{"title":"Which crossing number is it, anyway? [computational geometry]","authors":"J. Pach, G. Tóth","doi":"10.1109/SFCS.1998.743512","DOIUrl":"https://doi.org/10.1109/SFCS.1998.743512","url":null,"abstract":"A drawing of a graph G is a mapping which assigns to each vertex a point of the plane and to each edge a simple continuous arc connecting the corresponding two points. The crossing number of G is the minimum number of crossing points in any drawing of G. We define two new parameters, as follows. The pairwise crossing number (resp. the odd-crossing number) of G is the minimum number of pairs of edges that cross (resp. cross an odd number of times) over all drawings of G. We prove that the determination of each of these parameters is an NP-complete problem. We also prove that the largest of these numbers (the crossing number) cannot exceed twice the square of the smallest (the odd-crossing number). Our proof is based on the following generalization of an old result of Hanani, which is of independent interest. Let G be a graph and let E/sub 0/ be a subset of its edges such that there is a drawing of G, in which every edge belonging E/sub 0/ crosses any other edge an even number of times. Then G can be redrawn so that the element of E/sub 0/ are not involved in any crossing.","PeriodicalId":228145,"journal":{"name":"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117096075","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}
Martín Farach-Colton, P. Ferragina, S. Muthukrishnan
{"title":"Overcoming the memory bottleneck in suffix tree construction","authors":"Martín Farach-Colton, P. Ferragina, S. Muthukrishnan","doi":"10.1109/SFCS.1998.743441","DOIUrl":"https://doi.org/10.1109/SFCS.1998.743441","url":null,"abstract":"The suffix tree of a string is the fundamental data structure of string processing. Recent focus on massive data sets has sparked interest in overcoming the memory bottlenecks of known algorithms for building suffix trees. Our main contribution is a new algorithm for suffix tree construction in which we choreograph almost all disk accesses to be via the sort and scan primitives. This algorithm achieves optimal results in a variety of sequential and parallel computational models. Two of our results are: In the traditional external memory model, in which only the number of disk accesses is counted, we achieve an optimal algorithm, both for single and multiple disk cases. This is the first optimal algorithm known for either model. Traditional disk page access counting does not differentiate between random page accesses and block transfers involving several consecutive pages. This difference is routinely exploited by expert programmers to get fast algorithms on real machines. We adopt a simple accounting scheme and show that our algorithm achieves the same optimal tradeoff for block versus random page accesses as the one we establish for sorting.","PeriodicalId":228145,"journal":{"name":"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)","volume":"151 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132280883","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":"Decidability of bisimulation equivalence for equational graphs of finite out-degree","authors":"Géraud Sénizergues","doi":"10.1109/SFCS.1998.743435","DOIUrl":"https://doi.org/10.1109/SFCS.1998.743435","url":null,"abstract":"The bisimulation problem for equational graphs of finite out-degree is shown to be decidable. We reduce this problem to the /spl eta/-bisimulation problem for deterministic rational (vectors of) Boolean series on the alphabet of a dpda M. We then exhibit a complete formal system for deducing equivalent pairs of such vectors.","PeriodicalId":228145,"journal":{"name":"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)","volume":"98 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122648948","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":"Geometric separator theorems and applications","authors":"Warren D. Smith, N. Wormald","doi":"10.1109/SFCS.1998.743449","DOIUrl":"https://doi.org/10.1109/SFCS.1998.743449","url":null,"abstract":"We find a large number of \"geometric separator theorems\" such as: I: Given N disjoint isooriented squares in the plane, there exists a rectangle with /spl les/2N/3 squares inside, /spl les/2N/3 squares outside, and /spl les/(4+0(1))/spl radic/N partly in & out. II: There exists a rectangle that is crossed by the minimal spanning tree of N sites in the plane at /spl les/(4/spl middot/3/sup 1/4/+0(1))/spl radic/N points, having /spl les/2N/3 sites inside and outside. These theorems yield a large number of applications, such as subexponential algorithms for traveling salesman tour and rectilinear Steiner minimal tree in R/sup d/, new point location algorithms, and new upper and lower bound proofs for \"planar separator theorems\".","PeriodicalId":228145,"journal":{"name":"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116885448","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":"Fast Monte-Carlo algorithms for finding low-rank approximations","authors":"A. Frieze, R. Kannan, S. Vempala","doi":"10.1109/SFCS.1998.743487","DOIUrl":"https://doi.org/10.1109/SFCS.1998.743487","url":null,"abstract":"In several applications, the data consists of an m/spl times/n matrix A and it is of interest to find an approximation D of a specified rank k to A where, k is much smaller than m and n. Traditional methods like the Singular Value Decomposition (SVD) help us find the \"best\" such approximation. However, these methods take time polynomial in m, n which is often too prohibitive. In this paper, we develop an algorithm which is qualitatively faster provided we may sample the entries of the matrix according to a natural probability distribution. Indeed, in the applications such sampling is possible. Our main result is that we can find the description of a matrix D* of rank at most k so that /spl par/A-D*/spl par//sub F//spl les/min/D,rank(D)/spl les/k/spl par/A-D/spl par//sub F/+/spl epsiv//spl par/A/spl par//sub F/ holds with probability at least 1-/spl delta/. (For any matrix M, /spl par/M/spl par//sub F//sup 2/ denotes the sum of the squares of all the entries of M.) The algorithm takes time polynomial in k, 1//spl epsiv/, log(1//spl delta/) only, independent of m, n.","PeriodicalId":228145,"journal":{"name":"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115969183","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":"The complexity of the approximation of the bandwidth problem","authors":"Walter Unger","doi":"10.1109/SFCS.1998.743431","DOIUrl":"https://doi.org/10.1109/SFCS.1998.743431","url":null,"abstract":"The bandwidth problem has a long history and a number of important applications. It is the problem of enumerating the vertices of a given graph G such that the maximum difference between the numbers of adjacent vertices is minimal. We will show for any constant k/spl epsiv/N that there is no polynomial time approximation algorithm with an approximation factor of k. Furthermore, we will show that this result holds also for caterpillars, a class of restricted trees. We construct for any x,/spl epsiv//spl isin/R with x>1 and /spl epsiv/>0 a graph class for which an approximation algorithm with an approximation factor of x+/spl epsiv/ exists, but the approximation of the bandwidth problem within a factor of x-/spl epsiv/ is NP-complete. The best previously known approximation factors for the intractability of the bandwidth approximation problem were 1.5 for general graphs and 4/3 for trees.","PeriodicalId":228145,"journal":{"name":"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121460119","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":"Towards an optimal bit-reversal permutation program","authors":"L. Carter, K. Gatlin","doi":"10.1109/SFCS.1998.743505","DOIUrl":"https://doi.org/10.1109/SFCS.1998.743505","url":null,"abstract":"The speed of many computations is limited not by the number of arithmetic operations but by the time it takes to move and rearrange data in the increasingly complicated memory hierarchies of modern computers. Array transpose and the bit-reversal permutation-trivial operations on a RAM-present non-trivial problems, when designing highly-tuned scientific library functions, particular for the Fast Fourier Transform. We prove a precise bound for RoCol, a simple pebble-type game that is relevant to implementing these permutations. We use RoCol to give lower bounds on the amount of memory traffic in a computer with four-levels of memory (registers, cache, TLB, and memory), taking into account such \"messy\" features as block moves and set-associative caches. The insights from this analysis lead to a bit-reversal algorithm whose performance is close to the theoretical minimum. Experiments show that it performs significantly better than every program in a comprehensive study of 30 published algorithms.","PeriodicalId":228145,"journal":{"name":"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131717922","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":"Jitter control in QoS networks","authors":"Y. Mansour, B. Patt-Shamir","doi":"10.1109/SFCS.1998.743428","DOIUrl":"https://doi.org/10.1109/SFCS.1998.743428","url":null,"abstract":"We study jitter control in networks guaranteeing quality of service (QoS). Jitter measures variability of delivery times in packet streams. We propose on-line algorithms that control jitter and compare their performance to the best possible (by an off-line algorithm) for any given arrival sequence. For delay jitter, where the goal is to minimize the difference between delay times of different packets, we give an on-line algorithm using buffer size of 2B which guarantees the same delay-jitter as an off-line algorithm using buffer space B. We show that 2B space is the minimum space required by any on-line algorithm to provide delay-jitter related to the best possible delay-jitter using B buffer space. We also show that the guarantees made by our online algorithm hold even for distributed implementations, where the total buffer space is distributed along the path of the connection, provided that the input stream satisfies a certain simple property. For rate jitter, where the goal is to minimize the difference between inter-arrival times, we develop an on-line algorithm using a buffer of size 2B+h for any h/spl ges/1, and compare its jitter to the jitter of an optimal off-line algorithm using buffer size B. Our algorithm guarantees that the difference is bounded by a term proportional to B/h. We also prove that 2B space is necessary for on-line algorithms with non trivial guarantees for rate-jitter control.","PeriodicalId":228145,"journal":{"name":"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130670493","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":"Algorithms to tile the infinite grid with finite clusters","authors":"M. Szegedy","doi":"10.1109/SFCS.1998.743437","DOIUrl":"https://doi.org/10.1109/SFCS.1998.743437","url":null,"abstract":"We say that a subset T of Z/sup 2/, the two dimensional infinite grid, tiles Z/sup 2/ if we can cover Z/sup 2/ with non-overlapping translates of T. No algorithm is known to decide whether a finite T/spl sube/Z/sup 2/ tiles Z/sup 2/. Here we present two algorithms, one for the case when |T| is prime, and another for the case when |T|=4. Both algorithms generalize to the case, where we replace Z/sup 2/ with all arbitrary finitely generated Abelian group. As a by-product of our results we partially settle the Periodic Tiling Conjecture raised by J. Lagarias and Y. Wang (1997), and we also get the following generalization of a theorem of L.Redei (1965): Let G be a (finite or infinite) Abelian group G with a generator set T of prime cardinality such, that 0/spl isin/T, and there is a set T'/spl sube/G with the property that for every g/spl isin/G there are unique t/spl isin/T, t'/spl isin/T' such that g=t+t'. Then T' can be replaced with a subgroup of G, that also has the above property.","PeriodicalId":228145,"journal":{"name":"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114427059","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":"Probabilistically checkable proofs with low amortized query complexity","authors":"M. Sudan, L. Trevisan","doi":"10.1109/SFCS.1998.743425","DOIUrl":"https://doi.org/10.1109/SFCS.1998.743425","url":null,"abstract":"The error probability of Probabilistically Checkable Proof (PCP) systems can be made exponentially small in the number of queries by using sequential repetition. In this paper we are interested in determining the precise rate at which the error goes down in an optimal protocol, and we make substantial progress toward a tight resolution of this question. A PCP verifier uses q~ amortized query bits if, for some t, it makes q~t queries and has error probability at most 2/sup -t/. A PCP characterization of NP using 2.5 amortized query bits is known, and, unless P=NP, no such characterization is possible using 1 amortized query bits. We present a PCP characterization of NP that uses roughly 1.5 amortized query bits. Our result has two main implications. Separating PCP from 2-Provers 1-Round: In the 2-Provers 1-Round (2P1R) model the verifier has access to two oracles (or provers) and can make one query to each oracle. Each answer is a string of l bits (l is called the answer size). A 2P1R protocol with answer size l can be simulated by a PCP that reads 21 bits; we show that the converse does not hold for l/spl ges/7, unless P=NP. No such separation was known before. The Max kCSP problem: The Boolean constraint satisfaction problem with constraints involving at most k variables, usually called Max kCSP, is known to be hard to approximate within a factor 2/sup -4k/, and a 2.2/sup -k/-approximation algorithm is also known. We prove that Max kCSP is NP-hard to approximate within a factor of roughly 2/sup -2k/3/.","PeriodicalId":228145,"journal":{"name":"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131970049","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}