94,496 research outputs found

    Counting Proper Mergings of Chains and Antichains

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    A proper merging of two disjoint quasi-ordered sets PP and QQ is a quasi-order on the union of PP and QQ such that the restriction to PP and QQ yields the original quasi-order again and such that no elements of PP and QQ are identified. In this article, we consider the cases where PP and QQ are chains, where PP and QQ are antichains, and where PP is an antichain and QQ is a chain. We give formulas that determine the number of proper mergings in all three cases, and introduce two new bijections from proper mergings of two chains to plane partitions and from proper mergings of an antichain and a chain to monotone colorings of complete bipartite digraphs. Additionally, we use these bijections to count the Galois connections between two chains, and between a chain and a Boolean lattice respectively.Comment: 36 pages, 15 figures, 5 table

    Mergeable Dictionaries

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    A data structure is presented for the Mergeable Dictionary abstract data type, which supports the following operations on a collection of disjoint sets of totally ordered data: Predecessor-Search, Split and Union. While Predecessor-Search and Split work in the normal way, the novel operation is Union. While in a typical mergeable dictionary (e.g. 2-4 Trees), the Union operation can only be performed on sets that span disjoint intervals in keyspace, the structure here has no such limitation, and permits the merging of arbitrarily interleaved sets. Our data structure supports all operations, including Union, in O(log n) amortized time, thus showing that interleaved Union operations can be supported at no additional cost vis-a-vis disjoint Union operations

    Complexity of union-split-find problems

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    Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 45-46).In this thesis, we investigate various interpretations of the Union-Split-Find problem, an extension of the classic Union-Find problem. In the Union-Split Find problem, we maintain disjoint sets of ordered elements subject to the operations of constructing singleton sets, merging two sets together, splitting a set by partitioning it around a specified value, and finding the set that contains a given element. The different interpretations of this problem arise from the different assumptions made regarding when sets can be merged and any special properties the sets may have. We define and analyze the Interval, Cyclic, Ordered, and General Union-Split-Find problems. Previous work implies optimal solutions to the Interval and Ordered Union-Split-Find problems and an (log n/ log log n) lower bound for the Cyclic Union-Split-Find problem in the cell-probe model. We present a new data structure that achieves a matching upper bound of (log n/ log log n) for Cyclic Union-Split Find in the word RAM model. For General Union-Split-Find, no o(n) bound is known. We present a data structure which has an [Omega](log2 n) amortized lower bound in the worst case that we conjecture has polylogarithmic amortized performance. This thesis is the product of joint work with Erik Demaine.by Katherine Jane Lai.M.Eng

    Proper Mergings of Stars and Chains are Counted by Sums of Antidiagonals in Certain Convolution Arrays -- The Details

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    A proper merging of two disjoint quasi-ordered sets PP and QQ is a quasi-order on the union of PP and QQ such that the restriction to PP or QQ yields the original quasi-order again and such that no elements of PP and QQ are identified. In this article, we determine the number of proper mergings in the case where PP is a star (i.e. an antichain with a smallest element adjoined), and QQ is a chain. We show that the lattice of proper mergings of an mm-antichain and an nn-chain, previously investigated by the author, is a quotient lattice of the lattice of proper mergings of an mm-star and an nn-chain, and we determine the number of proper mergings of an mm-star and an nn-chain by counting the number of congruence classes and by determining their cardinalities. Additionally, we compute the number of Galois connections between certain modified Boolean lattices and chains.Comment: 27 pages, 7 figures, 1 table. Jonathan Farley has solved Problem 4.18; added Section 4.4 to describe his solutio

    Branch merging on continuum trees with applications to regenerative tree growth

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    We introduce a family of branch merging operations on continuum trees and show that Ford CRTs are distributionally invariant. This operation is new even in the special case of the Brownian CRT, which we explore in more detail. The operations are based on spinal decompositions and a regenerativity preserving merging procedure of (α,θ)(\alpha, \theta)-strings of beads, that is, random intervals [0,Lα,θ][0, L_{\alpha, \theta}] equipped with a random discrete measure dL1dL^{-1} arising in the limit of ordered (α,θ)(\alpha, \theta)-Chinese restaurant processes as introduced recently by Pitman and Winkel. Indeed, we iterate the branch merging operation recursively and give an alternative approach to the leaf embedding problem on Ford CRTs related to (α,2α)(\alpha, 2-\alpha)-regenerative tree growth processes.Comment: 40 pages, 5 figure

    Matching Tree-Level Matrix Elements with Interleaved Showers

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    We present an implementation of the so-called CKKW-L merging scheme for combining multi-jet tree-level matrix elements with parton showers. The implementation uses the transverse-momentum-ordered shower with interleaved multiple interactions as implemented in PYTHIA8. We validate our procedure using e+e--annihilation into jets and vector boson production in hadronic collisions, with special attention to details in the algorithm which are formally sub-leading in character, but may have visible effects in some observables. We find substantial merging scale dependencies induced by the enforced rapidity ordering in the default PYTHIA8 shower. If this rapidity ordering is removed the merging scale dependence is almost negligible. We then also find that the shower does a surprisingly good job of describing the hardness of multi-jet events, as long as the hardest couple of jets are given by the matrix elements. The effects of using interleaved multiple interactions as compared to more simplistic ways of adding underlying-event effects in vector boson production are shown to be negligible except in a few sensitive observables. To illustrate the generality of our implementation, we also give some example results from di-boson production and pure QCD jet production in hadronic collisions.Comment: 44 pages, 23 figures, as published in JHEP, including all changes recommended by the refere
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