We show that any randomized algorithm that runs in space S and time T and uses poly(S) random bits can be simulated using only O(S) random bits in space S and time T poly(S). A deterministic simulation in space S follows. Of independent interest is our main technical tool: a procedure which extracts randomness from a defective random source using a small additional number of truly random bits. 1

) Baruch Awerbuch Department of Mathematics and Laboratory for Computer Science M.I.T. Cambridge, MA 02139 Andrew V. Goldberg y Department of Computer Science Stanford University Stanford, CA 94305 Michael Luby z International Computer Science Institute Berkeley, CA 94704 Serge A. Plotkin x Department of Computer Science Stanford University Stanford, CA 94305 May 1989 Abstract We introduce a concept of network decomposition, the essence of which is to partition an arbitrary graph into small-diameter connected components, such that the graph created by contracting each component into a single node has low chromatic number. We present an efficient distributed algorithm for constructing such a decomposition, and demonstrate its use for design of efficient distributed algorithms. Our method yields new deterministic distributed algorithms for finding a maximal independent set and for (\Delta + 1)-coloring of graphs with maximum degree \Delta. These algorithms run...

Small sample spaces with almost independent random variables are applied to design efficient sequential deterministic algorithms for two problems. The first algorithm, motivated by the attempt to design efficient algorithms for the All Pairs Shortest Path problem using fast matrix multiplication, solves the problem of computing witnesses for the Boolean product of two matrices. That is, if A and B are two n by n matrices, and C = AB is their Boolean product, the algorithm finds for every entry Cij = 1 a witness: an index k so that Aik = Bkj = 1. Its running time exceeds that of computing the product of two n by n matrices with small integer entries by a polylogarithmic factor. The second algorithm is a nearly linear time deterministic procedure for constructing a perfect hash function for a given n-subset of {1,..., m}.

We present a technique for converting RNC algorithms into NC algorithms. Our approach is based on a parallel implementation of the method of conditional probabilities. This method was used to convert probabilistic proofs of existence of combinatorial structures into polynomial time deterministic algorithms. It has the apparent drawback of being extremely sequential in nature. We show certain general conditions under which it is possible to use this technique for devising deterministic parallel algorithms. We use our technique to devise an NC algorithm for the set balancing problem. This problem turns out to be a useful tool for parallel algorithms. Using our de-randomization method and the set balancing algorithm, we provide an NC algorithm for the lattice approximation problem. We also use the lattice approximation problem to bootstrap the set balancing algorithm, and the result is a more processor efficient algorithm. The set balancing algorithm also yields an NC algorithm for near-optimal edge coloring of simple graphs. Our methods also extend to the parallelization of various algorithms in computational geometry that rely upon the random sampling technique of Clarkson. Finally, our methods apply to constructing certain combinatorial structures, e.g. ...

. Certain types of routing, scheduling, and resource-allocation problems in a distributed setting can be modeled as edge-coloring problems. We present fast and simple randomized algorithms for edge coloring a graph in the synchronous distributed point-to-point model of computation. Our algorithms compute an edge coloring of a graph G with n nodes and maximum degree # with at most 1.6# +O(log 1+# n) colors with high probability (arbitrarily close to 1) for any fixed #>0; they run in polylogarithmic time. The upper bound on the number of colors improves upon the (2# - 1)-coloring achievable by a simple reduction to vertex coloring. To analyze the performance of our algorithms, we introduce new techniques for proving upper bounds on the tail probabilities of certain random variables. The Cherno#--Hoe#ding bounds are fundamental tools that are used very frequently in estimating tail probabilities. However, they assume stochastic independence among certain random variables, which may n...

We give fast randomized and deterministic parallel meth-ods for constructing convex hulls in IR d, for any fixed d. Our methods are for the weakest shared-memory model,the EREW PRAM, and have optimal work bounds (with high probability for the randomized methods). In partic-ular, we show that the convex hull of n points in IRd canbe constructed in O(log n) time using O(n log n + nbd=2c)work, with high probability. We also show that it can be constructed deterministically in O(log2 n) time using O(n log n) work for d = 3 and in O(log n) time using O(nbd=2c logc(dd=2e\Gamma bd=2c) n) work, for d * 4, where c? 0is a constant, which is optimal for even d * 4. We also showhow to make our 3-dimensional methods output-sensitive with only a small increase in running time.These methods can be applied to other problems as well. A variation of the convex hull algorithm for even dimen-sions deterministically constructs a (1=r)-cutting of n hy-perplanes in IR d in O(log n) time using optimal O(nrd\Gamma 1) work; when r = n, we obtain their arrangement and a pointlocation data structure for it. With appropriate modifications, our deterministic 3-dimensional convex hull algorithmcan be used to compute, in the same resource bounds, the intersection of n balls of equal radius in R³. This leads to asequential algorithm for computing the diameter of a point set in IR3 with running time O(n log³ n), which is arguablysimpler than an algorithm with the same running time by Brönnimann et al.

We present a fairly general method for finding deterministic constructions obeying what we call k- restrictions; this yields structures of size not much larger than the probabilistic bound. The structures constructed by our method include (n; k)-universal sets (a collection of binary vectors of length n such that for any subset of size k of the indices, all 2 configurations appear) and families of perfect hash functions. The near-optimal constructions of these objects imply the very efficient derandomization of algorithms in learning, of fixed-subgraph finding algorithms, and of near optimal \Sigma\Pi\Sigma threshold formulae. In addition, they derandomize the reduction showing the hardness of approximation of set cover. They also yield deterministic constructions for a local-coloring protocol, and for exhaustive testing of circuits.

This paper presents a distributed implementation of RAND, a randomized time slot scheduling algorithm, called DRAND. DRAND runs in O(δ) time and message complexity where δ is the maximum size of a two-hop neighborhood in a wireless network while message complexity remains O(δ), assuming that message delays can be bounded by an unknown constant. DRAND is the first fully distributed version of RAND. The algorithm is suitable for a wireless network where most nodes do not move, such as wireless mesh networks and wireless sensor networks. We implement the algorithm in TinyOS and demonstrate its performance in a real testbed of Mica2 nodes. The algorithm does not require any time synchronization and is shown to be effective in adapting to local topology changes without incurring global overhead in the scheduling. Because of these features, it can also be used even for other scheduling problems such as frequency or code scheduling (for FDMA or CDMA) or local identifier assignment for wireless networks where time synchronization is not enforced.

We present two techniques for approximating probability distributions. The first is a simple method for constructing the small-bias probability spaces introduced by Naor & Naor. We show how to efficiently combine this construction with the method of conditional probabilities to yield improved NC algorithms for many problems such as set discrepancy, finding large cuts in graphs, finding large acyclic subgraphs etc. The second is a construction of small probability spaces approximating general independent distributions, which is of smaller size than the constructions of Even, Goldreich, Luby, Nisan & Velickovi'c. Such approximations are useful, e.g., for the derandomization of certain randomized algorithms. Keywords. Derandomization, parallel algorithms, discrepancy, graph coloring, small sample spaces, explicit constructions. 1 Introduction Derandomization, the development of general tools to derive efficient deterministic algorithms from their randomized counterparts, has blossomed ...

f a The problem of constructing in parallel a maximal independent set o given graph is considered. A new deterministic NC -algorithm imple- - t mented in the EREW PRAM model is presented. On graphs with n ver ices and m edges, it uses O ((n +m )/logn ) processors and runs in O (log n ) 3 - c time. This reduces by a factor of logn both the running time and the pro essor count of the previously fastest deterministic algorithm which solves the problem using a linear number of processors. Key words: parallel computation, NC, graph, maximal independent set, 1 deterministic. . Introduction The problem of constructing in parallel a maximal independent set of a given graph, t MIS , has been investigated in several recent papers. Karp and Wigderson proved in [KW] hat the problem is in NC . Their algorithm finds a maximal independent set of an n - vertex graph in O (log n ) time and uses O (n /log n ) processors. In successive papers, the 4 3 3 - s authors proposed algorithms which either...