Spontaneous failure is an unavoidable aspect of all networks, particularly those with a physical basis such as communications networks or road networks. Whether due to malicious coordinated attacks or other causes, failures temporarily change the topology of the network and, as a consequence, its connectivity and distance metric. In this paper we look at the problem of efficiently answering connectivity, distance, and shortest route queries in the presence of two node or link failures. Our data structure uses Õ(n2) space and answers queries in Õ(1) time, which is within a polylogarithmic factor of optimal and nearly matches the single-failure distance oracles of Demestrescu et al. It may yet be possible to find distance/connectivity oracles capable of handling any fixed number of failures. However, the sheer complexity of our algorithm suggests that moving beyond dual-failures will require a fundamentally different approach to the problem. 1

The maximum cardinality and maximum weight matching problems can be solved in time Õ(m √ n), a bound that has resisted improvement despite decades of research. (Here m and n are the number of edges and vertices.) In this article we demonstrate that this “m √ n barrier ” is extremely fragile, in the following sense. For any ɛ> 0, we give an algorithm that computes a (1 − ɛ)-approximate maximum weight matching in O(mɛ −1 log ɛ −1) time, that is, optimal linear time for any fixed ɛ. Our algorithm is dramatically simpler than the best exact maximum weight matching algorithms on general graphs and should be appealing in all applications that can tolerate a negligible relative error. Our second contribution is a new exact maximum weight matching algorithm for integer-weighted bipartite graphs that runs in time O(m √ n log N). This improves on the O(Nm √ n)-time and O(m √ n log(nN))time algorithms known since the mid 1980s, for 1 ≪ log N ≪ log n. Here N is the maximum integer edge weight. 1

For many fundamental problems there exist randomized algorithms that are asymptotically optimal and are superior to the best known deterministic algorithm. Among these are the minimum spanning tree (MST) problem, the MST sensitivity analysis problem, the parallel connected components and parallel minimum spanning tree problems, and the local sorting and set maxima problems. (For the first two problems there are provably optimal deterministic algorithms with unknown, and possibly superlinear running times.) One downside of the randomized methods for solving these problems is that they use a number of random bits linear in the size of the input. In this paper we develop some general methods for reducing exponentially the consumption of random bits in comparison based algorithms. In some cases we are able to reduce the number of random bits from linear to nearly constant without affecting the expected running time. Most of our results are obtained by adjusting or reorganizing existing randomized algorithms to work well with a pairwise or O(1)-wise independent sampler. The prominent exception — and the main focus of this paper — is a linear-time randomized minimum spanning tree algorithm that is not derived from the well known Karger-Klein-Tarjan algorithm. In many ways it resembles more closely the deterministic minimum spanning tree algorithms based on Soft Heaps. Further,

The maximum cardinality and maximum weight matching problems can be solved in Õ(m√n) time, a bound that has resisted improvement despite decades of research. (Here m and n are the number of edges and vertices.) In this article, we demonstrate that this “m √ n barrier ” can be bypassed by approximation. For any ɛ> 0, we give an algorithm that computes a (1 − ɛ)-approximate maximum weight matching in O(mɛ−1 log ɛ−1) time, that is, optimal linear time for any fixed ɛ. Our algorithm is dramatically simpler than the best exact maximum weight matching algorithms on general graphs and should be appealing in all applications that can tolerate a negligible relative error.

We present a new scaling approach for the maximum weight perfect matching problem in general graphs, with running time O((m + n log n) n log(nN)), where n,m,N denote the number of vertices, number of edges, and largest magnitude of integral costs. Comparing with the complicated long-standing algorithm by [Gabow and Tarjan 1991] of running time O(m n log n log(nN)), our algorithm not only has a better time bound when m = ω(n log n), but is also dramatically simpler to describe and analyze. Our algorithm also matches the time bound O(m n log(nN)) of maximum weight perfect matching for bipartite graphs [Gabow and Tarjan 1989] when m = Ω(n log n). ∗Email:

In a densely connected communication network, represented by a graph G with non-negative edge weights, it is often advantageous to route all communication on a sparse spanning subnetwork, typically a spanning tree of G. We consider a tree spanner 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. When an edge of the communication tree T fails, network functionality may be restored byre-connecting the two separated parts of the tree with a swap edge. In situations where the failure can be repaired rapidly, such a quickfix is preferred over the recomputation of an entirely new minimum-stretch tree, 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 that minimizes the stretch of the new tree. We show how all these best swap edges

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