We attempt to determine the best order and search algorithm to store n comparable data items in an array, A, of length n so that we can, for any query value, x, quickly find the smallest value in A that is greater than or equal to x. In particular, we consider the important case where there are many such queries to the same array, A, which resides entirely in RAM. In addition to the obvious sorted order/binary search combination we consider the Eytzinger (BFS) layout normally used for heaps, an implicit B-tree layout that generalizes the Eytzinger layout, and the van Emde Boas layout commonly used in the cache-oblivious algorithms literature.After extensive testing and tuning on a wide variety of modern hardware, we arrive at the conclusion that, for small values of n, sorted order, combined with a good implementation of binary search is best. For larger values of n, we arrive at the surprising conclusion that the Eytzinger layout is usually the fastest. The latter conclusion is unexpected and goes counter to earlier experimental work by Brodal, Fagerberg, and Jacob (SODA 2003), who concluded that both the B-tree and van Emde Boas layouts were faster than the Eytzinger layout for large values of n. Our fastest C++ implementations, when compiled, use conditional moves to avoid branch mispredictions and prefetching to reduce cache latency.

International audienceWe consider the two-level uncapacitated facility location problem with single-assignment constraints (TUFLP-S), a problem that arises in industrial applications in freight transportation and telecommunications. We present a new Lagrangian relaxation approach for the TUFLP-S, based on solving a single-level uncapacitated facility location problem (UFLP) as the Lagrangian subproblem. We also develop a Lagrangian heuristic that includes a mixed-integer programming–based large neighborhood search heuristic exploring neighborhoods by solving a series of small UFLPs. The dual and primal bounds thus obtained are used within an enumerative algorithm that implements specialized branching rules. Our computational experiments on instances derived from an industrial application in freight transportation as well as on large, hard, artificial instances confirm the efficiency of the authors specialized branch-and-bound algorithm

We develop a variant of the variable neighborhood search (VNS) metaheuristic called the multilayer VNS (MLVNS).It consists in partitioning the neighborhood structures into multiple layers. For each layer l , a VNS defined on the associated neighborhood structures is invoked, each move being evaluated and completed by a recursive call to the MLVNS at layer l −1. A specific MLVNS is developed to solve approximately a class of two-level uncapacitated facility location problems with single assignment (TUFLPS), when only mild assumptions are imposed on the cost functions. Two special cases are used to illustrate the efficiency of the MLVNS: the classical TUFLPS and a problem with modular costs derived from a real-life case. To assess the efficiency of the MLVNS, computational results on a large set of instances are compared with those obtained by slope scaling heuristic methods and by solving integer programming models using a state-of-the-art commercial solver.

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