633 research outputs found
A Nearly Quadratic Bound for the Decision Tree Complexity of k-SUM
We show that the k-SUM problem can be solved by a linear decision tree of depth O(n^2 log^2 n),improving the recent bound O(n^3 log^3 n) of Cardinal et al. Our bound depends linearly on k, and allows us to conclude that the number of linear queries required to decide the n-dimensional Knapsack or SubsetSum problems is only O(n^3 log n), improving the currently best known bounds by a factor of n. Our algorithm extends to the RAM model, showing that the k-SUM problem can be solved in expected polynomial time, for any fixed k, with the above bound on the number of linear queries. Our approach relies on a new point-location mechanism, exploiting "Epsilon-cuttings" that are based on vertical decompositions in hyperplane arrangements in high dimensions.
A major side result of the analysis in this paper is a sharper bound on the complexity of the vertical decomposition of such an arrangement (in terms of its dependence on the dimension). We hope that this study will reveal further structural properties of vertical decompositions in hyperplane arrangements
The Order Dimension of the Poset of Regions in a Hyperplane Arrangement
We show that the order dimension of the weak order on a Coxeter group of type
A, B or D is equal to the rank of the Coxeter group, and give bounds on the
order dimensions for the other finite types. This result arises from a unified
approach which, in particular, leads to a simpler treatment of the previously
known cases, types A and B. The result for weak orders follows from an upper
bound on the dimension of the poset of regions of an arbitrary hyperplane
arrangement. In some cases, including the weak orders, the upper bound is the
chromatic number of a certain graph. For the weak orders, this graph has the
positive roots as its vertex set, and the edges are related to the pairwise
inner products of the roots.Comment: Minor changes, including a correction and an added figure in the
proof of Proposition 2.2. 19 pages, 6 figure
A vector partition function for the multiplicities of sl_k(C)
We use Gelfand-Tsetlin diagrams to write down the weight multiplicity
function for the Lie algebra sl_k(C) (type A_{k-1}) as a single partition
function. This allows us to apply known results about partition functions to
derive interesting properties of the weight diagrams. We relate this
description to that of the Duistermaat-Heckman measure from symplectic
geometry, which gives a large-scale limit way to look at multiplicity diagrams.
We also provide an explanation for why the weight polynomials in the boundary
regions of the weight diagrams exhibit a number of linear factors. Using
symplectic geometry, we prove that the partition of the permutahedron into
domains of polynomiality of the Duistermaat-Heckman function is the same as
that for the weight multiplicity function, and give an elementary proof of this
for sl_4(C) (A_3).Comment: 34 pages, 11 figures and diagrams; submitted to Journal of Algebr
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