Jaime Cohen

Numerical taxonomy on data: experimental results

We consider the problem of fitting an n x n distance matrix D by a tree metric T. This problem is NP-hard for most reasonable distance functions between D and T. Recently, an approximation algorithm was presented (Agarwala et al., 1996) which achieves a factor of 3 approximation to the L infinity best fitting tree. We call this method the Single Pivot (SP) heuristic. Within the biology community, the so-called Neighbor-Joining (NJ) heuristic (Saitou and Nei, 1987) has wide acceptance. In this paper, we introduced a new Double Pivot (DP) heuristic, which is an extension of the SP heuristic, and show that DP outperforms NJ on biological and random data.

Minimax relations for T-join packing problems

In this paper we present structural and algorithmic results for problems involving the packing of T-joins. We explore minimax relations that relate the size of a packing of T-joins with the size of a minimum T-cut in a graph. We present a new conjecture stating that if all T-cuts have the same parity then the maximum size of a family of T-joins that uses each edge at most twice equals the double of the size of a minimum T-cut. We show that this conjecture is equivalent to a famous conjecture for perfect matchings. We also prove a theorem for the case |T|/spl les/8 and describe a polynomial time algorithm for the maximization problem.

Delivering packets during the routing convergence latency interval through highly connected detours

Routing protocols present a convergence latency for all routers to update their tables after a fault occurs and the network topology changes. During this time interval, which in the Internet has been shown to be of up to minutes, packets may be lost before reaching their destinations. In order to allow nodes to continue communicating during the convergence latency interval, we propose the use of alternative routes called detours. In this work we introduce new criteria for selecting detours based on network connectivity. Detours are chosen without the knowledge of which node or link is faulty. Highly connected components present a larger number of distinct paths, thus increasing the probability that the detour will work correctly. Experimental results were obtained with simulation on random Internet-like graphs generated with the Waxman method. Results show that the fault coverage obtained through the usage of the best detour is up to 90%. When the three best detours are considered, the fault coverage is up to 98%.

Numerical taxonomy on data (abstract): experimental results

This definition does not specify a distance function. Since T induces a matrix of distances for S, we can take any standard distance function between matrices. such as L,, L2, or L,. Thatis,forsomekE {l.Z.....~},wewenttofindatreemetric T minimizing Lk(T D). We can similarly seek to minimize the relatrve error, which is defmed in terms of percent change, rather than absolute change. Another type of optimization criterion used in the biology literature is the so-called Mrnimum Eaolutton Cratenon (ME). Here we seek the tree T, such that T[i, j] > D[i,j]. for all i and j, and such that the sum of the weight of the edges in T is minimized. This is the Steiner Tree problem for tree metrics. The numerical taxonomy problems associated with most of the optimization criteria described above are NP hard [3, 5, 1. 41 In (11. the first positive result for numerical taxonomy was presented. They showed that if c is the distance to the closest tree metric under the L, norm. i.e.. c = min~{L,(T D)}, then it is possible to construct B tree T such that L,(T D) < 3c, that is, they geve a 3approximation algorithm for t& problem. We will refer to this algorithm as the Single Pxaol (SP) heuristic.

Parallel cut tree algorithms

Abstract A cut tree is a combinatorial structure that represents the edge-connectivity between all pairs of vertices of an undirected graph. Cut trees solve the all pairs minimum s – t -cut problem efficiently. Cut trees have a large number of applications including the solution of important combinatorial problems in fields such as graph clustering and graph connectivity. They have also been applied to scheduling problems, social network analysis, biological data analysis, among others. Two sequential algorithms to compute a cut tree of a capacitated undirected graph are well known: the Gomory–Hu algorithm and the Gusfield algorithm. In this work three parallel cut tree algorithms are presented, including parallel versions of Gusfield and Gomory–Hu algorithms. A hybrid algorithm that combines techniques from both algorithms is proposed which provides a more robust performance for arbitrary instances. Experimental results show that the three algorithms achieve significant speedups on real and synthetic graphs. We discuss the trade-offs between the alternatives, each of which presents better results given the characteristics of the input graph. On several instances the hybrid algorithm outperformed both other algorithms, being faster than the parallel Gomory–Hu algorithm on most instances.

A Parallel Implementation of Gomory-Hu's Cut Tree Algorithm

Cut trees are a compact representation of the edge-connectivity between every pair of vertices of an undirected graph, and have a large number of applications. In this work a parallel version of the well known Gomory-Hu cut tree algorithm is presented. The parallel strategy is based on the master/slave model. The strategy is optimistic in the sense that the master process manipulates the tree being constructed and the slaves solve minimum s-t-cuts independently. Another version is proposed that employs a heuristic that enumerates all (up to a limit) of the minimum s-t-cuts in order to choose the most balanced one. The algorithm was implemented and extensive experimental results are presented, including a comparison with Gusfieldâs cut tree algorithm. Parallel versions of these algorithms have achieved significant speedups on real and synthetic graphs. We discuss the trade-offs between the two alternatives, each of which presents better results given the characteristics of the input graph. In particular, the existence of balanced cuts clearly gives an advantage to Gomory-Huâsalgorithm.

Connectivity Criteria for Ranking Network Nodes

In this work we introduce a new quantitative criteria for assessing the connectivity of nodes based on the well known concept of edge-connectivity. We call the new criteria the connectivity numbers of a node. They consist of a hierarchy of measures that starts with a local measure that progressively becomes a global connectivity measure of the network. We show that the connectivity numbers can be computed in polynomial time. Experimental results are described showing how the proposed approach compares to other well known concepts involving connectivity and centrality of network nodes in real and synthetic networks.