31,271 research outputs found

    Cyclic classes and attraction cones in max algebra

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    In max algebra it is well-known that the sequence A^k, with A an irreducible square matrix, becomes periodic at sufficiently large k. This raises a number of questions on the periodic regime of A^k and A^k x, for a given vector x. Also, this leads to the concept of attraction cones in max algebra, by which we mean sets of vectors whose ultimate orbit period does not exceed a given number. This paper shows that some of these questions can be solved by matrix squaring (A,A^2,A^4, ...), analogously to recent findings concerning the orbit period in max-min algebra. Hence the computational complexity of such problems is of the order O(n^3 log n). The main idea is to apply an appropriate diagonal similarity scaling A -> X^{-1}AX, called visualization scaling, and to study the role of cyclic classes of the critical graph. For powers of a visualized matrix in the periodic regime, we observe remarkable symmetry described by circulants and their rectangular generalizations. We exploit this symmetry to derive a concise system of equations for attraction cpne, and we present an algorithm which computes the coefficients of the system.Comment: 38 page

    Minimum cycle and homology bases of surface embedded graphs

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    We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the 11-dimensional (Z2\mathbb{Z}_2)-homology classes) of an undirected graph embedded on a surface. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the 11-skeleton of any graph is exactly its minimum cycle basis. For the minimum cycle basis problem, we give a deterministic O(nω+22gn2+m)O(n^\omega+2^{2g}n^2+m)-time algorithm for graphs embedded on an orientable surface of genus gg. The best known existing algorithms for surface embedded graphs are those for general graphs: an O(mω)O(m^\omega) time Monte Carlo algorithm and a deterministic O(nm2/logn+n2m)O(nm^2/\log n + n^2 m) time algorithm. For the minimum homology basis problem, we give a deterministic O((g+b)3nlogn+m)O((g+b)^3 n \log n + m)-time algorithm for graphs embedded on an orientable or non-orientable surface of genus gg with bb boundary components, assuming shortest paths are unique, improving on existing algorithms for many values of gg and nn. The assumption of unique shortest paths can be avoided with high probability using randomization or deterministically by increasing the running time of the homology basis algorithm by a factor of O(logn)O(\log n).Comment: A preliminary version of this work was presented at the 32nd Annual International Symposium on Computational Geometr

    Systolic geometry of translation surfaces

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    Let SS be a translation surface of genus g>1g > 1 with nn cone points (pi)i=1,,n(p_i)_{i=1,\ldots,n} with cone angle 2π(ki+1)2\pi \cdot (k_i+1) at pip_i, where kiNk_i \in \mathbb{N}. In this paper we investigate the systolic landscape of these translation surfaces for fixed genus.Comment: 25 pages, 4 figures. Added explicit computations of systoles in the graph of saddle connections for origamis in H(1,1) and a criterion to decide whether such systoles define systoles on the translation surfac
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