Prasoon Tiwari

A deterministic algorithm for sparse multivariate polynomial interpolation

An efficient deterministic polynomial time algorithm is developed for the sparse polynomial interpolation problem. The number of evaluations needed by this algorithm is very small. The algorithm also has a simple NC implementation.

The electrical resistance of a graph captures its commute and cover times

View ann-vertex,m-edge undirected graph as an electrical network with unit resistors as edges. We extend known relations between random walks and electrical networks by showing that resistance in this network is intimately connected with thelengths of random walks on the graph. For example, thecommute time between two verticess andt (the expected length of a random walk froms tot and back) is precisely characterized by the effective resistanceRst betweens andt: commute time=2mRst. As a corollary, thecover time (the expected length of a random walk visiting all vertices) is characterized by the maximum resistanceR in the graph to within a factor of logn:mR<-cover time<-O(mRlogn). For many graphs, the bounds on cover time obtained in this manner are better than those obtained from previous techniques such as the eigenvalues of the adjacency matrix. In particular, we improve known bounds on cover times for high-degree graphs and expanders, and give new proofs of known results for multi-dimensional meshes. Moreover, resistance seems to provide an intuitively appealing and tractable approach to these problems.

The computational complexity of universal hashing

Mansour, Y., N. Nisan and P. Tiwari, The computational complexity of universal hashing, Theoretical Computer Science 107 (1993) 121-133. Any implementation of Carter-Wegman universal hashing from n-bit strings to m-bit strings requires a time-space tradeoff of TS=n(nm). The bound holds in the general boolean branching program model and, thus, in essentially any model of computation. As a corollary, computing a+ b * c in any field F requires a quadratic time-space tradeoff, and the bound holds for any representation of the elements of the field. Other lower bounds on the complexity of any implementation of universal hashing are given as well: quadratic AT’ bound for VLSI implementation; R(logn) parallel time bound on a CREW PRAM; and exponential size for constant-depth circuits.

Scheduling parallelizable tasks to minimize average response time

A <italic>parallelizable</italic> (or <italic>malleable</italic>) task is one which can be run on an arbitrary number of processors, with a task execution time that depends on the number of processors allotted to it. Consider a system of <italic>M</italic> independent parallelizable tasks which are to be scheduled without preemption on a parallel computer consisting of <italic>P</italic> identical processors. For each task, the execution time is a known function of the number of processors allotted to it. The goal is to find (1) for each task <italic>i</italic>, an allotment of processors β, and (2) overall, a non-preemptive schedule assigning the tasks to the processors which minimizes the average response time of the tasks. Equivalently, we can minimize the <italic>flow time</italic> which is the sum of the completion times of each of the tasks. In this paper we tackle the problem of finding a schedule with minimum average response time in the special case where each task in the system has sublinear speedup. This natural restriction on the task execution time means simply that the efficiency of a task decrease or remains constant as the number of processors allotted to it increases. The scheduling problem with sublinear speedups has been shown to be <inline-equation> <f> <ty><sc>NP</sc></ty></f> </inline-equation>-complete in the strong sense. We therefore focus on finding a polynomial time algorithm whose solution comes within a fixed multiplicative constant of optimal. In particular, we given algorithm which finds a schedule having a response time that is within 2 times that of the optimal schedule and which runs in O(M(M<supscrpt>2</supscrpt> + P)) time.

Lower bounds on communication complexity in distributed computer networks

The main result of this paper is a general technique for determining lower bounds on the communication complexity of problems on various distributed computer networks. This general technique is derived by simulating the general network by a linear array and then using a lower bound on the communication complexity of the problem on the linear array. Applications of this technique yield optimal bounds on the communication complexity of merging, ranking, uniqueness, and triangle-detection problems on a ring of processors. Nontrivial near-optimal lower bounds on the communication complexity of distinctness, merging, and ranking on meshes and complete binary trees are also derived.

Simple algorithms for approximating all roots of a polynomial with real roots

Abstract We present a simple algorithm for approximating all roots of a polynomial p(x) when it has only real roots. The algorithm is based on some interesting properties of the polynomials appearing in the Extended Euclidean Scheme for p(x) and p′(x). For example, it turns out that these polynomials are orthogonal; as a consequence, we are able to limit the precision required by our algorithm in intermediate steps. A parallel implementation of this algorithm yields a P-uniform NC2 circuit, and the bit complexity of its sequential implementation is within a polylog factor of the bit complexity of the best known algorithm for the problem.

A fast parallel algorithm for determining all roots of a polynomial with real roots

Given a polynomial

Lower bounds for computations with the floor operation

p(z)

A parallel MPEG-2 video encoder with look-ahead rate control

of degree n with m bit integer coefficients and an integer

A lower bound for integer greatest common divisor computations

\mu