43,651 research outputs found
Algorithms for the Problems of Length-Constrained Heaviest Segments
We present algorithms for length-constrained maximum sum segment and maximum
density segment problems, in particular, and the problem of finding
length-constrained heaviest segments, in general, for a sequence of real
numbers. Given a sequence of n real numbers and two real parameters L and U (L
<= U), the maximum sum segment problem is to find a consecutive subsequence,
called a segment, of length at least L and at most U such that the sum of the
numbers in the subsequence is maximum. The maximum density segment problem is
to find a segment of length at least L and at most U such that the density of
the numbers in the subsequence is the maximum. For the first problem with
non-uniform width there is an algorithm with time and space complexities in
O(n). We present an algorithm with time complexity in O(n) and space complexity
in O(U). For the second problem with non-uniform width there is a combinatorial
solution with time complexity in O(n) and space complexity in O(U). We present
a simple geometric algorithm with the same time and space complexities.
We extend our algorithms to respectively solve the length-constrained k
maximum sum segments problem in O(n+k) time and O(max{U, k}) space, and the
length-constrained maximum density segments problem in O(n min{k, U-L})
time and O(U+k) space. We present extensions of our algorithms to find all the
length-constrained segments having user specified sum and density in O(n+m) and
O(nlog (U-L)+m) times respectively, where m is the number of output.
Previously, there was no known algorithm with non-trivial result for these
problems. We indicate the extensions of our algorithms to higher dimensions.
All the algorithms can be extended in a straight forward way to solve the
problems with non-uniform width and non-uniform weight.Comment: 21 pages, 12 figure
Fast Monotone Summation over Disjoint Sets
We study the problem of computing an ensemble of multiple sums where the
summands in each sum are indexed by subsets of size of an -element
ground set. More precisely, the task is to compute, for each subset of size
of the ground set, the sum over the values of all subsets of size that are
disjoint from the subset of size . We present an arithmetic circuit that,
without subtraction, solves the problem using arithmetic
gates, all monotone; for constant , this is within the factor
of the optimal. The circuit design is based on viewing the summation as a "set
nucleation" task and using a tree-projection approach to implement the
nucleation. Applications include improved algorithms for counting heaviest
-paths in a weighted graph, computing permanents of rectangular matrices,
and dynamic feature selection in machine learning
Network Flow Algorithms for Structured Sparsity
We consider a class of learning problems that involve a structured
sparsity-inducing norm defined as the sum of -norms over groups of
variables. Whereas a lot of effort has been put in developing fast optimization
methods when the groups are disjoint or embedded in a specific hierarchical
structure, we address here the case of general overlapping groups. To this end,
we show that the corresponding optimization problem is related to network flow
optimization. More precisely, the proximal problem associated with the norm we
consider is dual to a quadratic min-cost flow problem. We propose an efficient
procedure which computes its solution exactly in polynomial time. Our algorithm
scales up to millions of variables, and opens up a whole new range of
applications for structured sparse models. We present several experiments on
image and video data, demonstrating the applicability and scalability of our
approach for various problems.Comment: accepted for publication in Adv. Neural Information Processing
Systems, 201
Linear-Space Data Structures for Range Mode Query in Arrays
A mode of a multiset is an element of maximum multiplicity;
that is, occurs at least as frequently as any other element in . Given a
list of items, we consider the problem of constructing a data
structure that efficiently answers range mode queries on . Each query
consists of an input pair of indices for which a mode of must
be returned. We present an -space static data structure
that supports range mode queries in time in the worst case, for
any fixed . When , this corresponds to
the first linear-space data structure to guarantee query time. We
then describe three additional linear-space data structures that provide
, , and query time, respectively, where denotes the
number of distinct elements in and denotes the frequency of the mode of
. Finally, we examine generalizing our data structures to higher dimensions.Comment: 13 pages, 2 figure
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