42,580 research outputs found
Sweep maps: A continuous family of sorting algorithms
We define a family of maps on lattice paths, called sweep maps, that assign
levels to each step in the path and sort steps according to their level.
Surprisingly, although sweep maps act by sorting, they appear to be bijective
in general. The sweep maps give concise combinatorial formulas for the
q,t-Catalan numbers, the higher q,t-Catalan numbers, the q,t-square numbers,
and many more general polynomials connected to the nabla operator and rational
Catalan combinatorics. We prove that many algorithms that have appeared in the
literature (including maps studied by Andrews, Egge, Gorsky, Haglund, Hanusa,
Jones, Killpatrick, Krattenthaler, Kremer, Orsina, Mazin, Papi, Vaille, and the
present authors) are all special cases of the sweep maps or their inverses. The
sweep maps provide a very simple unifying framework for understanding all of
these algorithms. We explain how inversion of the sweep map (which is an open
problem in general) can be solved in known special cases by finding a "bounce
path" for the lattice paths under consideration. We also define a generalized
sweep map acting on words over arbitrary alphabets with arbitrary weights,
which is also conjectured to be bijective.Comment: 21 pages; full version of FPSAC 2014 extended abstrac
Universal Sorting: Finding a DAG using Priced Comparisons
We resolve two open problems in sorting with priced information, introduced
by [Charikar, Fagin, Guruswami, Kleinberg, Raghavan, Sahai (CFGKRS), STOC
2000]. In this setting, different comparisons have different (potentially
infinite) costs. The goal is to sort with small competitive ratio (algorithmic
cost divided by cheapest proof).
1) When all costs are in , we give an algorithm that has
competitive ratio. Our algorithm generalizes the
algorithms for generalized sorting (all costs are either or ), a
version initiated by [Huang, Kannan, Khanna, FOCS 2011] and addressed recently
by [Kuszmaul, Narayanan, FOCS 2021].
2) We answer the problem of bichromatic sorting posed by [CFGKRS]: The input
is split into and , and and comparisons are more expensive
than an comparisons. We give a randomized algorithm with a O(polylog n)
competitive ratio.
These results are obtained by introducing the universal sorting problem,
which generalizes the existing framework in two important ways. We remove the
promise of a directed Hamiltonian path in the DAG of all comparisons. Instead,
we require that an algorithm outputs the transitive reduction of the DAG. For
bichromatic sorting, when and comparisons cost , this
generalizes the well-known nuts and bolts problem. We initiate an
instance-based study of the universal sorting problem. Our definition of
instance-optimality is inherently more algorithmic than that of the competitive
ratio in that we compare the cost of a candidate algorithm to the cost of the
optimal instance-aware algorithm. This unifies existing lower bounds, and opens
up the possibility of an -instance optimal algorithm for the bichromatic
version.Comment: 40 pages, 5 figure
Efficient parallel computation on multiprocessors with optical interconnection networks
This dissertation studies optical interconnection networks, their architecture, address schemes, and computation and communication capabilities. We focus on a simple but powerful optical interconnection network model - the Linear Array with Reconfigurable pipelined Bus System (LARPBS). We extend the LARPBS model to a simplified higher dimensional LAPRBS and provide a set of basic computation operations. We then study the following two groups of parallel computation problems on both one dimensional LARPBS\u27s as well as multi-dimensional LARPBS\u27s: parallel comparison problems, including sorting, merging, and selection; Boolean matrix multiplication, transitive closure and their applications to connected component problems. We implement an optimal sorting algorithm on an n-processor LARPBS. With this optimal sorting algorithm at disposal, we study the sorting problem for higher dimensional LARPBS\u27s and obtain the following results: • An optimal basic Columnsort algorithm on a 2D LARPBS. • Two optimal two-way merge sort algorithms on a 2D LARPBS. • An optimal multi-way merge sorting algorithm on a 2D LARPBS. • An optimal generalized column sort algorithm on a 2D LARPBS. • An optimal generalized column sort algorithm on a 3D LARPBS. • An optimal 5-phase sorting algorithm on a 3D LARPBS. Results for selection problems are as follows: • A constant time maximum-finding algorithm on an LARPBS. • An optimal maximum-finding algorithm on an LARPBS. • An O((log log n)2) time parallel selection algorithm on an LARPBS. • An O(k(log log n)2) time parallel multi-selection algorithm on an LARPBS. While studying the computation and communication properties of the LARPBS model, we find Boolean matrix multiplication and its applications to the graph are another set of problem that can be solved efficiently on the LARPBS. Following is a list of results we have obtained in this area. • A constant time Boolean matrix multiplication algorithm. • An O(log n)-time transitive closure algorithm. • An O(log n)-time connected components algorithm. • An O(log n)-time strongly connected components algorithm. The results provided in this dissertation show the strong computation and communication power of optical interconnection networks
Efficiency of linked cell algorithms
The linked cell list algorithm is an essential part of molecular simulation
software, both molecular dynamics and Monte Carlo. Though it scales linearly
with the number of particles, there has been a constant interest in increasing
its efficiency, because a large part of CPU time is spent to identify the
interacting particles. Several recent publications proposed improvements to the
algorithm and investigated their efficiency by applying them to particular
setups. In this publication we develop a general method to evaluate the
efficiency of these algorithms, which is mostly independent of the parameters
of the simulation, and test it for a number of linked cell list algorithms. We
also propose a combination of linked cell reordering and interaction sorting
that shows a good efficiency for a broad range of simulation setups.Comment: Submitted to Computer Physics Communications on 22 December 2009,
still awaiting a referee repor
Twenty-Five Comparators is Optimal when Sorting Nine Inputs (and Twenty-Nine for Ten)
This paper describes a computer-assisted non-existence proof of nine-input
sorting networks consisting of 24 comparators, hence showing that the
25-comparator sorting network found by Floyd in 1964 is optimal. As a
corollary, we obtain that the 29-comparator network found by Waksman in 1969 is
optimal when sorting ten inputs.
This closes the two smallest open instances of the optimal size sorting
network problem, which have been open since the results of Floyd and Knuth from
1966 proving optimality for sorting networks of up to eight inputs.
The proof involves a combination of two methodologies: one based on
exploiting the abundance of symmetries in sorting networks, and the other,
based on an encoding of the problem to that of satisfiability of propositional
logic. We illustrate that, while each of these can single handed solve smaller
instances of the problem, it is their combination which leads to an efficient
solution for nine inputs.Comment: 18 page
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