{"title":"On the Composition of Public-Coin Zero-Knowledge Protocols","authors":"R. Pass, Wei-Lung Dustin Tseng, Douglas Wikström","doi":"10.1137/100811465","DOIUrl":"https://doi.org/10.1137/100811465","url":null,"abstract":"We show that only languages in BPP have public-coin, black-box zero-knowledge protocols that are secure under an unbounded (polynomial) number of parallel repetitions. This result holds both in the plain model (without any set-up) and in the Bare Public-Key Model (where the prover and the verifier have registered public keys). We complement this result by showing the existence of a public-coin black-box zero-knowledge proof that remains secure under any a-priori bounded number of concurrent executions.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134178341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Better Algorithm for Random k-SAT","authors":"A. Coja-Oghlan","doi":"10.1137/09076516X","DOIUrl":"https://doi.org/10.1137/09076516X","url":null,"abstract":"Let *** be a uniformly distributed random k -SAT formula with n variables and m clauses. We present a polynomial time algorithm that finds a satisfying assignment of *** with high probability for constraint densities $m/n , where *** k ***0. Previously no efficient algorithm was known to find solutions with non-vanishing probability beyond m /n = 1.817·2 k /k [Frieze and Suen, Journal of Algorithms 1996].","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"103 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115506194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Line Transversals of Convex Polyhedra in R3","authors":"Haim Kaplan, Natan Rubin, M. Sharir","doi":"10.1137/080744694","DOIUrl":"https://doi.org/10.1137/080744694","url":null,"abstract":"We establish a bound of O(n2k1+e), for any e > 0, on the combinatorial complexity of the set T of line transversals of a collection P of k convex polyhedra in R3 with a total of n facets, and present a randomized algorithm which computes the boundary of T in comparable expected time. Thus, when k L n, the new bounds on the complexity (and construction cost) of T improve upon the previously best known bounds, which are nearly cubic in n. To obtain the above result, we study the set Tl0 of line transversals which emanate from a fixed line l0, establish an almost tight bound of O(nk1+e) on the complexity of Tl0, and provide a randomized algorithm which computes Tl0 in comparable expected time. Slightly improved combinatorial bounds for the complexity of Tl0, and comparable improvements in the cost of constructing this set, are established for two special cases, both assuming that the polyhedra of P are pairwise disjoint: the case where l0 is disjoint from the polyhedra of P, and the case where the polyhedra of P are unbounded in a direction parallel to l0.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124866946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Bit-Complexity of Lempel-Ziv Compression","authors":"P. Ferragina, Igor Nitto, Rossano Venturini","doi":"10.1137/120869511","DOIUrl":"https://doi.org/10.1137/120869511","url":null,"abstract":"One of the most famous and investigated lossless data-compression schemes is the one introduced by Lempel and Ziv about 30 years ago [37]. This compression scheme is known as \"dictionary-based compressor\" and consists of squeezing an input string by replacing some of its substrings with (shorter) codewords which are actually pointers to a dictionary of phrases built as the string is processed. Surprisingly enough, although many fundamental results are nowadays known about the speed and effectiveness of this compression process (see e.g. [23, 28] and references therein), \"we are not aware of any parsing scheme that achieves optimality when the LZ77-dictionary is in use under any constraint on the codewords other than being of equal length\" [28, pag. 159]. Here optimality means to achieve the minimum number of bits in compressing each individual input string, without any assumption on its generating source. In this paper we investigate three issues pertaining to the bit-complexity of LZ-based compressors, and we design algorithms which achieve bit-optimality in the compressed output size by taking efficient/optimal time and optimal space. These theoretical results will be sustained by some experiments that will compare our novel LZ-based compressors against the most popular compression tools (like gzip, bzip2) and state-of-the-art compressors (like the booster of [14, 13]).","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116982062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Preprocessing Imprecise Points and Splitting Triangulations","authors":"M. V. Kreveld, M. Löffler, Joseph S. B. Mitchell","doi":"10.1137/090753620","DOIUrl":"https://doi.org/10.1137/090753620","url":null,"abstract":"Traditional algorithms in computational geometry assume that the input points are given precisely. In practice, data is usually imprecise, but information about the imprecision is often available. In this context, we investigate what the value of this information is. We show here how to preprocess a set of disjoint regions in the plane of total complexity $n$ in $O(nlog n)$ time so that if one point per set is specified with precise coordinates, a triangulation of the points can be computed in linear time. In our solution, we solve another problem which we believe to be of independent interest. Given a triangulation with red and blue vertices, we show how to compute a triangulation of only the blue vertices in linear time.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124956170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, David P. Woodruff
{"title":"Transitive-Closure Spanners","authors":"Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, David P. Woodruff","doi":"10.1137/110826655","DOIUrl":"https://doi.org/10.1137/110826655","url":null,"abstract":"Given a directed graph $G = (V,E)$ and an integer $k geq 1$, a $k$-transitive-closure-spanner ($k$-TC-spanner) of $G$ is a directed graph $H = (V, E_H)$ that has (1) the same transitive-closure as $G$ and (2) diameter at most $k$. These spanners were implicitly studied in the context of circuit complexity, data structures, property testing, and access control, and properties of these spanners have been rediscovered over the span of 20 years. We abstract the common task implicitly tackled in these diverse applications as the problem of constructing sparse TC-spanners. We initiate the study of approximability of the size of the sparsest $k$-TC-spanner of a given directed graph. We completely resolve the approximability of $2$-TC-spanners, showing that it is $Theta(log n)$ unless $textsf{P} = textsf{NP}$. For $k>2$, we present a polynomial time algorithm that finds a $k$-TC-spanner with size within $O((n log n)^{1-1/k})$ of the optimum. Our techniques also yield algorithms with the first nontrivial app...","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129957552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An Expansion Tester for Bounded Degree Graphs","authors":"Satyen Kale, Santa Clara, C. Seshadhri","doi":"10.1137/100802980","DOIUrl":"https://doi.org/10.1137/100802980","url":null,"abstract":"We consider the problem of testing graph expansion (eithervertex or edge) in the bounded degree model [10]. We give aproperty tester that given a graph with degree bound d, anexpansion bound α, and a parametere> 0, accepts the graph with high probabilityif its expansion is more than α, and rejects it withhigh probability if it is e-far from any graphwith expansion α' with degree bound d,where α' 0.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121188663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
L. A. Goldberg, Martin Grohe, M. Jerrum, M. Thurley
{"title":"A Complexity Dichotomy for Partition Functions with Mixed Signs","authors":"L. A. Goldberg, Martin Grohe, M. Jerrum, M. Thurley","doi":"10.1137/090757496","DOIUrl":"https://doi.org/10.1137/090757496","url":null,"abstract":"emph{Partition functions}, also known as emph{homomorphism functions}, form a rich family of graph invariants that contain combinatorial invariants such as the number of $k$-colourings or the number of independent sets of a graph and also the partition functions of certain ``spin glass'' models of statistical physics such as the Ising model. \u0000 \u0000Building on earlier work by Dyer and Greenhill (2000) and Bulatov and Grohe (2005), we completely classify the computational complexity of partition functions. Our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or #P-complete. Partition functions are described by symmetric matrices with real entries, and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or #P-complete. \u0000 \u0000While in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases, for partition functions described by a Hadamard matrices --- these turn out to be central in our proofs --- we obtain a simple algebraic tractability criterion, which says that the tractable cases are those ``representable'' by a quadratic polynomial over the field $ensuremath{mathbb{F}_2}$.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128905633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
C. Daskalakis, R. Karp, Elchanan Mossel, Samantha J. Riesenfeld, Elad Verbin
{"title":"Sorting and Selection in Posets","authors":"C. Daskalakis, R. Karp, Elchanan Mossel, Samantha J. Riesenfeld, Elad Verbin","doi":"10.1137/070697720","DOIUrl":"https://doi.org/10.1137/070697720","url":null,"abstract":"Classical problems of sorting and searching assume an underlying linear ordering of the objects being compared. In this paper, we study these problems in the context of partially ordered sets, in which some pairs of objects are incomparable. This generalization is interesting from a combinatorial perspective, and it has immediate applications in ranking scenarios where there is no underlying linear ordering, e.g., conference submissions. It also has applications in reconstructing certain types of networks, including biological networks. Our results represent significant progress over previous results from two decades ago by Faigle and Turan. In particular, we present the first algorithm that sorts a width-$w$ poset of size $n$ with query complexity $O(n(w+log n))$ and prove that this query complexity is asymptotically optimal. We also describe a variant of Mergesort with query complexity $O(wnlogfrac{n}{w})$ and total complexity $O(w^{2}nlogfrac{n}{w})$; an algorithm with the same query complexity was given by Faigle and Turan, but no efficient implementation of that algorithm is known. Both our sorting algorithms can be applied with negligible overhead to the more general problem of reconstructing transitive relations. We also consider two related problems: finding the minimal elements, and its generalization to finding the bottom $k$ “levels,” called the $k$-selection problem. We give efficient deterministic and randomized algorithms for finding the minimal elements with query complexity and total complexity $O(wn)$. We provide matching lower bounds for the query complexity up to a factor of 2 and generalize the results to the $k$-selection problem. Finally, we present efficient algorithms for computing a linear extension of a poset and computing the heights of all elements.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131993173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hierarchical Unambiguity","authors":"Holger Spakowski, Rahul Tripathi","doi":"10.1137/07068196X","DOIUrl":"https://doi.org/10.1137/07068196X","url":null,"abstract":"We develop techniques to investigate relativized hierarchical unambiguous computation. We apply our techniques to push forward some known constructs involving relativized unambiguity based complexity classes (UP and Promise- UP) to new constructs involving arbitrary levels of the relativized unambiguous polynomial hierarchy (UPH). Our techniques are developed on constraints imposed by hierarchical assembly of unambiguous nondeterministic polynomial-time Turing machines, and so our techniques differ substantially, in applicability and in nature, from standard techniques (such as the switching lemma [Has87]), which are known to play roles in carrying out similar generalizations. \u0000 \u0000Aside from achieving these generalizations, we resolve a question posed by Cai, Hemachandra, and Vyskoc [CHV93] on an issue related to nonadaptive Turing access to UP and adaptive smart Turing access to Promise-UP.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130593354","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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