The Thinnest Path Problem
Abstract
We formulate and study the thinnest path problem in wireless ad hoc networks. The objective is to find a path from a source to its destination that results in the minimum number of nodes overhearing the message by a judicious choice of relaying nodes and their corresponding transmission power. We adopt a directed hypergraph model of the problem and establish the NPcompleteness of the problem in 2D networks. We then develop two polynomialtime approximation algorithms that offer $\sqrt{\frac{n}{2}}$ and $\frac{n}{2\sqrt{n1}}$ approximation ratios for general directed hypergraphs (which can model nonisomorphic signal propagation in space) and constant approximation ratios for ring hypergraphs (which result from isomorphic signal propagation). We also consider the thinnest path problem in 1D networks and 1D networks embedded in 2D field of eavesdroppers with arbitrary unknown locations (the socalled 1.5D networks). We propose a linearcomplexity algorithm based on nested backward induction that obtains the optimal solution to both 1D and 1.5D networks. This algorithm does not require the knowledge of eavesdropper locations and achieves the best performance offered by any algorithm that assumes complete location information of the eavesdroppers.
 Publication:

arXiv eprints
 Pub Date:
 May 2013
 arXiv:
 arXiv:1305.3688
 Bibcode:
 2013arXiv1305.3688G
 Keywords:

 Computer Science  Networking and Internet Architecture;
 Computer Science  Data Structures and Algorithms