Results 1  10
of
34
Are bitvectors optimal?
"... ... We show lower bounds that come close to our upper bounds (for a large range of n and ffl): Schemes that answer queries with just one bitprobe and error probability ffl must use \Omega ( nffl log(1=ffl) log m) bits of storage; if the error is restricted to queries not in S, then the scheme must u ..."
Abstract

Cited by 57 (7 self)
 Add to MetaCart
... We show lower bounds that come close to our upper bounds (for a large range of n and ffl): Schemes that answer queries with just one bitprobe and error probability ffl must use \Omega ( nffl log(1=ffl) log m) bits of storage; if the error is restricted to queries not in S, then the scheme must use \Omega ( n2ffl2 log(n=ffl) log m) bits of storage. We also
On The Upper Bound Of The Size Of The rCoverFree Families
, 1994
"... Let T (r; n) denote the maximum number of subsets of an nset satisfying the condition in the title. It is proved in a purely combinatorial way, that for n sufficiently large log 2 T (r; n) n 8 \Delta log 2 r r 2 holds. 1. Introduction The notion of the rcoverfree families was introduced by ..."
Abstract

Cited by 44 (2 self)
 Add to MetaCart
Let T (r; n) denote the maximum number of subsets of an nset satisfying the condition in the title. It is proved in a purely combinatorial way, that for n sufficiently large log 2 T (r; n) n 8 \Delta log 2 r r 2 holds. 1. Introduction The notion of the rcoverfree families was introduced by Kautz and Singleton in 1964 [17]. They initiated investigating binary codes with the property that the disjunction of any r (r 2) codewords are distinct (UD r codes). This led them to studying the binary codes with the property that none of the codewords is covered by the disjunction of r others (Superimposed codes, ZFD r codes; P. Erdos, P. Frankl and Z. Furedi called the correspondig set system rcoverfree in [7]). Since that many results have been proved about the maximum size of these codes. Various authors studied these problems basically from three different points of view, and these three lines of investigations were almost independent of each other. This is why many results were ...
An O(k 3 log n)Approximation Algorithm for VertexConnectivity Survivable Network Design
, 2008
"... In the Survivable Network Design problem (SNDP), we are given an undirected graph G(V, E) with costs on edges, along with a connectivity requirement r(u, v) for each pair u, v of vertices. The goal is to find a minimumcost subset E ∗ of edges, that satisfies the given set of pairwise connectivity r ..."
Abstract

Cited by 24 (0 self)
 Add to MetaCart
In the Survivable Network Design problem (SNDP), we are given an undirected graph G(V, E) with costs on edges, along with a connectivity requirement r(u, v) for each pair u, v of vertices. The goal is to find a minimumcost subset E ∗ of edges, that satisfies the given set of pairwise connectivity requirements. In the edgeconnectivity version we need to ensure that there are r(u, v) edgedisjoint paths for every pair u, v of vertices, while in the vertexconnectivity version the paths are required to be vertexdisjoint. The edgeconnectivity version of SNDP is known to have a 2approximation. However, no nontrivial approximation algorithm has been known so far for the vertex version of SNDP, except for special cases of the problem. We present an extremely simple algorithm to achieve an O(k 3 log n)approximation for this problem, where k denotes the maximum connectivity requirement, and n denotes the number of vertices. We also give a simple proof of the recently discovered O(k 2 log n)approximation result for the singlesource version of vertexconnectivity SNDP. We note that in both cases, our analysis in fact yields slightly better guarantees in that the log n term in the approximation guarantee can be replaced with a log τ term where τ denotes the number of distinct vertices that participate in one or more pairs with a positive connectivity requirement.
New Bounds for the Language Compression Problem
, 2000
"... The CD complexity of a string x is the length of the shortest polynomial time program which accepts only the string x. The language compression problem consists of giving an upper bound on the CD A n complexity of all strings x in some set A. The best known upper bound for this problem is 2 log(j ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
The CD complexity of a string x is the length of the shortest polynomial time program which accepts only the string x. The language compression problem consists of giving an upper bound on the CD A n complexity of all strings x in some set A. The best known upper bound for this problem is 2 log(jjA n jj) + O(log(n)), due to Buhrman and Fortnow. We show that the constant factor 2 in this bound is optimal. We also give new bounds for a certain kind of random sets R ` f0; 1g n , for which we show an upper bound of log(jjR n jj) + O(log(n)). 1 Introduction Kolmogorov complexity is a notion that measures the amount of regularity in a finite string. It has turned out to be a very useful tool in theoretical computer science. A simple counting argument showing that for each length there exist random strings, i.e. strings with no regularity, has had many applications (see [LV97]). Early in the history of computational complexity resource bounded notions of Kolmogorov complexity were...
The quantum complexity of set membership
 In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... We study the quantum complexity of the static set membership problem: given a subset S (S  ≤ n) of a universe of size m ( ≫ n), store it as a table, T: {0,1} r → {0,1}, of bits so that queries of the form ‘Is x in S? ’ can be answered. The goal is to use a small table and yet answer queries using ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
We study the quantum complexity of the static set membership problem: given a subset S (S  ≤ n) of a universe of size m ( ≫ n), store it as a table, T: {0,1} r → {0,1}, of bits so that queries of the form ‘Is x in S? ’ can be answered. The goal is to use a small table and yet answer queries using few bit probes. This problem was considered recently by Buhrman, Miltersen, Radhakrishnan and Venkatesh [BMRV00], who showed lower and upper bounds for this problem in the classical deterministic and randomised models. In this paper, we formulate this problem in the “quantum bit probe model”. We assume that access to the table T is provided by means of a black box (oracle) unitary transform OT that takes the basis state y,b 〉 to the basis state y,b⊕T(y)〉. The query algorithm is allowed to apply OT on any superposition of basis states. We show tradeoff results between space (defined as 2 r) and number of probes (oracle calls) in this model. Our results show that the lower bounds shown in [BMRV00] for the classical model also hold (with minor differences) in the quantum bit probe model. These bounds almost match the classical upper bounds. Our lower bounds are proved using linear algebraic arguments.
NonAdaptive Fault Diagnosis for AllOptical Networks via Combinatorial Group Testing on Graphs
"... Abstract—We consider the fault diagnosis problem in alloptical networks, focusing on probing schemes to detect faults. Our work concentrates on nonadaptive probing schemes, in order to meet the stringent time requirements for fault recovery. This fault diagnosis problem motivates a new technical fr ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
Abstract—We consider the fault diagnosis problem in alloptical networks, focusing on probing schemes to detect faults. Our work concentrates on nonadaptive probing schemes, in order to meet the stringent time requirements for fault recovery. This fault diagnosis problem motivates a new technical framework that we introduce: group testing with graphbased constraints. Using this framework, we develop several new probing schemes to detect network faults. The efficiency of our schemes often depends on the network topology; in many cases we can show that our schemes are nearoptimal by providing tight lower bounds. I.
Group Testing with Probabilistic Tests: Theory, Design and Application
"... Identification of defective members of large populations has been widely studied in the statistics community under the name of group testing. It involves grouping subsets of items into different pools and detecting defective members based on the set of test results obtained for each pool. In a class ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
Identification of defective members of large populations has been widely studied in the statistics community under the name of group testing. It involves grouping subsets of items into different pools and detecting defective members based on the set of test results obtained for each pool. In a classical noiseless group testing setup, it is assumed that the sampling procedure is fully known to the reconstruction algorithm, in the sense that the existence of a defective member in a pool results in the test outcome of that pool to be positive. However, this may not be always a valid assumption in some cases of interest. In particular, we consider the case where the defective items in a pool can become independently inactive with a certain probability. Hence, one may obtain a negative test result in a pool despite containing some defective items. As a result, any sampling and reconstruction method should be able to cope with two different types of uncertainty, i.e., the unknown set of defective items and the partially unknown, probabilistic testing procedure. In this work, motivated by the application of detecting infected people in viral epidemics, we design nonadaptive sampling procedures that allow successful identification of the defective items through a set of probabilistic tests. Our design requires only a small number of tests to single out the defective items.
On Greedy Algorithms in Coding Theory
, 1996
"... We study a wide class of problems in coding theory for which we consider two different formulations: in terms of incidence matrices and in terms of hypergraphs. These problems are dealt with using a greedy algorithm due to Stein and Lov'asz. Some examples, including constructing covering codes, code ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
We study a wide class of problems in coding theory for which we consider two different formulations: in terms of incidence matrices and in terms of hypergraphs. These problems are dealt with using a greedy algorithm due to Stein and Lov'asz. Some examples, including constructing covering codes, codes for conflict resolution, separating systems, source encoding with distortion, etc., are given a unified treatment. Under certain conditions derandomization can be performed, leading to an essential reduction in the complexity of the constructions.
CoverFree Families and Superimposed Codes: Constructions, Bounds, and Applications to Cryptography and Group Testing
"... This paper deals with (s; `)coverfree families or superimposed (s; `)codes. They generalize the concept of superimposed scodes and have several applications for cryptography and group testing. We present a new asymptotic bound on the rate of optimal codes and develop some constructions. ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
This paper deals with (s; `)coverfree families or superimposed (s; `)codes. They generalize the concept of superimposed scodes and have several applications for cryptography and group testing. We present a new asymptotic bound on the rate of optimal codes and develop some constructions.
Almost optimal explicit selectors
 FCT 2005. LNCS
, 2005
"... We understand selection by intersection as distinguishing a single element of a set by the uniqueness of its occurrence in some other set. More precisely, given two sets A and B, if A ∩ B = {z}, then element z ∈ A is selected by set B. Selectors are such families S of sets B of some domain that allo ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We understand selection by intersection as distinguishing a single element of a set by the uniqueness of its occurrence in some other set. More precisely, given two sets A and B, if A ∩ B = {z}, then element z ∈ A is selected by set B. Selectors are such families S of sets B of some domain that allow to select many elements from sufficiently small subsets A of the domain. Selectors are used in communication protocols for the multipleaccess channel, in implementations of distributedcomputing primitives in radio networks, and in algorithms for group testing. We give new explicit (n, k, r)selectors of size O(min [ k n, 2 k−r+1 polylog n]), for any parameters r ≤ k ≤ n. We establish a lower bound Ω(min [ k n, 2 k−r+1 · log(n/k)) on the length log(k/(k−r+1)) of (n, k, r)selectors, which demonstrates that our construction is within a polylog n factor close to optimal. The new selectors are applied to develop explicit implementations of selection resolution on the multipleaccess channel, gossiping in radio networks and an algorithm for group testing with inhibitors.