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Listing triangles
 In Automata, Languages, and Programming  41st International Colloquium, ICALP 2014
"... Abstract. We present new algorithms for listing triangles in dense and sparse graphs. The running time of our algorithm for dense graphs is Õ(nω + n3(ω−1)/(5−ω)t2(3−ω)/(5−ω)), and the running time of the algorithm for sparse graphs is Õ(m2ω/(ω+1) + m3(ω−1)/(ω+1)t(3−ω)/(ω+1)), where n is the numbe ..."
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Cited by 9 (0 self)
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Abstract. We present new algorithms for listing triangles in dense and sparse graphs. The running time of our algorithm for dense graphs is Õ(nω + n3(ω−1)/(5−ω)t2(3−ω)/(5−ω)), and the running time of the algorithm for sparse graphs is Õ(m2ω/(ω+1) + m3(ω−1)/(ω+1)t(3−ω)/(ω+1)), where n
Reducing 3XOR to listing triangles, an exposition
, 2011
"... The 3SUM problem asks if there are three integers a, b, c summing to 0 in a given set of n integers of magnitude poly(n). This problem can be easily solved in time Õ(n2). (Throughout this note, Õ and ˜ Ω hide subpolynomial factors no(1).) It seems natural to believe that this problem also requires t ..."
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SUM to the problem of listing triangles of a graph.[Pǎt10] In this note we present this reduction by Pǎtra¸scu but for a variant of the 3SUM problem which we call 3XOR. The problem 3XOR is like 3SUM except that integer summation is replaced with bitwise xor. So one can think of 3XOR as asking if a
Quantum Gravity
, 2004
"... We describe the basic assumptions and key results of loop quantum gravity, which is a background independent approach to quantum gravity. The emphasis is on the basic physical principles and how one deduces predictions from them, at a level suitable for physicists in other areas such as string theor ..."
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Cited by 566 (11 self)
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theory, cosmology, particle physics, astrophysics and condensed matter physics. No details are given, but references are provided to guide the interested reader to the literature. The present state of knowledge is summarized in a list of 35 key results on topics including the hamiltonian and path
The Encyclopedia of Integer Sequences
"... This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs ..."
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Cited by 866 (15 self)
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This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae
Stream Control Transmission Protocol
, 2007
"... This document is an InternetDraft and is in full conformance with all provisions of Section 10 of RFC 2026. InternetDrafts are working documents of the Internet Engineering Task Force (IETF), its areas, and its working groups. Note that other groups may also distribute working documents as Interne ..."
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Cited by 570 (22 self)
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as InternetDrafts. InternetDrafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet Drafts as reference material or to cite them other than as ‘‘work in progress.’’ The list of current
Multiresolution Analysis of Arbitrary Meshes
, 1995
"... In computer graphics and geometric modeling, shapes are often represented by triangular meshes. With the advent of laser scanning systems, meshes of extreme complexity are rapidly becoming commonplace. Such meshes are notoriously expensive to store, transmit, render, and are awkward to edit. Multire ..."
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Cited by 605 (16 self)
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In computer graphics and geometric modeling, shapes are often represented by triangular meshes. With the advent of laser scanning systems, meshes of extreme complexity are rapidly becoming commonplace. Such meshes are notoriously expensive to store, transmit, render, and are awkward to edit. Multiresolution analysis offers a simple, unified, and theoretically sound approach to dealing with these problems. Lounsbery et al. have recently developed a technique for creating multiresolution representations for a restricted class of meshes with subdivision connectivity. Unfortunately, meshes encountered in practice typically do not meet this requirement. In this paper we present a method for overcoming the subdivision connectivity restriction, meaning that completely arbitrary meshes can now be converted to multiresolution form. The method is based on the approximation of an arbitrary initial mesh M by a mesh M that has subdivision connectivity and is guaranteed to be within a specified tolerance. The key
Why a diagram is (sometimes) worth ten thousand words
 Cognitive Science
, 1987
"... We distinguish diagrammatic from sentential paperandpencil representationsof information by developing alternative models of informationprocessing systems that are informationally equivalent and that can be characterized as sentential or diagrammatic. Sentential representations are sequential, li ..."
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Cited by 777 (2 self)
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We distinguish diagrammatic from sentential paperandpencil representationsof information by developing alternative models of informationprocessing systems that are informationally equivalent and that can be characterized as sentential or diagrammatic. Sentential representations are sequential, like the propositions in a text. Dlogrammotlc representations ore indexed by location in a plane. Diogrommatic representations also typically display information that is only implicit in sententiol representations and that therefore has to be computed, sometimes at great cost, to make it explicit for use. We then contrast the computational efficiency of these representotions for solving several illustrative problems in mothematics and physics. When two representotions are informationally equivolent, their computational efficiency depends on the informationprocessing operators that act on them. Two sets of operators may differ in their copobilities for recognizing patterns, in the inferences they con carry out directly, and in their control strategies (in portitular. the control of search). Diogrommotic ond sentential representations sup
Primitives for the manipulation of general subdivisions and the computations of Voronoi diagrams
 ACM Tmns. Graph
, 1985
"... The following problem is discussed: Given n points in the plane (the sites) and an arbitrary query point 4, find the site that is closest to q. This problem can be solved by constructing the Voronoi diagram of the given sites and then locating the query point in one of its regions. Two algorithms ar ..."
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Cited by 543 (11 self)
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The following problem is discussed: Given n points in the plane (the sites) and an arbitrary query point 4, find the site that is closest to q. This problem can be solved by constructing the Voronoi diagram of the given sites and then locating the query point in one of its regions. Two algorithms are given, one that constructs the Voronoi diagram in O(n log n) time, and another that inserts a new site in O(n) time. Both are based on the use of the Voronoi dual, or Delaunay triangulation, and are simple enough to be of practical value. The simplicity of both algorithms can be attributed to the separation of the geometrical and topological aspects of the problem and to the use of two simple but powerful primitives, a geometric predicate and an operator for manipulating the topology of the diagram. The topology is represented by a new data structure for generalized diagrams, that is, embeddings of graphs in twodimensional manifolds. This structure represents simultaneously an embedding, its dual, and its mirror image. Furthermore, just two operators are sufficient for building and modifying arbitrary diagrams.
An Extended Set of Fortran Basic Linear Algebra Subprograms
 ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
, 1986
"... This paper describes an extension to the set of Basic Linear Algebra Subprograms. The extensions are targeted at matrixvector operations which should provide for efficient and portable implementations of algorithms for high performance computers. ..."
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Cited by 526 (72 self)
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This paper describes an extension to the set of Basic Linear Algebra Subprograms. The extensions are targeted at matrixvector operations which should provide for efficient and portable implementations of algorithms for high performance computers.
Results 1  10
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255,860