Reliably Detecting Connectivity using Local Graph Traits

Local distributed algorithms can only gather sufficient information to identify local graph traits, that is, properties that hold within the local neighborhood of each node. However, it is frequently the case that global graph properties (connectivity, diameter, girth, etc) have a large influence on the execution of a distributed algorithm. This paper studies local graph traits and their relationship with global graph properties. Specifically, we focus on graph k-connectivity. First we prove a negative result that shows there does not exist a local graph trait which perfectly captures graph k-connectivity. We then present three different local graph traits which can be used to reliably predict the k-connectivity of a graph with varying degrees of accuracy. As a simple application of these results, we present upper and lower bounds for a local distributed algorithm which determines if a graph is k-connected. As a more elaborate application of local graph traits, we describe, and prove the correctness of, a local distributed algorithm that preserves k-connectivity in mobile ad hoc networks while allowing nodes to move independently whenever possible

Approximating minimum power covers of intersecting families and directed edge-connectivity problems

AbstractGiven a (directed) graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of its nodes. Let G=(V,E) be a graph with edge costs {c(e):e∈E} and let k be an integer. We consider problems that seek to find a min-power spanning subgraph G of G that satisfies a prescribed edge-connectivity property. In the Min-Powerk-Edge-Outconnected Subgraph problem we are given a root r∈V, and require that G contains k pairwise edge-disjoint rv-paths for all v∈V−r. In the Min-Powerk-Edge-Connected Subgraph problem G is required to be k-edge-connected. For k=1, these problems are at least as hard as the Set-Cover problem and thus have an Ω(ln|V|) approximation threshold. For k=Ω(nε), they are unlikely to admit a polylogarithmic approximation ratio [15]. We give approximation algorithms with ratio O(kln|V|). Our algorithms are based on a more general O(ln|V|)-approximation algorithm for the problem of finding a min-power directed edge-cover of an intersecting set-family; a set-family F is intersecting if X∩Y,X∪Y∈F for any intersecting X,Y∈F, and an edge set I covers F if for every X∈F there is an edge in I entering X

Power assignment for k-connectivity in wireless ad hoc networks

Abstract — The problem Min-Power k-Connectivity seeks a power assignment to the nodes in a given wireless ad hoc network such that the produced network topology is k-connected and the total power is the lowest. In this paper, we present several approximation algorithms for this problem. Specifically, we propose a 3k-approximation algorithm for any k ≥ 3, a(k +12H (k))-approximation algorithm for k (2k − 1) ≤ n where n is the network size, a (k +2⌈(k +1)/2⌉)-approximation algorithm for 2 ≤ k ≤ 7, a6-approximation algorithm for k =3, and a 9-approximation algorithm for k =4. Index Terms — k-connectivity, power assignment, wireless ad hoc sensor networks I

Local distributed algorithms for multi-robot systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 165-173) and index.The field of swarm robotics focuses on controlling large populations of simple robots to accomplish tasks more effectively than what is possible using a single robot. This thesis develops distributed algorithms tailored for multi-robot systems with large populations. Specifically we focus on local distributed algorithms since their performance depends primarily on local parameters on the system and are guaranteed to scale with the number of robots in the system. The first part of this thesis considers and solves the problem of finding a trajectory for each robot which is guaranteed to preserve the connectivity of the communication graph, and when feasible it also guarantees the robots advanced towards a goal defined by an arbitrary motion planner. We also describe how to extend our proposed approach to preserve the k-connectivity of a communication graph. Finally, we show how our connectivity-preserving algorithm can be combined with standard averaging procedures to yield a provably correct flocking algorithm. The second part of this thesis considers and solves the problem of having each robot localize an arbitrary subset of robots in a multi-robot system relying only on sensors at each robot that measure the angle, relative to the orientation of each robot, towards neighboring robots in the communication graph. We propose a distributed localization algorithm that computes the relative orientations and relative positions, up to scale, of an arbitrary subset of robots. For the case when the robots move in between rounds we show how to use odometry information to allow each robot to compute the relative positions complete with scale, of an arbitrary subset of robots. Finally we describe how to use the our localization algorithm to design a variety of multi-robot tasks.by Alejandro Cornejo.Ph.D