeneous, which in the long run may appear as a serious flaw. CLUSTERPOW was designed to overcome the shortcomings of COMPOW by introducing clusters. However, it still suffers from a significant message overhead. Cone-Based Topology Control (CBTC), generates a graph structure similar to the one proposed by Yao [7]. Suitable initial power level and the increment of power level at each step are the driving force of CBTC, and plays a pioneer role in determining the number of overhead messages. Rodoplu [8] introduce the notion of relay region based on a specific power model. Their algorithm was later modified by Li [2] to trim some unnecessary edges. Our work closely relates to these two studies. N. Li [9] propose a distributed topology control algorithm (called LMST) based on constructing minimum spanning trees. Their algorithm achieves three goals: 1) connectivity, 2) bounded degree ( 6) and 3) bi-directional links. In LMST, each u collects the position information of all neighbors reachable at the maximum power. Based on this information, u creates its own local minimum spanning tree among the set of neighbors, where the weight of an edge is the necessary transmission power between its two ends. Once the tree has been constructed, u contributes to the final topology those s that are its neighbors in the spanning tree. One problem with LMST is that the resulting graph does not preserve the minimum-energy paths. III. ALGORITHM A. Power Model We are using the same power model as [1]. Here we assume the well known, generic, two-way, channel path loss model, where the minimum transmission power is a function of distance [1]. To send a packet from x to y, separated by distance d(x,y), the minimum necessary transmission power is approximated by P trans (x, y) =t d α (x, y), (3) where α 2 isthepath loss factor and t is a constant. Signal reception is assumed to cost a fixed amount of power denoted by r. Thus, the total power required for one-hop transmission between x and y becomes P total (x, y) =t d α (x, y)+r, (4) The model assumes that each is aware of its own position with a reasonable accuracy, e.g., via a GPS device. B. Previous approach to constructing G 2 An algorithm was presented in [8] based on relay region. Theconceptislikethis: Given a u and another v within u s communication range (at P max ), the relay region of v as perceived by u (with respect to u), R u v, is the collection of points such that relaying through v to any point in R u v takes less energy than a direct transmission to that point (see 1)). Given Fig. 1. u (a) The relay region of v with respect to u (b) The enclosure of the definition of relay region, the algorithm for constructing G 2 becomes straightforward. Construction of G 2 efficiently in sparse and moderately dense networks with some assistance of the Medium Access Control (MAC) layer was presented in [1].

Algorithm: 1. /* s main */ 2. /* ℵ is the set of discovered so far */ 3. /* ℵ s Neighbor set of s*/ 4. /* Suppose, s got a reply from r*/ 5. begin 6. ℵ s = ℵ s r 7. for each n in ℵ{ 8. /*get list of common neighbors 9. between n and r*/ 1. D=getCommonNeighbor(n,r,ℵ) 11. /*get list of common neighbors 12. between s and r*/ 13. G=getCommonNeighbor(s,r,ℵ) 14. Segment 1: 15. if r is in ℵ s { 16. /* case 1(a): */ 17. /* check for two length path 18. between s and r relaying through n*/ 19. if (cost(s, r) cost(s, n)+cost(n, r)){ 2. ℵ s = ℵ s {r} 21. goto Segment 2 22. } 23. /* case 1(b): */ 24. /* check for three length path between s 25. and r relaying n and d*/ 26. for each d in D{ 27. if (cost(s, r) cost(s, n) 28. +cost(n, d)+cost(d, r)){ 29. ℵ s = ℵ s {r} 3. goto Segment 2 31. } 32. } 33. } C. Algorithm We propose an algorithm for constructing close approximation of G 2 G 3 with much less overhead which works in a distributed way. We shall refer to our algorithm as the Local Minimum Energy Communication Network algorithm or algorithm in short through this paper. In algorithm, function cost(x, y) takes two s x and y as parameters and calculates the cost of direct transmission between them according to the equation 4 assuming t, the predetection threshold equal to 1 7 mw, additional receive power, r = 2mW and path loss factor, α = 4. Function getcommonneighbor(x,y,ℵ) determines the common neighbor set of x and y having information of set of s ℵ. The operation of the above algorithm is described from the viewpoint of one s. Heres broadcasts a single neighbor discovery message (NDM) at the maximum power P max. While s collects the replies of its neighbors, it learns their identities and locations. It constructs set ℵ and ℵ s. Initially, all those sets are empty (s doesn t even know what neighbors there are). Here N is defined as the set of all s that s has 34. Segment 2: 35. if n is in ℵ s { 36. /* case 2(a): */ 37. /* check for two length path 38. between s and n relaying through r*/ 39. if (cost(s, n) cost(s, r)+cost(r, n)){ 4. ℵ s = ℵ s {n} 41. goto Segment 3 42. } 43. /* case 2(b): */ 44. /* check for three length path between 45. s and n relaying r and d*/ 46. for each d in D{ 47. if (cost(s, n) 48. cost(s, r)+cost(r, d)+cost(d, n)){ 49. ℵ s = ℵ s {n} 5. goto Segment 3 51. } 52. } 53. /* case 2(c): */ 54. /* check for three length path between s 55. and n relaying through g and r*/ 56. for each g in G{ 57. if (cost(s, n) cost(s, g) 58. +cost(g, r)+cost(r, n)){ 59. ℵ s = ℵ s {n} 6. goto Segment 3 61. } 62. } 63. } 64. Segment 3: 65. continue 66. } 67. ℵ = ℵ r 68. End discovered so far and ℵ s is the set of its neighbors. Node s has to maintain direct link or direct communication with each in ℵ s, ℵ s ℵ. When s gets a reply from r it is added in the neighbor set ℵ s, then it considers 2 cases for each in ℵ. First it tries to find out a n in ℵ such that cost(s, r) >= cost(s, n)+cost(n, r), thenr is discarded from ℵ s because n can be used as a relay to transmit to r (see figure 2 case 1(a)). If cost(s, r) <cost(s, n)+cost(n, r) then there is no path of length two that can be used to transmit r with less energy than direct transmission between s and r. So,r will not be discarded from neighbor set ℵ s. Thus G 2 is forming here. Now a three length path that costs less energy than a direct transmission between s and r is searched on. If d is a common neighbor between n and r then cost(s, r) >= cost(s, n) + cost(n, d) + cost(d, r) condition is checked. If there is a that fulfils the above condition then r is discarded from ℵ s because there is path of length three which costs less than direct transmission (see Figure 3

Fig. 2. Case 1(a) Fig. 5. Case 2(b) Fig. 3. Case 1(b) case 1(b)). Now our algorithm search for the s that should be removed from neighbor set ℵ s after including r. First it discards all the s for which r can be used as relay in case of two length path (see figure 4). Thus G 2 is formed here. Then search for a d in common neighbor set of n and r and for a g in common neighbor set of s and r is done to find out a three length path that is less costly than direct transmission between s and n (see figure 5 and figure 6). This algorithm constructs G 2 in a proper way and then tries to construct G 3 from available (partial) information. As we know that construction of G 3 requires 2 hop information from neighbors which causes more messages to be exchanged and increases energy overheads. So we tried to improve performance of G 2 here. We constructed G 3 using Fig. 4. Case 2(a) Fig. 6. Case 2(c) the local information found by exchanging 1 hop information like G 2. We refer to the graph found by as G throughout this paper. D. Why G close to G 2 In this section we present scenario where local information from immediate neighbors are not enough to construct G 3. In Figure 7 the circle represents R max neighborhood of a. b and c is inside its neighbor region but d is outside of its sensing range. Suppose d is situated very close to R max i.e., near the boundary of a. As a has only local information in, it will see that there is no two or three length path that costs less than direct communication between a and b, therefore it maintains link ab. But in actual case it might happen that relaying through c and d may cost less than direct communication between a and b. So,acdb is the three length path that is considered in G 3 but not in. This is a case where produces more link than G 2. Lemma 1: G G 2. Proof: A link between two s u, v V is deleted in only when there is another minimum energy path of two or three length between the s, i.e. applies G 2 and locally G 3, thus, it does not destroy any minimum energy path having length two or three. Besides, produces some extra links as described in previous subsection. So the resultant graph G is definitely a superset of G 2.

7 6 original Fig. 7. An Extraneous Edge 5 4 3 2 Lemma 2: preserves connectivity. Proof: We presented G min as, 1 1 2 3 4 5 6 7 G min = G 2 G4... Gn 1 According to [2] G min is the smallest subgraph with the minimum energy property. G min has no redundant edges and also preserves connectivity. Thereby, each of G i G min,for any i = 2, 3,..., n-1 is a minimum-energy path preserving graph and also preserves connectivity. G 2 G 3 is a superset of G min according to Lemma 2 G G 2 G 3 and thus, preserves connectivity. A. Topological Analysis IV. EXPERIMENTAL RESULTS By simulation we evaluate the performance of our algorithm. Initially we deployed 25-3 s over a flat square area of 67m 67m according to uniform distribution. Figure 8 shows a typical deployment of 75 s, each having a maximum communication range of 25m. This is the starting graph G for our algorithm. The graph found by G 2 is figure 9 which shows a great amount of savings than original graph G. The next graph at figure 1 is found by G 3,thisalsohas a large amount of savings with respect to original graph G at figure 8. So, both G 2 and G 3 shows good results. Now the graph that is closer to G min than G 2 and also G 3 is G 2 G 3 is shown in figure 11. This graph has the least number of links among the graphs mentioned above. Figure 12 shows the graph we found by. Here the graph has just one more linkthanthatofg 2 G 3. This implies that the performance of is much closer to G 2 G 3 but incurs much less overhead than constructing G 3 (2 hop information) to calculate G 2 G 3. We perform more simulation on our algorithm deploying 25-1 s according to zipf distribution, figure 13 shows the starting graph G of our algorithm. It is a typical deployment of 75 s, each having a maximum communication range of 25m. The next two fig shows the graph found by G 2 (figure 14) and G 3 (figure 15) where both shows good savings. In uniform distribution G 2 G 3 (figure 16) and G (figure 17) both shows same no of links. We have got the same result here without increasing the overhead, it is definitely a good achievement. Fig. 8. Original Graph (uniform distribution) no. of s 75 7 G2 6 5 4 3 2 1 1 2 3 4 5 6 7 Fig. 9. G 2 (uniform distribution) no. of s 75 7 6 5 4 3 2 1 1 2 3 4 5 6 7 Fig. 1. G 3 (uniform distribution) no. of s 75

7 6 G2Int 7 6 G2 5 5 4 4 3 3 2 2 1 1 1 2 3 4 5 6 7 Fig. 11. G 2 G 3 (uniform distribution) no. of s 75 1 2 3 4 5 6 7 Fig. 14. G 2 (Zipf distribution) no. of s 75 7 6 7 6 5 5 4 4 3 3 2 2 1 1 1 2 3 4 5 6 7 1 2 3 4 5 6 7 Fig. 12. G (uniform distribution) no. of s 75 Fig. 15. G 3 (Zipf distribution) no. of s 75 7 6 original 7 6 G2Int 5 5 4 4 3 3 2 2 1 1 1 2 3 4 5 6 7 1 2 3 4 5 6 7 Fig. 13. Original Graph (Zipf distribution) no. of s 75 Fig. 16. G 2 G 3 (Zipf distribution) no. of s 75

7 6 5 4 3 no. of links 3 25 2 15 G2 2 1 1 2 3 4 5 6 7 Fig. 17. G (Zipf distribution) no. of s 75 Fig. 19. 1 5 2 3 4 5 6 7 8 9 1 no. of s comparison between G 2 and G in Zipf distribution no. of links 1 9 8 7 6 5 4 3 2 1 G2 5 1 15 2 25 3 no. of s TABLE I COMPARISON BETWEEN G 2 AND G IN UNIFORM DISTRIBUTION No of Edges Edges Nodes in G 2 in G 25 61.6 6. 35 85.2 81. 5 133.6 124.8 75 22.8 19.8 1 278. 262. 15 441.6 412.4 2 596. 562.8 3 948.8 92.8 Fig. 18. Comparison Between G 2 and G in Uniform distribution B. Comparison between G 2 and G Here we deployed 25-3 s over a flat square area of 67m 67m according to uniform distribution and simulate the network both for G 2 and. For illustration, Table I shows a comparison between the number of edges in G 2 and in. For a specific number of s there are five different scenarios. We have taken average number of edges or links for each set. We found here a saving of 4.85% without increasing overhead( see figure 18). Again, we simulate on a network of 25-1 s deployed over a flat square area of 67m 67m according to Zipf distribution. Table II shows a comparison between the number of edges in G 2 and in. Up to 6.18% of links is saved here (see figure 19). In both distribution the savings is quite satisfactory and considerably higher for denser networks. C. Overhead Analysis A s broadcasts NDM at first to know about its neighbors. In case of constructing G 2, the number of reply messages for any s is equal to NodeDegree(s). Nodedegree is defined as the number of neighbors it has within the region having radius R max. For constructing G 2 G 3, the number of reply messages is equal to, NodeDegree(s)+ n neighbor(s) NodeDegree(n). Now, we took the number of reply messages as overhead. To analysis on overhead cost by G 3 and G (or G 2 ) we simulate on networks of 25-3 s deployed over a flat square area of 67m 67m spread under uniform distribution ( 2) and 25-1 s deployed over a flat square area of 67m 67m according to Zipf distribution ( 21). For a specific number of s there are five different scenarios. We have taken average number of overhead messages for each set. We found that construction of G 3 causes huge amount of overhead with respect to G in both the cases. TABLE II COMPARISON BETWEEN G 2 AND G IN ZIPF DISTRIBUTION No of Edges Edges Nodes in G 2 in G 25 57.6 55.4 3 75.2 7.8 35 91.2 85.2 4 15.2 99.6 45 118. 111. 5 131.6 125.2 75 27.6 198.2 1 284.8 267.2

No. of overhead messages 3e+6 2.5e+6 2e+6 1.5e+6 1e+6 5 V. CONCLUSION The algorithm we have presented is a distributed algorithm that ensures minimum energy path preserving graphs in adhoc wireless networks. Our construction of is assisted by the rudimentary protocol of constructing G 2. Its need of information for an additional hop is served with no additional overhead and only through the previous information gained through G 2. Our studies assisted by some simulation results, have demonstrated the superiority and robustness of the new algorithm over the previous solutions for networks with moderate and low density of s. 5 1 15 2 25 3 No. of s Fig. 2. Overhead comparison between G 3 and G in Uniform distribution No. of overhead messages 25 2 15 1 5 2 3 4 5 6 7 8 9 1 No. of s Fig. 21. Overhead comparison between G 3 and G in Zipf distribution TABLE III OVERHEAD COMPARISON BETWEEN G 3 AND G IN UNIFORM DISTRIBUTION No of Overhead msg. Overhead msg. Nodes in G in G 3 25 16.8 1342.4 35 334.8 391.8 5 81.4 15613.6 75 1637.2 4113.8 1 382. 16442.8 15 6978.8 361218. 2 12421.6 852938.4 3 27777.6 2821319.2 REFERENCES [1] A. Rahman and P. Gburzynski, Mac-assisted topology control for ad-hoc wireless networks. International Journal of Communication Systems, vol. 19, no. 9, pp. 955 976, 26. [2] L. Li and J. Halpern, Minimum energy mobile wireless networks revisited, Proc. IEEE International Conference on Communications (ICC), june, 21. [3] J. P. Monks, V. Bhargavan, and W. M. Hwu, A power controlled MAC protocol for wireless packet networks, in Proceedings of INFOCOM, 21, pp. 219 228. [4] S. Singh and C. Raghavendra, Power efficient MAC protocol for multihop radio networks, in Proceedings of PIMRC, 1998, pp. 153 157. [5] R. Ramanathan and R. Rosales-Hain, Topology control of multihop wireless networks using transmit power adjustment, in In Proc. of IEEE INFOCOM 2, Tel Aviv, Israel, March 2, pp. 44 413. [6] S. Narayanaswamy, V. Kawadia, R. S. Sreenivas, and P. R. Kumar, Power control in ad-hoc networks: Theory, architecture, algorithm and implementation of the COMPOW protocol, in Proceedings of the European Wireless Conference - Next Generation Wireless Networks: Technologies, Protocols, Services and Applications, Florence, Italy, February 22, pp. 156 162. [7] Y.-S. C. h. Sze-Yao Ni, Yu-Chee Tseng and J.-P. Sheu, The broadcast storm problem in a mobile ad hoc network, in Mobicom, 1999. [8] V. Rodoplu and T. Meng, Minimum energy mobile wireless networks, IEEE Journal of Selected Areas in Communications, vol. 17, pp. 1333 1344, 1999. [9] N. Li, J. C. Hou, and L. Sha, Design and analysis of an MST-based topology control algorithm, in Proceedings of INFOCOM, 23. [1] T. Rappaport, Wireless communications: principles and practice, pp. 219 228, 1996. TABLE IV OVERHEAD COMPARISON BETWEEN G 3 AND G IN ZIPF DISTRIBUTION No of Overhead msg. Overhead msg. Nodes in G in G 3 25 38.8 4799.6 3 357.6 5274.4 35 515.2 956.4 4 675.2 13656.4 45 93.8 2331.2 5 1146.4 24828.4 75 2592.4 14965.6 1 4377.2 22126.8