46,345 research outputs found
Continuous Multiclass Labeling Approaches and Algorithms
We study convex relaxations of the image labeling problem on a continuous
domain with regularizers based on metric interaction potentials. The generic
framework ensures existence of minimizers and covers a wide range of
relaxations of the originally combinatorial problem. We focus on two specific
relaxations that differ in flexibility and simplicity -- one can be used to
tightly relax any metric interaction potential, while the other one only covers
Euclidean metrics but requires less computational effort. For solving the
nonsmooth discretized problem, we propose a globally convergent
Douglas-Rachford scheme, and show that a sequence of dual iterates can be
recovered in order to provide a posteriori optimality bounds. In a quantitative
comparison to two other first-order methods, the approach shows competitive
performance on synthetical and real-world images. By combining the method with
an improved binarization technique for nonstandard potentials, we were able to
routinely recover discrete solutions within 1%--5% of the global optimum for
the combinatorial image labeling problem
Searching a bitstream in linear time for the longest substring of any given density
Given an arbitrary bitstream, we consider the problem of finding the longest
substring whose ratio of ones to zeroes equals a given value. The central
result of this paper is an algorithm that solves this problem in linear time.
The method involves (i) reformulating the problem as a constrained walk through
a sparse matrix, and then (ii) developing a data structure for this sparse
matrix that allows us to perform each step of the walk in amortised constant
time. We also give a linear time algorithm to find the longest substring whose
ratio of ones to zeroes is bounded below by a given value. Both problems have
practical relevance to cryptography and bioinformatics.Comment: 22 pages, 19 figures; v2: minor edits and enhancement
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