Abstract. Let G = (V, E) be a 2-edge connected, undirected and nonnegatively weighted graph, and let S(r) be a single source shortest paths tree (SPT) of G rooted at r ∈ V. Whenever an edge e in S(r) fails, we are interested in reconnecting the nodes now disconnected from the root by means of a single edge e ′ crossing the cut created by the removal of e. Such an edge e ′ is named a swap edge for e. Let Se/e ′(r) be the swap tree (no longer an SPT, in general) obtained by swapping e with e ′ , and let Se be the set of all possible swap trees with respect to e. Let F be a function defined over Se that expresses some feature of a swap tree, such as the average length of a path from the root r to all the nodes below edge e, or the maximum length, or one of many others. A best swap edge for e with respect to F is a swap edge f such that F(Se/f (r)) is minimum. In this paper we present efficient algorithms for the problem of finding a best swap edge, for each edge e of S(r), with respect to several objectives. Our work is motivated by a scenario in which individual connections in a communication network suffer transient failures. As a consequence of an edge failure, the shortest paths to all the nodes below the failed edge might completely change, and it might be desirable to avoid an expensive switch to a new SPT, because the failure is only temporary. As an aside, what we get is not even far from a new SPT: our analysis shows that the trees obtained from the swapping have features very similar to those of the corresponding SPTs rebuilt from scratch. Key Words. Network survivability, Single source shortest paths tree, Swap algorithms. 1. Introduction. Survivability

Abstract. In network communication systems, frequently messages are routed along a minimum diameter spanning tree (MDST) of the network, to minimize the maximum delay in delivering a message. When a transient edge failure occurs, it is important to choose a temporary replacement edge which minimizes the diameter of the new spanning tree. Such an optimal replacement is called the best swap. As a natural extension, the all-best-swaps (ABS) problem is the problem of finding the best swap for every edge of the MDST. Given a weighted graph G =(V, E), where |V | = n and |E | = m,wesolvetheABSprobleminO(n √ m)time and O(m + n) space, thus improving previous bounds for m = o(n 2). 1

The problem of verifying a Minimum Spanning Tree (MST) was introduced by Tarjan in a sequential setting. Given a graph and a tree that spans it, the algorithm is required to check whether this tree is an MST. This paper investigates the problem in the distributed setting, where the input is given in a distributed manner, i.e., every node “knows ” which of its own emanating edges belong to the tree. Informally, the distributed MST verification problem is the following. Label the vertices of the graph in such a way that for every node, given (its own label and) the labels of its neighbors only, the node can detect whether these edges are indeed its MST edges. In this paper we present such a verification scheme with a maximum label size of O(log n log W), where n is the number of nodes and W is the largest weight of an edge. We also give a matching lower bound of Ω(log n log W) (except when W ≤ log n). Both our bounds improve previously known bounds for the problem. Our techniques (both for the lower bound and for the upper bound) may indicate a strong relation between the fields of proof labeling schemes and implicit labeling schemes. For the related problem of tree sensitivity also presented by Tarjan, our method yields rather efficient schemes for both the distributed and the sequential settings.

We describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a near-linear-time randomized combinatorial construction that transforms any graph on n vertices into an O(n log n)-edge graph on the same vertices whose cuts have approximately the same value as the original graph’s. In this new graph, for example, we can run the Õ(m3/2)-time maximum flow algorithm of Goldberg and Rao to find an s– t minimum cut in Õ(n3/2) time. This corresponds to a (1 + ɛ)-times minimum s–t cut in the original graph. A related approach leads to a randomized divide and conquer algorithm producing an approximately maximum flow in Õ(m √ n) time. Our algorithm is also used to improve the running time of sparsest cut algorithms from Õ(mn) to Õ(n²). Our approach also accelerates several other recent cut and flow algorithms. Our algorithms are based on a general theorem analyzing the concentration of cut values near their expectation in random graphs.

We present a deterministic parallel algorithm on the EREW PRAM model to verify a minimum spanning tree of a graph. The algorithm runs on a graph with n vertices and m edges in O(log n) time and O(m + n) work. The algorithm is a parallelization of King's linear time sequential algorithm for the problem. 1 Introduction The problem of verifying if a given spanning tree in an edge-weighted graph is a minimum spanning tree for the graph is an important problem which is closely related to the problem of finding a minimum spanning tree of a graph. Recently Dixon, Rauch and Tarjan ([DRT92]) and King ([Kin95]) have developed sequential linear time algorithms for the problem. Dixon and Tarjan have also given an optimal CREW algorithm ([DT94]). In this paper, we present a parallel algorithm which runs in optimal time and work bounds on the weaker EREW PRAM model. This resolves an open question posed in [DT94]. The high-level structure of the algorithm has been adapted from [DT94] and [Kin95]. W...

Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in combinatorial optimization, and demonstrate the effectiveness of these paradigms for two optimization problems on matroids: finding an optimum matroid basis and packing disjoint matroid bases. Applications of these ideas to the graphic matroid led to fast algorithms for minimum spanning trees and minimum cuts. An optimum matroid basis is typically found by a greedy algorithm that grows an independent set into an the optimum basis one element at a time. This continuous change in the independent set can make it hard to perform the independence tests needed by the greedy algorithm. We simplify matters by using sampling to reduce the problem of finding an optimum matroid basis to the problem of verifying that a given fixed basis is optimum, showing that the two problems can be solved in roughly the same ...

We show that for any edge-weighted graph G there is an equivalent graph EG such that the minimum spanning trees of G correspond one-for-one with the spanning trees of EG. The equivalent graph can be constructed in time O(m + n log n) given a single minimum spanning tree of G. As a consequence we can count the minimum spanning trees of G in time O(m + n 2.376 ), generate a random minimum spanning tree in time O(mn), and list all minimum spanning trees in time O(m+n log n+k) where k denotes the number of minimum spanning trees generated. We also discuss similar equivalent graph constructions for shortest paths, minimum cost flows, and bipartite matching.

Updating a minimum spanning tree (MST) is a basic problem for communication networks. In this paper, we consider single node deletions in MSTs. Let G = (V; E) be an undirected graph with n nodes and m edges, and let T be the MST of G. For each node v in V , the node replacement for v is the minimum weight set of edges R(v) that connect the components of T \Gamma v. We present a sequential algorithm and a parallel algorithm that find R(v) for all V simultaneously. The sequential algorithm takes O(m log n) time, but only O(mff(m; n)) time when the edges of E are presorted by weight. The parallel algorithm takes O(log 2 n) time using m processors on a CREW PRAM. 2 1 INTRODUCTION For communication networks, minimum spanning trees (MSTs) are used for basic network tasks such as broadcast, leader election, and synchronization. Updating the MST after changes in network topology is a fundamental problem. In this paper, we update MSTs after single node deletions. In a graph G with ...

The dynamic trees problem is that of maintaining a forest that changes over time through edge insertions and deletions. We can associate data with vertices or edges and manip-ulate this data, individually or in bulk, with operations that deal with whole paths or trees. Efficient solutions to this problem have numerous applications, particularly in algo-rithms for network flows and dynamic graphs in general. Several data structures capable of logarithmic-time dynamic tree operations have been proposed. The first was Sleator and Tarjan’s ST-tree, which represents a partition of the tree into paths. Although reasonably fast in practice, adapting ST-trees to different applications is nontrivial. Frederickson’s topology trees, Alstrup et al.’s top trees, and Acar et al.’s RC-trees are based on tree contractions: they progressively combine vertices or edges to obtain a hierarchical represen-tation of the tree. This approach is more flexible in theory, but all known implementations assume the trees have bounded degree; arbitrary trees are supported only after ternar-ization. This thesis shows how these two approaches can be combined (with very little overhead) to produce a data structure that is at least as generic as any other, very easy to

We investigate the single link failure recovery problem and its application to the alternate path routing problem for ATM networks. Specifically, given a 2-connected graph G, a specified node s, and a shortest paths tree T s = fe 1 ; e 2 ; : : : ; e n\Gamma1 g of s, where e i = (x i ; y i ) and x i = parent Ts (y i ), find a shortest path from y i to s in the graph Gne i for 1 i n \Gamma 1. We present an O(m + n log n) time algorithm for this problem and a linear time algorithm for the case when all weights are equal. When the edge weights are integers, we present an algorithm that takes O(m+ T sort (n)) time where T sort (n) is the time required to sort n integers. We show that any solution to the single link recovery problem can adapted to solve the alternate path routing problem in ATM networks.