Computing Best Swaps in Optimal Tree Spanners

Abstract

Abstract. In a densely connected communication network, represented by a graph G with nonnegative edge-weights, it is often advantageous to route all communication on a sparse, spanning subnetwork, typically a spanning tree of G. With the communication overhead in mind, we consider a spanning tree T of G which guarantees that for any two nodes, their distance in T is at most k times their distance in G, where k, called the stretch, is as small as possible. Such a spanning tree which minimizes the stretch is called an optimal tree spanner, and it can be used for efficient routing. However, for a communication tree, the failure of an edge is catastrophic; it disconnects the tree. Functionality can be restored by connecting both parts of the tree with another edge, while leaving the two parts themselves untouched. In situations where the failure can be repaired rapidly, such a quick fix is preferred over the recomputation of an entirely new optimal tree spanner, because it is much closer to the previous solution and hence requires far fewer adjustments in the routing scheme. We are therefore interested in the problem of finding for any possibly failing edge in the spanner T a best swap edge to replace it. The objective here is naturally to minimize the stretch of the new tree. We show how all these best swap edges can be computed in total time O(m 2 log n) in graphs with arbitrary nonnegative edge weights. For graphs with unit weight edges (also called unweighted graphs), we present an O(n 3 ) time algorithm. Furthermore, we present a distributed algorithm for computing the best swap for each edge in the spanner.

Distributed Minimum Spanning Tree Maintenance for Transient Node Failures

“…For example, shortest paths tree have been studied in [1], [17], optimal tree spanners in [5], minimum diameter spanning trees in [14], [19]. Distributed algorithms have been devised for determining swap edges for shortest paths spanning trees in [10] and [11], and for minimum diameter spanning trees in [15].…”

Linear time distributed swap edge algorithms

“…One recent technique, particularly efficient in case of transient faults, consists in pre-computing a replacement spanning tree for each possible link or node failure, by computing the best replacement edge (or edges) which reconnects the tree. A number of studies have been done for this problem, both for the sequential [1][2][3][4][5] and distributed [6][7][8][9] models of computation, for different types of spanning trees and failures.…”

Linear Time Distributed Swap Edge Algorithms

“…A number of studies have been done for this problem, both for the sequential [1][2][3][4][5][6] and distributed [7][8][9][10] In this paper, we consider the all best swap edges problem in the distributed setting. We are given a positively weighted 2-edge connected network X of processes, where w(x, y) denotes the weight of any edge {x, y} of X, together with a spanning tree T of X, rooted at a process r. Suppose that all communication between processes is routed through T .…”