Efficient Approximations for Thinnest Path Problem in Grid Graph

In this paper, we study the thinnest path problem for secure communication in wireless sensor networks. In real situations, there may exist potential eavesdroppers in the sensor networks and the sensors usually cannot afford the expensive encryption and decryption algorithms. An intuitive idea is to find a path from a source to its destination with minimum number of nodes overhearing the message (the nodes directly connects to the source-destination path). We refer such problem as the Thinnest Path (TP) problem, and discuss it under a graph model named Grid Graph (GG). We first prove that a modified Dijkstra algorithm has an approximation ratio of 7 for any given source-destination pair. Next, we define a new concept of "shortest thinnest path (STP)", and propose a novel design named Pseudo-Dijkstra algorithm based on STP. We then prove that Pseudo-Dijkstra algorithm yields an optimal solution while scarifies time-complexity. Finally, we provide some simulation experiments to exhibit the performance of our observations and designs.