33,568 research outputs found

    Succinct Partial Sums and Fenwick Trees

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    We consider the well-studied partial sums problem in succint space where one is to maintain an array of n k-bit integers subject to updates such that partial sums queries can be efficiently answered. We present two succint versions of the Fenwick Tree - which is known for its simplicity and practicality. Our results hold in the encoding model where one is allowed to reuse the space from the input data. Our main result is the first that only requires nk + o(n) bits of space while still supporting sum/update in O(log_b n) / O(b log_b n) time where 2 <= b <= log^O(1) n. The second result shows how optimal time for sum/update can be achieved while only slightly increasing the space usage to nk + o(nk) bits. Beyond Fenwick Trees, the results are primarily based on bit-packing and sampling - making them very practical - and they also allow for simple optimal parallelization

    A bandwidth theorem for approximate decompositions

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    We provide a degree condition on a regular nn-vertex graph GG which ensures the existence of a near optimal packing of any family H\mathcal H of bounded degree nn-vertex kk-chromatic separable graphs into GG. In general, this degree condition is best possible. Here a graph is separable if it has a sublinear separator whose removal results in a set of components of sublinear size. Equivalently, the separability condition can be replaced by that of having small bandwidth. Thus our result can be viewed as a version of the bandwidth theorem of B\"ottcher, Schacht and Taraz in the setting of approximate decompositions. More precisely, let δk\delta_k be the infimum over all δ≥1/2\delta\ge 1/2 ensuring an approximate KkK_k-decomposition of any sufficiently large regular nn-vertex graph GG of degree at least δn\delta n. Now suppose that GG is an nn-vertex graph which is close to rr-regular for some r≥(δk+o(1))nr \ge (\delta_k+o(1))n and suppose that H1,…,HtH_1,\dots,H_t is a sequence of bounded degree nn-vertex kk-chromatic separable graphs with ∑ie(Hi)≤(1−o(1))e(G)\sum_i e(H_i) \le (1-o(1))e(G). We show that there is an edge-disjoint packing of H1,…,HtH_1,\dots,H_t into GG. If the HiH_i are bipartite, then r≥(1/2+o(1))nr\geq (1/2+o(1))n is sufficient. In particular, this yields an approximate version of the tree packing conjecture in the setting of regular host graphs GG of high degree. Similarly, our result implies approximate versions of the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree.Comment: Final version, to appear in the Proceedings of the London Mathematical Societ

    Analysis of the Min-Sum Algorithm for Packing and Covering Problems via Linear Programming

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    Message-passing algorithms based on belief-propagation (BP) are successfully used in many applications including decoding error correcting codes and solving constraint satisfaction and inference problems. BP-based algorithms operate over graph representations, called factor graphs, that are used to model the input. Although in many cases BP-based algorithms exhibit impressive empirical results, not much has been proved when the factor graphs have cycles. This work deals with packing and covering integer programs in which the constraint matrix is zero-one, the constraint vector is integral, and the variables are subject to box constraints. We study the performance of the min-sum algorithm when applied to the corresponding factor graph models of packing and covering LPs. We compare the solutions computed by the min-sum algorithm for packing and covering problems to the optimal solutions of the corresponding linear programming (LP) relaxations. In particular, we prove that if the LP has an optimal fractional solution, then for each fractional component, the min-sum algorithm either computes multiple solutions or the solution oscillates below and above the fraction. This implies that the min-sum algorithm computes the optimal integral solution only if the LP has a unique optimal solution that is integral. The converse is not true in general. For a special case of packing and covering problems, we prove that if the LP has a unique optimal solution that is integral and on the boundary of the box constraints, then the min-sum algorithm computes the optimal solution in pseudo-polynomial time. Our results unify and extend recent results for the maximum weight matching problem by [Sanghavi et al.,'2011] and [Bayati et al., 2011] and for the maximum weight independent set problem [Sanghavi et al.'2009]
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