The arboricity of a graph is the minimum number of forests it can be partitioned into. Previous approximation schemes were nonconstructive, i.e., they only approximated the arboricity as a value without computing a corresponding forest partition, as they operate on the related pseudoforest partitions or the dual problem. We propose an algorithm for converting a partition of @math pseudoforests into a partition of @math forests in @math time, where @math is the inverse Ackermann function, when @math expected time is allowed for pre-computation of a perfect hash function. Without perfect hashing, we obtain @math with a data structure by Brodal and Fagerberg that stores graphs of arboricity @math . For every fixed @math , the latter result implies a constructive @math -approximation algorithm with runtime @math by using Kowalik's approximation scheme for pseudoforest partitions. Our algorithm might help in designing a faster exact arboricity algorithm. We also make several remarks on approximation algorithms for the pseudoarboricity and the equivalent graph orientations with smallest maximum indegree, and correct some mistakes made in the literature.
@cite_24 compute a @math -orientation (assuming @math ) with a variant of the greedy orientation algorithm in @math , but the orientation produced may contain cycles.
@cite_24
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Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090. Abstract This paper presents a novel version of the five-point relative orientation algorithm given in Nister [Nister, D., 2004. An efficient solution to the five-point relative pose problem, IEEE Transactions on Pattern Analysis and Machine Intelligence, 26 (6), 756–770]. The name of the algorithm arises from the fact that it can operate even on the minimal five-point correspondences required for a finite number of solutions to relative orientation. For the minimal five correspondences, the algorithm returns up to 10 real solutions. The algorithm can also operate on many points. Like the previous version of the five-point algorithm, our method can operate correctly even in the face of critical surfaces, including planar and ruled quadric scenes. The paper presents comparisons with other direct methods, including the previously developed five-point method, two different six-point methods, the seven-point method, and the eight-point method. It is shown that the five-point method is superior in most cases among the direct methods. The new version of the algorithm was developed from the perspective of algebraic geometry and is presented in the context of computing a Grobner basis. The constraints are formulated in terms of polynomial equations in the entries of the fundamental matrix. The polynomial equations generate an algebraic ideal for which a Grobner basis is computed. The Grobner basis is used to compute the action matrix for multiplication by a single variable monomial. The eigenvectors of the action matrix give the solutions for all the variables and thereby also relative orientation. Using a Grobner basis makes the solution clear and easy to explain. In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an @math round matrix multiplication algorithm, where @math is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -- triangle and 4-cycle counting in @math rounds, improving upon the @math triangle detection algorithm of [DISC 2012], -- a @math -approximation of all-pairs shortest paths in @math rounds, improving upon the @math -round @math -approximation algorithm of Nanongkai [STOC 2014], and -- computing the girth in @math rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles. The greedy sequential algorithm for maximal independent set (MIS) loops over the vertices in an arbitrary order adding a vertex to the resulting set if and only if no previous neighboring vertex has been added. In this loop, as in many sequential loops, each iterate will only depend on a subset of the previous iterates (i.e. knowing that any one of a vertex's previous neighbors is in the MIS, or knowing that it has no previous neighbors, is sufficient to decide its fate one way or the other). This leads to a dependence structure among the iterates. If this structure is shallow then running the iterates in parallel while respecting the dependencies can lead to an efficient parallel implementation mimicking the sequential algorithm. In this paper, we show that for any graph, and for a random ordering of the vertices, the dependence length of the sequential greedy MIS algorithm is polylogarithmic (O(log^2 n) with high probability). Our results extend previous results that show polylogarithmic bounds only for random graphs. We show similar results for greedy maximal matching (MM). For both problems we describe simple linear-work parallel algorithms based on the approach. The algorithms allow for a smooth tradeoff between more parallelism and reduced work, but always return the same result as the sequential greedy algorithms. We present experimental results that demonstrate efficiency and the tradeoff between work and parallelism.