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17
Computing with Very Weak Random Sources
, 1994
"... For any fixed 6> 0, we show how to simulate RP algorithms in time nO(‘Ogn) using the output of a 6source wath minentropy R‘. Such a weak random source is asked once for R bits; it outputs an Rbit string such that any string has probability at most 2Rc. If 6> 1 l/(k + l), our BPP simulatio ..."
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Cited by 76 (7 self)
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For any fixed 6> 0, we show how to simulate RP algorithms in time nO(‘Ogn) using the output of a 6source wath minentropy R‘. Such a weak random source is asked once for R bits; it outputs an Rbit string such that any string has probability at most 2Rc. 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 polynomialtime BPP simulation using ChorGoldreich sources of minentropy Ro(’), which is optimal. We present applications to timespace tradeoffs, expander constructions, and the hardness of approximation. Also of interest is our randomnessefficient Leflover Hash Lemma, found independently by Goldreich & Wigderson.
Approximation Algorithms for Finding Highly Connected Subgraphs
, 1996
"... Contents 1 Introduction 2 1.1 Outline of Chapter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 EdgeConnectivity Problems 3 2.1 Weighted EdgeConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.2 Unweighted EdgeConnectivity : : : : : ..."
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Cited by 63 (1 self)
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Contents 1 Introduction 2 1.1 Outline of Chapter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 EdgeConnectivity Problems 3 2.1 Weighted EdgeConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2.2 Unweighted EdgeConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2.1 2 EdgeConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2.2 EdgeConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 3 VertexConnectivity Problems 11 3.1 Weighted VertexConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 3.2 Unweighted VertexConnectivity : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 3.2.1 2 VertexConnectivity : : : : : : : : : : : : : : : : :
Improved Algorithms via Approximations of Probability Distributions
 Journal of Computer and System Sciences
, 1997
"... We present two techniques for approximating probability distributions. The first is a simple method for constructing the smallbias probability spaces introduced by Naor & Naor. We show how to efficiently combine this construction with the method of conditional probabilities to yield improved NC ..."
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Cited by 24 (2 self)
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We present two techniques for approximating probability distributions. The first is a simple method for constructing the smallbias 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 ...
Nontree routing for reliability and yield improvement
 In Proc. Int. Conf. on Computer Aided Design
, 2002
"... Abstract—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 ..."
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Cited by 13 (0 self)
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Abstract—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 postprocessing, treeaugmentation approach, which can be easily integrated in current physical design flows. Our contributions are as follows. 1) We formulate the problem as a variant of the classical twoedgeconnectivity augmentation problem in which we take into account such practical issues as wirelength increase budget, routing obstacles, and the use of Steiner points.2) 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. 3) 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. 4) We give a wellscaling greedy algorithm which has a practical runtime
Explicit ordispersers with polylogarithmic degree
 J. ACM
, 1998
"... An (N,M,T)ORdisperser is a bipartite multigraph G = (V,W,E) withV  = 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 ( ..."
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Cited by 13 (1 self)
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An (N,M,T)ORdisperser is a bipartite multigraph G = (V,W,E) withV  = 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)ORdisperser such that the outdegree of any vertex in V is at most polylogarithmic in N. Using this with known applications of ORdispersers yields several results. First, our construction implies that the complexity class StrongRP defined by Sipser, equals RP. Second, for any fixed η>0, we give the first polynomialtime simulation of RP algorithms using the output of any “ηminimally random ” source. For any integral R>0, such a source accepts a single request for an Rbit 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
Minimizing Randomness in Minimum Spanning Tree, Parallel Connectivity, and Set Maxima Algorithms
 In Proc. 13th Annual ACMSIAM Symposium on Discrete Algorithms (SODA'02
, 2001
"... 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 conne ..."
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Cited by 8 (4 self)
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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.
ON IMPROVING CONNECTIVITY OF STATIC ADHOC NETWORKS BY ADDING NODES ∗
"... 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 ..."
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Cited by 5 (0 self)
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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
Undirected VertexConnectivity Structure and Smallest FourVertexConnectivity Augmentation
 Proc. 6th ISAAC
, 1995
"... In this paper, we study properties for the structure of an undirected graph that is not 4vertexconnected. We also study the evolution of this structure when an edge is added to optimally increase the vertexconnectivity of the underlying graph. Several properties reported here can be extended t ..."
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Cited by 4 (0 self)
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In this paper, we study properties for the structure of an undirected graph that is not 4vertexconnected. We also study the evolution of this structure when an edge is added to optimally increase the vertexconnectivity of the underlying graph. Several properties reported here can be extended to the case of a graph that is not kvertex connected, for an arbitrary k. Using properties obtained here, we solve the problem of finding a smallest set of edges whose addition 4vertexconnects 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 4vertexconnects 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 3vertexconnecte...
A note on two source location problems
"... We consider Source Location (SL) problems: given a capacitated network G = (V,E), cost c(v) and a demand d(v) for every v ∈ V, choose a mincost S ⊆ V so that λ(v,S) ≥ d(v) holds for every v ∈ V, where λ(v,S) is the maximum flow value from v to S. In the directed variant, we have demands d in (v) an ..."
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Cited by 3 (1 self)
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We consider Source Location (SL) problems: given a capacitated network G = (V,E), cost c(v) and a demand d(v) for every v ∈ V, choose a mincost S ⊆ V so that λ(v,S) ≥ d(v) holds for every v ∈ V, where λ(v,S) is the maximum flow value from v to S. In the directed variant, we have demands d in (v) and d out (v) and we require λ(S,v) ≥ d in (v) and λ(v,S) ≥ dout (v). Undirected SL is (weakly) NPhard on stars with r(v) = 0 for all v except the center. But, it is known to be polynomially solvable for uniform costs and uniform demands. For general instances, both directed an undirected SL admit a (ln D+1)approximation algorithms, where D is the sum of the demands; up to constant this is tight, unless P=NP. We give a pseudopolynomial algorithm for undirected SL on trees with running time O(V  ∆ 3), where ∆ = maxv∈V d(v). This algorithm is used to derive a linear time algorithm for undirected SL with ∆ ≤ 3. We also consider the Single Assignment Source Location (SASL) where every v ∈ V should be assigned to a single node s(v) ∈ S. While the undirected SASL is in P, we give a (ln V  + 1)approximation algorithm for the directed case, and show that this is tight, unless P=NP. 1
Optimal BiLevel Augmentation for Selectively Enhancing Graph Connectivity with Applications
 in Proc. 2nd International Symp. on Computing and Combinatorics, vol. LNCS #1090
, 1996
"... 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 bilevel augmentation problem is that of a ..."
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Cited by 1 (1 self)
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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 bilevel augmentation problem is that of adding to G the smallest number of edges such that G contains two internally vertexdisjoint paths between every pair of vertices in H 1 and two edgedisjoint 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 bilevel 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...