163,015 research outputs found
An efficient rounding boundary test for pow(x,y) in double precision
18 pagesThe correct rounding of the function pow: (x,y) -> x^y is currently based on Ziv's iterative approximation process. In order to ensure its termination, cases when x^y falls on a rounding boundary must be filtered out. Such rounding boundaries are floating-point numbers and midpoints between two consecutive floating-point numbers. Detecting rounding boundaries for pow is a difficult problem. Previous approaches use repeated square root extraction followed by repeated square and multiply. This article presents a new rounding boundary test for pow in double precision which resumes to a few comparisons with pre-computed constants. These constants are deduced from worst cases for the Table Maker's Dilemma, searched over a small subset of the input domain. This is a novel use of such worst-case bounds. The resulting algorithm has been designed for a fast-on-average correctly rounded implementation of pow, considering the scarcity of rounding boundary cases. It does not stall average computations for rounding boundary detection. The article includes its correction proof and experimental results
A New Dynamic Programming Approach for Spanning Trees with Chain Constraints and Beyond
Short spanning trees subject to additional constraints are important building
blocks in various approximation algorithms. Especially in the context of the
Traveling Salesman Problem (TSP), new techniques for finding spanning trees
with well-defined properties have been crucial in recent progress. We consider
the problem of finding a spanning tree subject to constraints on the edges in
cuts forming a laminar family of small width. Our main contribution is a new
dynamic programming approach where the value of a table entry does not only
depend on the values of previous table entries, as it is usually the case, but
also on a specific representative solution saved together with each table
entry. This allows for handling a broad range of constraint types.
In combination with other techniques -- including negatively correlated
rounding and a polyhedral approach that, in the problems we consider, allows
for avoiding potential losses in the objective through the randomized rounding
-- we obtain several new results. We first present a quasi-polynomial time
algorithm for the Minimum Chain-Constrained Spanning Tree Problem with an
essentially optimal guarantee. More precisely, each chain constraint is
violated by a factor of at most , and the cost is no larger than
that of an optimal solution not violating any chain constraint. The best
previous procedure is a bicriteria approximation violating each chain
constraint by up to a constant factor and losing another factor in the
objective. Moreover, our approach can naturally handle lower bounds on the
chain constraints, and it can be extended to constraints on cuts forming a
laminar family of constant width.
Furthermore, we show how our approach can also handle parity constraints (or,
more precisely, a proxy thereof) as used in the context of (Path) TSP and one
of its generalizations, and discuss implications in this context.Comment: A short version of this work appeared in the proceedings of the 30th
annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2019
An Improved Algorithm for Fixed-Hub Single Allocation Problem
This paper discusses the fixed-hub single allocation problem (FHSAP). In this
problem, a network consists of hub nodes and terminal nodes. Hubs are fixed and
fully connected; each terminal node is connected to a single hub which routes
all its traffic. The goal is to minimize the cost of routing the traffic in the
network. In this paper, we propose a linear programming (LP)-based rounding
algorithm. The algorithm is based on two ideas. First, we modify the LP
relaxation formulation introduced in Ernst and Krishnamoorthy (1996, 1999) by
incorporating a set of validity constraints. Then, after obtaining a fractional
solution to the LP relaxation, we make use of a geometric rounding algorithm to
obtain an integral solution. We show that by incorporating the validity
constraints, the strengthened LP often provides much tighter upper bounds than
the previous methods with a little more computational effort, and the solution
obtained often has a much smaller gap with the optimal solution. We also
formulate a robust version of the FHSAP and show that it can guard against data
uncertainty with little cost
Improving Image Restoration with Soft-Rounding
Several important classes of images such as text, barcode and pattern images
have the property that pixels can only take a distinct subset of values. This
knowledge can benefit the restoration of such images, but it has not been
widely considered in current restoration methods. In this work, we describe an
effective and efficient approach to incorporate the knowledge of distinct pixel
values of the pristine images into the general regularized least squares
restoration framework. We introduce a new regularizer that attains zero at the
designated pixel values and becomes a quadratic penalty function in the
intervals between them. When incorporated into the regularized least squares
restoration framework, this regularizer leads to a simple and efficient step
that resembles and extends the rounding operation, which we term as
soft-rounding. We apply the soft-rounding enhanced solution to the restoration
of binary text/barcode images and pattern images with multiple distinct pixel
values. Experimental results show that soft-rounding enhanced restoration
methods achieve significant improvement in both visual quality and quantitative
measures (PSNR and SSIM). Furthermore, we show that this regularizer can also
benefit the restoration of general natural images.Comment: 9 pages, 6 figure
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