{"title":"Reconfiguring a hypercube in the presence of faults","authors":"J. Håstad, F. Leighton, M. Newman","doi":"10.1145/28395.28425","DOIUrl":"https://doi.org/10.1145/28395.28425","url":null,"abstract":"We consider the computational power of a hypercube containing a potentially large number of randomly located faulty components. In particular, we describe algorithms for embedding an N/2-node hypercube in an N-node hypercube with faulty processors. Provided that the processors of the N-node hypercube are faulty with probability p < 1/2, and that the faults are independently distributed, we show that with high probability, adjacent cells in the N/2-node hypercube are mapped to functioning cells at distance 3 or less apart in the N-node hypercube. The algorithm is deterministic, easy to implement and runs in &Ogr;(log N) steps using only local control. We also describe ways to produce embeddings which allow for low delay simulations, as well as ways to use a faulty hypercube to efficiently simulate a completely functioning hypercube of the same size.","PeriodicalId":161795,"journal":{"name":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114774108","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":"Threshold spectra for random graphs","authors":"S. Shelah, J. Spencer","doi":"10.1145/28395.28440","DOIUrl":"https://doi.org/10.1145/28395.28440","url":null,"abstract":"Let G = G(n, p) be the random graph with n vertices and edge probability p and ƒ(n, p, A) be the probability that G has A, where A is a first order property of graphs. The evolution of the random graph is discussed in terms of a spectrum of p = p(n) where ƒ(n, p, A) changes. A partial characterization of possible spectra is given. When p = n-a, a irrational, and A is any first order statement, it is shown that lim ƒ(n, p, A) = 0 or 1.","PeriodicalId":161795,"journal":{"name":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121439237","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":"Parallel symmetry-breaking in sparse graphs","authors":"A. Goldberg, Serge A. Plotkin, Gregory E. Shannon","doi":"10.1145/28395.28429","DOIUrl":"https://doi.org/10.1145/28395.28429","url":null,"abstract":"We describe efficient deterministic techniques for breaking symmetry in parallel. The techniques work well on rooted trees and graphs of constant degree or genus. Our primary technique allows us to 3-color a rooted tree in &Ogr;(lg*n) time on an EREW PRAM using a linear number of processors. We apply these techniques to construct fast linear processor algorithms for several problems, including (&Dgr; + 1)-coloring constant-degree graphs, 5-coloring planar graphs, and finding depth-first-search trees in planar graphs. We also prove lower bounds for 2-coloring directed lists and for finding maximal independent sets in arbitrary graphs.","PeriodicalId":161795,"journal":{"name":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124656976","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 learnability of Boolean formulae","authors":"M. Kearns, Ming Li, L. Pitt, L. Valiant","doi":"10.1145/28395.28426","DOIUrl":"https://doi.org/10.1145/28395.28426","url":null,"abstract":"We study the computational feasibility of learning boolean expressions from examples. Our goals are to prove results and develop general techniques that shed light on the boundary between the classes of expressions that are learnable in polynomial time and those that are apparently not. The elucidation of this boundary, for boolean expressions and possibly other knowledge representations, is an example of the potential contribution of complexity theory to artificial intelligence. We employ the distribution-free model of learning introduced in /lo]. A more complete discussion and justification of this model can be found in [4,10,11,12]. [4] includes some discussion that is relevant more particularly to infinite representations, such as geometric ones, rather than the finite case of boolean functions. For other recent related work see [1,2,7,&g]. The results of this paper fall into three categories: closure properties of learnable classes, negative results, and distribution-specific positive results. The closure properties are of two kinds. In section 3 we discuss closure under boolean operations on the members of the learnable classes. The assumption that the classes are learnable from positive or negative ex-","PeriodicalId":161795,"journal":{"name":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","volume":"183 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121939010","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 page number of genus g graphs is (g)","authors":"Lenwood S. Heath, S. Istrail","doi":"10.1145/28395.28437","DOIUrl":"https://doi.org/10.1145/28395.28437","url":null,"abstract":"This paper disproves the conjecture that graphs of fixed genus g ≤ 1 have unbounded pagenumber (Bernhart and Kainen, 1979). We show that genus g graphs can be embedded in &Ogr;(g) pages, and derive an &OHgr;(√g) lower bound. We present the first algorithm in the literature for embedding an arbitrary graph in a book with a non-trivial upper bound on the number of pages. We first compute the genus g of a graph using the algorithm of Filotti, Miller, Reif (1979), and then apply our (optimal-time) algorithm for obtaining an &Ogr;(g) page embedding. An important aspect of our construction is a new decomposition theorem, of independent interest, for a graph embedded on a surface. Book embedding has application in several areas, two of which are directly related to the results we obtain: fault-tolerant VLSI and complexity theory.","PeriodicalId":161795,"journal":{"name":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1987-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130307391","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":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","authors":"A. Aho","doi":"10.1145/28395","DOIUrl":"https://doi.org/10.1145/28395","url":null,"abstract":"","PeriodicalId":161795,"journal":{"name":"Proceedings of the nineteenth annual ACM symposium on Theory of computing","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127984313","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}