Shiva Chaudhuri

Shortest paths in digraphs of small treewidth. Part I, Sequential algorithms

Abstract. We consider the problem of preprocessing an n -vertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give algorithms that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms can answer distance queries in O(α(n)) time after O(n) preprocessing. This improves upon previously known results for the same problem. We also give a dynamic algorithm which, after a change in an edge weight, updates the data structure in time O(nβ) , for any constant 0 < β < 1 . Furthermore, an algorithm of independent interest is given: computing a shortest path tree, or finding a negative cycle in linear time.

Approximate and Exact Deterministic Parallel Selection

The selection problem of size n is, given a set of n elements drawn from an ordered universe and an integer r with 1 0 asks for any element whose true rank differs from r by at most An. Our main results are: (1) For all t≥(log log n)4, approximate selection problems of size n can be solved in O(t) time with optimal speedup with relative accuracy \(2^{{{ - t} \mathord{\left/{\vphantom {{ - t} {\left( {\log \log n} \right)}}} \right.\kern-\nulldelimiterspace} {\left( {\log \log n} \right)}}^4 }\); no deterministic PRAM algorithm for approximate selection with a running time below Ο(log n/log log n) was previously known. (2) Exact selection problems of size n can be solved in O(log n/log log n) time with O(n log log n/log n) processors. This running time is the best possible (using only a polynomial number of processors), and the number of processors is optimal for the given running time (optimal speedup); the best previous algorithm achieves optimal speedup with a running time of O(log n log*n/log log n).

The p -neighbor k -center problem

The k-center problem with triangle inequality is that of placing k center nodes in a weighted undirected graph in which the edge weights obey the triangle inequality, so that the maximum distance of any node to its nearest center is minimized. In this paper, we consider a generalization of this problem where, given a number p, we wish to place k centers so as to minimize the maximum distance of any non-center node to its pth closest center. We derive a best possible approximation algorithm for this problem.

Shortest paths in digraphs of small treewidth. Part II: optimal parallel algorithms

We consider the problem of preprocessing an n-vertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give parallel algorithms for the EREW PRAM model of computation that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms can answer distance queries in O(α(n)) time using a single processor, after a preprocessing of O(log 2 n) time and O(n) work, where α(n) is the inverse of Ackermanns function. The class of constant treewidth graphs contains outerplanar graphs and series-parallel graphs, among others. To the best of our knowledge, these are the first parallel algorithms which achieve these bounds for any class of graphs except trees. We also give a dynamic algorithm which, after a change in an edge weight, updates our data structures in O(log n) time using O(n β ) work, for any constant 0<β< I. Moreover, we give an algorithm of independent interest: computing a shortest path tree, or finding a negative cycle in O(log 2 n) time using O(n) work.

Computing Mimicking Networks

Abstract. A mimickingnetwork for a k -terminal network, N , is one whose realizable external flows are the same as those of N . Let S(k) denote the minimum size of a mimicking network for a k -terminal network. In this paper we give new constructions of mimicking networks and prove the following results (the values in brackets are the previously best known results): S(4)=5 [216] , S(5)=6 [232] . For bounded treewidth networks we show S(k)= O(k) [2^ 2k ] , and for outerplanar networks we show S(k)

Probabilistic recurrence relations revisited

\leq

The Randomized Complexity of Maintaining the Minimum

10k-6 [k22k+2] .

Sensitive functions and approximate problems

Abstract The performance attributes of a broad class of randomised algorithms can be described by a recurrence relation of the form T(x) = a(x) + T(H(x)), where a is a function and H(x) is a random variable. For instance, T(x) may describe the running time of such an algorithm on a problem of size x. Then T(x) is a random variable, whose distribution depends on the distribution of H(x). To give high probability guarantees on the performance of such randomised algorithms, it suffices to obtain bounds on the tail of the distribution of T(x). Karp derived tight bounds on this tail distribution, when the distribution of H(x) satisfies certain restrictions. In this paper, we give a simple proof of bounds similar to that of Karp using standard tools from elementary probability theory, such as Markovs inequality, stochastic dominance and a variant of Chernoff bounds applicable to unbounded geometrically distributed variables. Further, we extend the results, showing that similar bounds hold under weaker restrictions on H(x). As an application, we derive performance bounds for an interesting class of algorithms that was outside the scope of the previous results.

A lower bound for area-universal graphs

The complexity of maintaining a set under the operations Insert, Delete and FindMin is considered. In the comparison model it is shown that any randomized algorithm with expected amortized cost t comparisons per Insert and Delete has expected cost at least n/(e22t)-1 comparisons for FindMin. If FindMin is replaced by a weaker operation. FindAny, then it is shown that a randomized algorithm with constant expected cost per operation exists; in contrast, it is shown that no deterministic algorithm can have constant cost per operation. Finally, a deterministic algorithm with constant amortized cost per operation for an offline version of the problem is given.

A lower bound for linear approximate compaction

We investigate properties of functions that are good measures of the CRCW PRAM complexity of computing them. While the block sensitivity is known to be a good measure of the CREW PRAM complexity, no such measure is known for CRCW PRAMs. We show that the complexity of computing a function is related to its everywhere sensitivity, introduced by Vishkin and Wigderson (1985). Specifically we show that the time required to compute a function f:D/sup n//spl rarr/R of everywhere sensitivity es(f) with P/spl ges/n processors and unbounded memory is /spl Omega/(log[log es(f)/(log 4P|D|- log es(f))]). This improves previous results of Azar (1992), and Vishkin and Wigderson. We use this lower bound to derive new lower bounds for some approximate problems. These problems can often be solved faster than their exact counterparts and for many applications, it is sufficient to solve the approximate problem. We show that approximate selection requires time /spl Omega/(log[log n/log k]) with kn, processors and approximate counting with accuracy /spl lambda//spl ges/2 requires time /spl Omega/(log[log n/(log k+log /spl lambda/)]) with kn processors. In particular, for constant accuracy, no lower bounds were known for these problems.<<ETX>>