Computational Aspects of Vlsi

The NP-completeness column: An ongoing guide

Abstract: This is the fourteenth edition of a quarterly column that provides continuing coverage of new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book “Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., New York, 1979 (hereinafter referred to as “[GJ previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomialtime-solvability, etc.), or open problems they would like publicized, should send them to David S. Johnson, Room 2C-355, AT&T Bell Laboratories, Murray Hill, NJ 07974 (CSNET address: dsj.btl@csnet-relay). Please include details, or at least sketches, of any new proofs (full papers are preferred). If the results are unpublished, please state explicitly that you would like them to be mentioned in the column. Comments and corrections are also welcome. For more details on the nature of the column and the form of desired submissions, see the December 1981 issue of this Journal.

Approximating the permanent

Abstract: A randomised approximation scheme for the permanent of a 0–1s presented. The task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph; the latter is accomplished by simulating a Markov chain whose states are the matchings in the graph. For a wide class of 0–1 matrices the approximation scheme is fully-polynomial, i.e., runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output. This class includes all dense matrices (those that contain sufficiently many 1’s) and almost all sparse matrices in some reasonable probabilistic model for 0–1 matrices of given density.For the approach sketched above to be computationally efficient, the Markov chain must be rapidly mixing: informally, it must converge in a short time to its stationary distribution. A major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing, apparently the first such result for a Markov chain with genuinely c...

How to draw a planar graph on a grid

Abstract: Answering a question of Rosenstiehl and Tarjan, we show that every plane graph withn vertices has a Fary embedding (i.e., straight-line embedding) on the 2n−4 byn−2 grid and provide anO(n) space,O(n logn) time algorithm to effect this embedding. The grid size is asymptotically optimal and it had been previously unknown whether one can always find a polynomial sized grid to support such an embedding. On the other hand we show that any setF, which can support a Fary embedding of every planar graph of sizen, has cardinality at leastn+(1−o(1))√n which settles a problem of Mohar.

The design and analysis of parallel algorithms

Abstract: Kurskod: 5DV050 Inrättad: 2008-03-31 Inrättad av: teknisk-naturvetenskapliga fakultetsnämnden Reviderad: 2012-02-29 Reviderad av: teknisk-naturvetenskapliga fakultetsnämnden Kursplan giltig från: 2012, vecka 10 Ansvarig enhet: Inst för datavetenskap SCB-ämnesrubrik: Informatik/Dataoch systemvetenskap Huvudområden och successiv fördjupning: Beräkningsteknik: Avancerad nivå, har endast kurs/er på grundnivå som förkunskapskrav (A1N) , Datavetenskap: Avancerad nivå, har endast kurs/er på grundnivå som förkunskapskrav (A1N) Betygsskala: För denna kurs ges betygen 5 Med beröm godkänd, 4 Icke utan beröm godkänd, 3 Godkänd, VG Väl godkänd, G Godkänd, U Underkänd Utbildningsnivå: Avancerad nivå

On the complexity of VLSI implementations and graph representations of Boolean functions with application to integer multiplication

Abstract: Lower-bound results on Boolean-function complexity under two different models are discussed. The first is an abstraction of tradeoffs between chip area and speed in very-large-scale-integrated (VLSI) circuits. The second is the ordered binary decision diagram (OBDD) representation used as a data structure for symbolically representing and manipulating Boolean functions. The lower bounds demonstrate the fundamental limitations of VLSI as an implementation medium, and that of the OBDD as a data structure. It is shown that the same technique used to prove that any VLSI implementation of a single output Boolean function has area-time complexity AT/sup 2/= Omega (n/sup 2/) also proves that any OBDD representation of the function has Omega (c/sup n/) vertices for some c>1 but that the converse is not true. An integer multiplier for word size n with outputs numbered 0 (least significant) through 2n-1 (most significant) is described. For the Boolean function representing either output i-1 or output 2n-i-1, where 1 >