Computational Experiments for the Problem of Sensor-Mission Assignment in Wireless Sensor Networks

Transcription

1 Adv. Studies Theor. Phys., Vol. 7, 2013, no. 3, HIKARI Ltd, Computational Experiments for the Problem of Sensor-Mission Assignment in Wireless Sensor Networks Anna Gorbenko Department of Intelligent Systems and Robotics Ural Federal University Ekaterinburg, Russia Vladimir Popov Department of Intelligent Systems and Robotics Ural Federal University Ekaterinburg, Russia Abstract In this paper we consider an approach to solve the problem of sensormission assignment in wireless sensor networks. Our approach is based on usage of local search algorithms to solve a logical model for the problem. PACS: Fb, Dz Keywords: sensor-mission assignment, wireless sensor networks, satisfiability problem, NP-complete Many planning problems are extensively studied recently (see e.g. [1] [4]). In particular, the problem of sensor-mission assignment in wireless sensor networks [5] has natural applications in the design of experiments. The problem of sensor-mission assignment in wireless sensor networks: Instance: A weighted bipartite graph G =(S; M; P ; E) and a positive integer K where S M is the set of nodes, S = {S[1];...; S[n]} is a collection of sensors, M = {M[1];...; M[m]} is a collection of missions, P = {p[1];...; p[m]}

2 136 A. Gorbenko and V. Popov is a collection of positive mission profits, and E is a collection of non-negative weights for the edges S M. Question: Is there a subset F of E such that no two chosen edges share the same sensor and M[j] A p[j] K where A is a subset of M such that (i,j) F e[i, j] 1 for each M[j] from A? Note that the problem of sensor-mission assignment in wireless sensor networks is NP-complete. Encoding hard problems as instances of SAT and solving them with different efficient satisfiability algorithms has caused considerable interest (see e.g. [6] [14]). In this paper, we consider an approach to solve the problem of sensor-mission assignment in wireless sensor networks. Our approach is based on an explicit reduction from the problem to the satisfiability problem. We consider different local search algorithms to solve a logical model for the problem. We assume that p[0] = 0, p[j] = d q=0 w[j, q]2 q, K = d q=0 w[q]2 q, w[j, d + 1] = w[d + 1] = 0, for all 1 j m. Let ϕ[1] = 1 i n, 1 j[1]<j[2] m ( x[i, j[1]] x[i, j[2]]), ϕ[2] = 1 j m ( y[j] ( 1 i n,e[i,j]>0 x[i, j])), ϕ[3] = 0 q d+1, z[1,j,q]=w[j, q], 0 j m ϕ[4] = 1 j m z[3,j,0], ϕ[5] = 0 q d+1 ( y[1] z[2, 1,q]=z[1, 1,q]), ϕ[6] = 0 q d+1 (y[1] z[2, 1,q]), ϕ[7] = 0 q d+1, ( z[2,j 1,d+1] z[2,j 1,q]=z[2,j,q]), ϕ[8] = 0 q d+1, (z[2,j 1,d+1] y[j] (z[2,j 1,q] z[3,j,q] z[1,j,q] z[2,j,q])), ϕ[9] = 0 q d+1, (z[2,j 1,d+1] y[j] (z[2,j 1,q] z[3,j,q] z[1,j,q] z[3,j,q+ 1])), ϕ[10] = 0 q d+1, (z[2,j 1,d+1] y[j] ( z[2,j 1,q] z[3,j,q] z[1,j,q] z[2,j,q])), ϕ[11] = 0 q d+1, (z[2,j 1,d+1] y[j] ( z[2,j 1,q] z[3,j,q] z[1,j,q] z[3,j,q+ 1])), ϕ[12] = 0 q d+1, (z[2,j 1,d+1] y[j] (z[2,j 1,q]

3 Computational experiments for sensor-mission assignment 137 z[3,j,q] z[1,j,q] z[2,j,q])), ϕ[13] = 0 q d+1, (z[2,j 1,d+1] y[j] (z[2,j 1,q] z[3,j,q] z[1,j,q] z[3,j,q+ 1])), ϕ[14] = 0 q d+1, (z[2,j 1,d+1] y[j] (z[2,j 1,q] z[3,j,q] z[1,j,q] z[2,j,q])), ϕ[15] = 0 q d+1, (z[2,j 1,d+1] y[j] (z[2,j 1,q] z[3,j,q] z[1,j,q] z[3,j,q+ 1])), ϕ[16] = 0 q d+1, (z[2,j 1,d+1] y[j] ( z[2,j 1,q] z[3,j,q] z[1,j,q] z[2,j,q])), ϕ[17] = 0 q d+1, (z[2,j 1,d+1] y[j] ( z[2,j 1,q] z[3,j,q] z[1,j,q] z[3,j,q+ 1])), ϕ[18] = 0 q d+1, (z[2,j 1,d+1] y[j] ( z[2,j 1,q] z[3,j,q] z[1,j,q] z[2,j,q])), ϕ[19] = 0 q d+1, (z[2,j 1,d+1] y[j] ( z[2,j 1,q] z[3,j,q] z[1,j,q] z[3,j,q+ 1])), ϕ[20] = 0 q d+1, (z[2,j 1,d+1] y[j] (z[2,j 1,q] z[3,j,q] z[1,j,q] z[2,j,q])), ϕ[21] = 0 q d+1, (z[2,j 1,d+1] y[j] (z[2,j 1,q] z[3,j,q] z[1,j,q] z[3,j,q+ 1])), ϕ[22] = 0 q d+1, (z[2,j 1,d+1] y[j] ( z[2,j 1,q] z[3,j,q] z[1,j,q] z[2,j,q])), ϕ[23] = 0 q d+1, (z[2,j 1,d+1] y[j] ( z[2,j 1,q] z[3,j,q] z[1,j,q] z[3,j,q+ 1])), ϕ[24] = 0 q d+1, (y[j] z[2,j 1,q]=z[2,j,q]), ϕ[25] = 1 q d+1 (z[2,m,q]=w[q] z[q]) z[q 1], ϕ[26] = 1 q d+1 (z[2,m,q] >w[q] z[q]) z[q 1],

4 138 A. Gorbenko and V. Popov ϕ[27] = 1 q d+1 (z[2,m,q] >w[q] z[q]) u, ϕ[28] = 1 q d+1 (z[2,m,q] <w[q] z[q]) z[q 1], ϕ[29] = 1 q d+1 (z[2,m,q] <w[q] z[q]) u, ϕ[30] = 1 q d+1 z[q] z[q 1], ϕ[31] = z[d +1], ϕ[32] = (z[2,m,0] >w[0] z[0]) u, ϕ[33] = (z[2,m,0] = w[0] z[0]) u, ϕ[34] = (z[2,m,0] <w[0] z[0]) u. Let ξ = 34 i=1ϕ[i]. It is easy to check that ξ is satisfiable if and only if there is a subset F of E such that no two chosen edges share the same sensor and M[j] A p[j] K where A is a subset of M such that (i,j) F e[i, j] 1 for each M[j] from A. Using standard transformations (see e.g. [15]) we can easily obtain an explicit transformation ξ into ζ such that ξ ζ and ζ is a 3-CNF. It is clear that ζ gives us an explicit reduction from the problem of sensor-mission assignment in wireless sensor networks to 3SAT. We have created a generator of natural instances for the problem of sensor-mission assignment in wireless sensor networks. We have used algorithms from [16]: fgrasp and posit. We have used a heterogeneous cluster for solution of the problem. Each test was run on a cluster of at least 100 nodes. Due to restrictions on computation time (20 hours) we used savepoints. Selected experimental results are given in Table 1. time average max best fgrasp hr hr 9.48 min posit hr hr min Table 1: Experimental results for 3SAT ACKNOWLEDGEMENTS. The work was partially supported by Analytical Departmental Program Developing the scientific potential of high school References [1] A. Gorbenko and V. Popov, Footstep Planning for Humanoid Robots, Applied Mathematical Sciences, 6 (2012), [2] A. Gorbenko and V. Popov, Programming for Modular Reconfigurable Robots, Programming and Computer Software, 38 (2012)

5 Computational experiments for sensor-mission assignment 139 [3] A. Gorbenko, V. Popov, and A. Sheka, Localization on Discrete Grid Graphs, Lecture Notes in Electrical Engineering, 107 (2012), [4] A. Gorbenko, M. Mornev, and V. Popov, Planning a Typical Working Day for Indoor Service Robots, IAENG International Journal of Computer Science, 38 (2011) [5] H. Rowaihy, M.P. Johnson, O. Liu, A. Bar-Noy, T. Brown, and T. La Porta, Sensor-mission assignment in wireless sensor networks, ACM Transactions on Sensor Networks, 6 (2010) 36. [6] A. Gorbenko and V. Popov, The Problem of Finding Two Edge-Disjoint Hamiltonian Cycles, Applied Mathematical Sciences, 6 (2012), [7] A. Gorbenko and V. Popov, Hamiltonian Alternating Cycles with Fixed Number of Color Appearances, Applied Mathematical Sciences, 6 (2012), [8] A. Gorbenko and V. Popov, The Hamiltonian Alternating Path Problem, IAENG International Journal of Applied Mathematics, 42 (2012) [9] A. Gorbenko and V. Popov, On the Longest Common Subsequence Problem, Applied Mathematical Sciences, 6 (2012), [10] A. Gorbenko and V. Popov, On the Problem of Sensor Placement, Advanced Studies in Theoretical Physics, 6 (2012) [11] A. Gorbenko and V. Popov, The Longest Common Parameterized Subsequence Problem, Applied Mathematical Sciences, 6 (2012), [12] A. Gorbenko and V. Popov, The set of parameterized k-covers problem, Theoretical Computer Science, 423 (2012), [13] A. Gorbenko and V. Popov, The Far From Most String Problem, Applied Mathematical Sciences, 6 (2012), [14] A. Gorbenko and V. Popov, Multi-agent Path Planning, Applied Mathematical Sciences, 6 (2012), [15] A. Gorbenko and V. Popov, Task-resource Scheduling Problem, International Journal of Automation and Computing, 9 (2012), [16] SATLIB The Satisfiability Library. [Online]. Available: hoos/satlib/index-ubc.html Received: December 3, 2012