For any fixed 6> 0, we show how to simulate RP algorithms in time nO(‘Ogn) using the output of a 6-source wath min-entropy R‘. Such a weak random source is asked once for R bits; it outputs an R-bit string such that any string has probability at most 2-Rc. If 6> 1- l/(k + l), our BPP simulations take time no(‘og(k)n) (log(k) is the logarithm iterated k times). We also gave a polynomial-time BPP simulation using Chor-Goldreich sources of min-entropy Ro(’), which is optimal. We present applications to time-space tradeoffs, expander constructions, and the hardness of approximation. Also of interest is our randomness-efficient Leflover Hash Lemma, found independently by Goldreich & Wigderson.

Contents 1 Introduction 2 1.1 Outline of Chapter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 Edge-Connectivity Problems 3 2.1 Weighted Edge-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.2 Unweighted Edge-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2.1 2 Edge-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2.2 Edge-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 3 Vertex-Connectivity Problems 11 3.1 Weighted Vertex-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 3.2 Unweighted Vertex-Connectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 3.2.1 2 Vertex-Connectivity : : : : : : : : : : : : : : : : :

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 ...

An (N,M,T)-OR-disperser is a bipartite multigraph G = (V,W,E) with|V | = N, and |W | = M, having the following expansion property: any subset of V having at least T vertices has a neighbor set of size at least M/2. For any pair of constants ξ,λ,1 ≥ ξ>λ ≥ 0, any sufficiently large N, andforany (log N)ξ (log N)λ T ≥ 2, M ≤ 2, we give an explicit elementary construction of an (N,M,T)-OR-disperser such that the out-degree of any vertex in V is at most polylogarithmic in N. Using this with known applications of OR-dispersers yields several results. First, our construction implies that the complexity class Strong-RP defined by Sipser, equals RP. Second, for any fixed η>0, we give the first polynomial-time simulation of RP algorithms using the output of any “η-minimally random ” source. For any integral R>0, such a source accepts a single request for an R-bit string and generates the string according to a distribution that assigns probability at most 2−Rη to any string. It is minimally random in the sense that any weaker source is

We propose to introduce redundant interconnects for manufacturing yield and reliability improvement. By introducing redundant interconnects, the potential for open faults is reduced at the cost of increased potential for short faults; overall, manufacturing yield and fault tolerance can be improved. We focus on a post-processing, tree augmentation approach which can be easily integrated in current physical design flows. Our contributions are as follows: We formulate the problem as a variant of the classical 2-edge-connectivity augmentation problem in which we take into account such practical issues as wirelength increase budget, routing obstacles, and use of Steiner points. We show that an optimum solution can always be found on the Hanan grid defined by the terminals and the corners of the feasible routing region. We give a compact integer program formulation which is solved in practical runtime by the commercial optimization package CPLEX for nets with up to 100 terminals. We give a well-scaling greedy algorithm which has practical runtime up to 1,000 terminals, and comes on the average within 1-2 % of the optimum computed by CPLEX. We give a comprehensive experimental study comparing the solution quality and runtime of our methods with the best methods reported in the literature for the related 2-edge-connectivity augmentation problem, including a sophisticated heuristic based on minimum-weight branchings [11] and a recent genetic algorithm [17]. Experiments on randomly generated and industry testcases show that our greedy augmentation method achieves significant increase in reliability (as measured by the percentage of biconnected tree edges) with very small increase in wirelength. For example, on 1,000 terminal nets the average percentage of biconnected tree edges is 34 ¡ 19 % for a wirelength increase of only 1%, and 87 ¡ 73 % for a wirelength increase of 20%. SPICE simulations on industry routed nets show that non-tree routing has the additional benefit of reducing maximum sink delay by an average of 28 ¡ 26 % compared to Steiner routing, and by an average of 3 ¡ 72 % compared to timing optimized routing. SPICE simulations further imply that non-tree routing has smaller delay variation due to process variability. I.

There are several fundamental problems whose deterministic complexity remains unresolved, but for which there exist randomized algorithms whose complexity is equal to known lower bounds. Among such problems are the minimum spanning tree problem, the set maxima problem, the problem of computing connected components and (minimum) spanning trees in parallel, and the problem of performing sensitivity analysis on shortest path trees and minimum spanning trees. However, while each of these problems has a randomized algorithm whose performance meets a known lower bound, all of these randomized algorithms use a number of random bits which is linear in the number of operations they perform. We address the issue of reducing the number of random bits used in these randomized algorithms. For each of the problems listed above, we present randomized algorithms that have optimal performance but use only a polylogarithmic number of random bits; for some of the problems our optimal algorithms use only log n random bits. Our results represent an exponential savings in the amount of randomness used to achieve the same optimal performance as in the earlier algorithms. Our techniques are general and could likely be applied to other problems.

In this paper, we study properties for the structure of an undirected graph that is not 4-vertex-connected. We also study the evolution of this structure when an edge is added to optimally increase the vertex-connectivity of the underlying graph. Several properties reported here can be extended to the case of a graph that is not k-vertex- connected, for an arbitrary k. Using properties obtained here, we solve the problem of finding a smallest set of edges whose addition 4-vertex-connects an undirected graph. This is a fundamental problem in graph theory and has applications in network reliability and in statistical data security. We give an O(n \Delta log n + m)-time algorithm for finding a set of edges with the smallest cardinality whose addition 4-vertex-connects an undirected graph, where n and m are the number of vertices and edges in the input graph, respectively. This is the first polynomial time algorithm for this problem when the input graph is not 3-vertex-connecte...

Our main problem is abstracted from several optimization problems for protecting information in cross tabulated tables and for improving the reliability of communication networks. Given an undirected graph G and two vertex subsets H 1 and H 2 , the smallest bi-level augmentation problem is that of adding to G the smallest number of edges such that G contains two internally vertex-disjoint paths between every pair of vertices in H 1 and two edge-disjoint paths between every pair of vertices in H 2 . We give a data structure to represent essential connectivity information of H 1 and H 2 simultaneously. Using this data structure, we solve the bi-level augmentation problem in O(n + m) time, where n and m are the numbers of vertices and edges in G. Our algorithm can be parallelized to run in O(log 2 n) time using n +m processors on an EREW PRAM. By properly setting G, H 1 and H 2 , our augmentation algorithm also subsumes several existing optimal algorithms for graph augmentation. 1 Int...

Ad hoc networks are by nature constructed “automatically”, by the nodes adapting to the neighboring nodes and building up a network. In this context, the network topology is random, and in particular, no connectivity is guaranteed: the nodes may be so sparsely located that they are unable to make up a

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