48 research outputs found
Assigning channels via the meet-in-the-middle approach
We study the complexity of the Channel Assignment problem. By applying the
meet-in-the-middle approach we get an algorithm for the -bounded Channel
Assignment (when the edge weights are bounded by ) running in time
. This is the first algorithm which breaks the
barrier. We extend this algorithm to the counting variant, at the
cost of slightly higher polynomial factor.
A major open problem asks whether Channel Assignment admits a -time
algorithm, for a constant independent of . We consider a similar
question for Generalized T-Coloring, a CSP problem that generalizes \CA. We
show that Generalized T-Coloring does not admit a
-time algorithm, where is the
size of the instance.Comment: SWAT 2014: 282-29
Dynamic Range Majority Data Structures
Given a set of coloured points on the real line, we study the problem of
answering range -majority (or "heavy hitter") queries on . More
specifically, for a query range , we want to return each colour that is
assigned to more than an -fraction of the points contained in . We
present a new data structure for answering range -majority queries on a
dynamic set of points, where . Our data structure uses O(n)
space, supports queries in time, and updates in amortized time. If the coordinates of the points are integers,
then the query time can be improved to . For constant values of , this improved query
time matches an existing lower bound, for any data structure with
polylogarithmic update time. We also generalize our data structure to handle
sets of points in d-dimensions, for , as well as dynamic arrays, in
which each entry is a colour.Comment: 16 pages, Preliminary version appeared in ISAAC 201
Partial Sums on the Ultra-Wide Word RAM
We consider the classic partial sums problem on the ultra-wide word RAM model
of computation. This model extends the classic -bit word RAM model with
special ultrawords of length bits that support standard arithmetic and
boolean operation and scattered memory access operations that can access
(non-contiguous) locations in memory. The ultra-wide word RAM model captures
(and idealizes) modern vector processor architectures.
Our main result is a new in-place data structure for the partial sum problem
that only stores a constant number of ultraword in addition to the input and
supports operations in doubly logarithmic time. This matches the best known
time bounds for the problem (among polynomial space data structures) while
improving the space from superlinear to a constant number of ultrawords. Our
results are based on a simple and elegant in-place word RAM data structure,
known as the Fenwick tree. Our main technical contribution is a new efficient
parallel ultra-wide word RAM implementation of the Fenwick tree, which is
likely of independent interest.Comment: Extended abstract appeared at TAMC 202
Dynamic nested brackets
We consider the problem of maintaining a string of n brackets '('or')' under the operation reverse(i) that changes the ith bracket from '('to')' or vice versa, and returns 'yes' if and only if the resulting string is properly balanced. We show that this problem can be solved on the RAM in time O(log n/log log n) per operation using linear space and preprocessing. Moreover, we show that this is optimal in the sense that every data structure supporting reverse (no matter its space and preprocessing complexity) needs time Omega(Iog n/log log n) per operation in the cell probe model. (C) 2004 Elsevier Inc. All rights reserved
Multivariate Analysis of Orthogonal Range Searching and Graph Distances Parameterized by Treewidth
We show that the eccentricities, diameter, radius, and Wiener index of an undirected -vertex graph with nonnegative edge lengths can be computed in time , where is the treewidth of the graph. For every , this bound is , which matches a hardness result of Abboud, Vassilevska Williams, and Wang (SODA 2015) and closes an open problem in the multivariate analysis of polynomial-time computation. To this end, we show that the analysis of an algorithm of Cabello and Knauer (Comp. Geom., 2009) in the regime of non-constant treewidth can be improved by revisiting the analysis of orthogonal range searching, improving bounds of the form to , as originally observed by Monier (J. Alg. 1980). We also investigate the parameterization by vertex cover number
The first parameterized algorithms and computational experiments challenge
\u3cbr/\u3eIn this article, the steering committee of the Parameterized Algorithms and Computational Experiments challenge (PACE) reports on the first iteration of the challenge. Where did PACE come from, how did it go, who won, and what's next
Sparsification upper and lower bounds for graphs problems and not-all-equal SAT
We present several sparsification lower and upper bounds for classic problems in graph theory and logic. For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time algorithm that reduces any n-vertex input to an equivalent instance, of an arbitrary problem, with bitsize O(n^{2-epsilon}) for epsilon > 0, unless NP is a subset of coNP/poly and the polynomial-time hierarchy collapses. These results imply that existing linear-vertex kernels for k-Nonblocker and k-Max Leaf Spanning Tree (the parametric duals of (Connected) Dominating Set) cannot be improved to have O(k^{2-epsilon}) edges, unless NP is a subset of NP/poly. We also present a positive result and exhibit a non-trivial sparsification algorithm for d-Not-All-Equal-SAT. We give an algorithm that reduces an n-variable input with clauses of size at most d to an equivalent input with O(n^{d-1}) clauses, for any fixed d. Our algorithm is based on a linear-algebraic proof of Lovász that bounds the number of hyperedges in critically 3-chromatic d-uniform n-vertex hypergraphs by binom{n}{d-1}. We show that our kernel is tight under the assumption that NP is not a subset of NP/poly