Distributed approximation algorithms for minimum spanning trees and other related problems with applications to wireless ad hoc networks

Md Abdul Maleq Khan, Purdue University

Abstract

Due to the advent of various advanced network technologies, distributed algorithms have become an important and rapidly growing field of research. Many emerging networks such as wireless ad hoc networks and peer-to-peer networks operate under inherent resource constraints (energy, bandwidth, etc.). Topologies of these networks can also change dynamically. For these networks, it is necessary to develop efficient (in both time and message complexities) distributed algorithms even if the solutions are sub-optimal (approximate). In this dissertation, we develop and analyze a class of distributed approximation algorithms to solve two fundamental network optimization problems: minimum spanning trees (MST) and minimum-cost k-connected subgraphs. We design and analyze a simple randomized scheme called Nearest Neighbor Tree (NNT) for efficient construction of approximate MSTs. We show that our scheme constructs a O(log n)-approximate MST in any weighted graph and a constant approximation for uniform distribution of nodes on a plane. Then we apply the NNT scheme to design local distributed algorithms for MST in complete networks, arbitrary networks, and wireless ad hoc networks. Our main contribution is the first non-trivial distributed O(log n)-approximation algorithm for MST in an arbitrary network which takes Õ(D + L) time and Õ(E) messages, where L is a parameter called the local shortest path diameter and D is the (unweighted) diameter of the graph. L always lies between 1 and n but can be much smaller than [special characters omitted] in most of the graphs. In addition, we develop an algorithm for a complete graph that takes O(log n) time and expected O(n log n) messages and an algorithm for wireless ad hoc network that takes O(log2 n) time and expected O(n) messages. We also perform extensive simulations of our algorithms for wireless ad hoc networks. Simulations validate the theoretical results and show that the bounds can be much better in practice. We extend the NNT scheme to develop a simple randomized scheme for constructing low-cost k-connected spanning subgraphs in a weighted complete graph. We show that our algorithm gives an approximation ratio of O(klog n) for a metric graph, O(k) for a random graph with nodes uniformly randomly distributed in [0,1]2 and O(log [special characters omitted]) for a graph with random edge weights U(0,1). We then design an efficient local distributed algorithm for constructing a k-connected spanning subgraph (for any k ≥= 1) in a point-to-point distributed model, where the processors form a complete network. This algorithm takes O(log [special characters omitted]) time and expected O(nk log [special characters omitted]) messages.

Degree

Ph.D.

Advisors

Pandurangan, Purdue University.

Subject Area

Computer science

COinS