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Towards Optimal Locality in MeshIndexings
, 1997
"... The efficiency of many data structures and algorithms relies on "localitypreserving" indexing schemes for meshes. We concentrate on the case in which the maximal distance between two mesh nodes indexed i and j shall be a slowgrowing function of ji jj. We present a new 2D indexing scheme we call H ..."
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Cited by 31 (4 self)
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The efficiency of many data structures and algorithms relies on "localitypreserving" indexing schemes for meshes. We concentrate on the case in which the maximal distance between two mesh nodes indexed i and j shall be a slowgrowing function of ji jj. We present a new 2D indexing scheme we call Hindexing , which has superior (possibly optimal) locality in comparison with the wellknown Hilbert indexings. Hindexings form a Hamiltonian cycle and we prove that they are optimally localitypreserving among all cyclic indexings. We provide fairly tight lower bounds for indexings without any restriction. Finally, illustrated by investigations concerning 2D and 3D Hilbert indexings, we present a framework for mechanizing upper bound proofs for locality.
Optimal Linear Arrangement of a Rectangular Grid
 Discrete Mathematics
"... An optimal linear arrangement of a finite simple graph G = (V; E) with vertex set V , edge set E, and jV j = N , is a map f from V onto f1; 2; : : : ; Ng that minimizes P fu;vg2E jf(u) \Gamma f(v)j. We determine optimal linear arrangements for m \Theta n rectangular grids where V = f1; 2; : : : ..."
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Cited by 8 (0 self)
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An optimal linear arrangement of a finite simple graph G = (V; E) with vertex set V , edge set E, and jV j = N , is a map f from V onto f1; 2; : : : ; Ng that minimizes P fu;vg2E jf(u) \Gamma f(v)j. We determine optimal linear arrangements for m \Theta n rectangular grids where V = f1; 2; : : : ; mg \Theta f1; 2; : : : ; ng and E = ff(i; j); (k; `)g : ji \Gamma kj + jj \Gamma `j = 1g. When m n 5, they are disjoint from bandwidthminimizing arrangements for which f minimizes the maximum jf(u) \Gamma f(v)j over E. The different solutions to the bandwidth and linear arrangement problems for rectangular grids is reminiscent of Harper's result ([7],[8]) of different bandwidth and linear arrangement solutions for the hypercube graph with vertex set f0; 1g n and edge set ff(x 1 ; x 2 ; : : : ; x n ), (y 1 ; y 2 ; : : : ; y n )g : P i jx i \Gamma y i j = 1g. 1 Introduction In this work, we report an optimal labeling (with integers 1; : : : ; mn) of the vertices of an m \Theta n gr...
Random Geometric Problems on [0, 1]²
 RANDOMIZATION AND APPROXIMATION TECHNIQUES IN COMPUTER SCIENCE, NUMBER 1518 IN LECTURE NOTES IN COMPUTER SCIENCE
, 1998
"... In this paper we survey the work done for graphs on random geometric models. We present some heuristics for the problem of the Minimal Linear Arrangement on [0, 1]² and we conclude with a collection of open problems. ..."
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Cited by 5 (2 self)
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In this paper we survey the work done for graphs on random geometric models. We present some heuristics for the problem of the Minimal Linear Arrangement on [0, 1]² and we conclude with a collection of open problems.
Convergence Theorems for Some Layout Measures on Random Lattice and Random Geometric Graphs
"... This work deals with convergence theorems and bounds on the cost of several layout measures for lattice graphs, random lattice graphs and sparse random geometric graphs. Specifically, we consider the following problems: Minimum Linear Arrangement, Cutwidth, Sum Cut, Vertex Separation, Edge Bisection ..."
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Cited by 5 (0 self)
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This work deals with convergence theorems and bounds on the cost of several layout measures for lattice graphs, random lattice graphs and sparse random geometric graphs. Specifically, we consider the following problems: Minimum Linear Arrangement, Cutwidth, Sum Cut, Vertex Separation, Edge Bisection and Vertex Bisection. For full square lattices, we give optimal layouts for the problems still open. For arbitrary lattice graphs, we present best possible bounds disregarding a constant factor. We apply percolation theory to the study of lattice graphs in a probabilistic setting. In particular, we deal with the subcritical regime that this class of graphs exhibits and characterize the behavior of several layout measures in this space of probability. We extend the results on random lattice graphs to random geometric graphs, which are graphs whose nodes are spread at random in the unit square and whose edges connect pairs of points which are within a given distance. We also characterize the behavior of several layout measures on random geometric graphs in their subcritical regime. Our main results are convergence theorems that can be viewed as an analog of the Beardwood, Halton and Hammersley theorem for the Euclidean TSP on random points in the unit square.
Fundamental Limits for Information Retrieval
, 1999
"... The fundamental limits of performance for a general model of information retrieval from databases are studied. In the scenarios considered a large quantity of information is to be stored on some physical storage device. Requests for information are modeled as a randomly generated sequence with a kno ..."
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Cited by 2 (1 self)
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The fundamental limits of performance for a general model of information retrieval from databases are studied. In the scenarios considered a large quantity of information is to be stored on some physical storage device. Requests for information are modeled as a randomly generated sequence with a known distribution. The requests are assumed to be "contextdependent", i.e., to vary according to the sequence of previous requests. The state of the physical storage device is also assumed to depend on the history of previous requests. In general the logical structure of the information to be stored does not match the physical structure of the storage device, and consequently there are nontrivial limits on the minimum achievable average access times, where the average is over the possible sequences of user requests. The paper applies basic informationtheoretic methods to establish these limits and demonstrates constructive procedures that approach them, for a wide class of systems. Allowing ...
Approximating Layout Problems on Geometric Random Graphs (Extended abstract)
 Algorithms and Complexity in Information Technology
, 1998
"... We show two simple algorithms that, with high probability, approximate within a constant several layout problems for geometric random graphs drawn from the Gn (r) model with r = c p log n=n for any constant c 6. The layout problems that we consider are: Bandwidth, Minimum Linear Arrangement, Mini ..."
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Cited by 1 (0 self)
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We show two simple algorithms that, with high probability, approximate within a constant several layout problems for geometric random graphs drawn from the Gn (r) model with r = c p log n=n for any constant c 6. The layout problems that we consider are: Bandwidth, Minimum Linear Arrangement, Minimum Cut, Minimum Sum Cut, Vertex Separation and Bisection. This research was partially supported by ESPRIT LTR Project no. 20244  ALCOMIT, CICYT Project TIC971475CE, and CIRIT project 1997SGR00366. y Departament de Llenguatges i Sistemes Inform`atics. Universitat Polit`ecnica de Catalunya. Campus Nord C6. c/ Jordi Girona 13. 08034 Barcelona (Spain). fdiaz,jpetit,mjsernag@lsi.upc.es z Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, England. Mathew.Penrose@durham.ac.uk 1 Introduction Several wellknown optimization problems on graphs can be formulated as Layout Problems. Their goal is to find a layout (linear ordering) of the nodes of an...
Heuristics for the MinLA Problem: An Empirical and Theoretical Analysis (Extended Abstract)
, 1998
"... ) Josep D'iaz Jordi Petit i Silvestre Mar'ia Jos'e Serna Departament de Llenguatges i Sistemes Inform`atics (LSI) Universitat Polit`ecnica de Catalunya (UPC) Campus Nord  M`odul C6 Jordi Girona Salgado, 13 08034 Barcelona, Catalonia (Spain) Fax: +34 934 017 014 fdiaz,jpetit,mjsernag@lsi.upc.es ..."
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Cited by 1 (1 self)
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) Josep D'iaz Jordi Petit i Silvestre Mar'ia Jos'e Serna Departament de Llenguatges i Sistemes Inform`atics (LSI) Universitat Polit`ecnica de Catalunya (UPC) Campus Nord  M`odul C6 Jordi Girona Salgado, 13 08034 Barcelona, Catalonia (Spain) Fax: +34 934 017 014 fdiaz,jpetit,mjsernag@lsi.upc.es Paul Spirakis Computer Technology Institute (CTI) Kolokotroni 3 GR 262 21 Patras (Greece) spirakis@cti.gr Supported by ESPRIT LTR Project no. 20244  ALCOMIT, CICYT Project TIC971475CE, DGES Project PB9J0787Koala and CIRIT 1997SGR00366. The implementation and evaluation of these heuristics has been done at the C 4 (Centre de Computaci'o i Comunicacions de Catalunya). 1 Introduction Several wellknown optimization problems on graphs can be formulated as Linear Arrangement Problems. Their goal is to find a layout (ordering) of the nodes of an input graph such that a certain function is minimized. Linear arrangement problems are an important class of problems with many differe...
Linear Orderings of Random Geometric Graphs (Extended Abstract)
, 1999
"... ) Josep D'iaz y Mathew D. Penrose z Jordi Petit y Mar'ia Serna y April 8, 1999 Abstract In random geometric graphs, vertices are randomly distributed on [0; 1] 2 and pairs of vertices are connected by edges whenever they are sufficiently close together. Layout problems seek a linear orde ..."
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) Josep D'iaz y Mathew D. Penrose z Jordi Petit y Mar'ia Serna y April 8, 1999 Abstract In random geometric graphs, vertices are randomly distributed on [0; 1] 2 and pairs of vertices are connected by edges whenever they are sufficiently close together. Layout problems seek a linear ordering of the vertices of a graph such that a certain measure is minimized. In this paper, we study several layout problems on random geometric graphs: Bandwidth, Minimum Linear Arrangement, Minimum Cut, Minimum Sum Cut, Vertex Separation and Bisection. We first prove that some of these problems remain NPcomplete even for geometric graphs. Afterwards, we compute lower bounds that hold with high probability on random geometric graphs. Finally, we characterize the probabilistic behavior of the lexicographic ordering for our layout problems on the class of random geometric graphs. This research was partially supported by ESPRIT LTR Project no. 20244  ALCOMIT, CICYT Project TIC971475CE, ...