19,558 research outputs found
The Folklore of Sorting Algorithms
The objective of this paper is to review the folklore knowledge seen in research work devoted on synthesis, optimization, and effectiveness of various sorting algorithms. We will examine sorting algorithms in the folklore lines and try to discover the tradeoffs between folklore and theorems. Finally, the folklore knowledge on complexity values of the sorting algorithms will be considered, verified and subsequently converged in to theorems
Synchronous Counting and Computational Algorithm Design
Consider a complete communication network on nodes, each of which is a
state machine. In synchronous 2-counting, the nodes receive a common clock
pulse and they have to agree on which pulses are "odd" and which are "even". We
require that the solution is self-stabilising (reaching the correct operation
from any initial state) and it tolerates Byzantine failures (nodes that
send arbitrary misinformation). Prior algorithms are expensive to implement in
hardware: they require a source of random bits or a large number of states.
This work consists of two parts. In the first part, we use computational
techniques (often known as synthesis) to construct very compact deterministic
algorithms for the first non-trivial case of . While no algorithm exists
for , we show that as few as 3 states per node are sufficient for all
values . Moreover, the problem cannot be solved with only 2 states per
node for , but there is a 2-state solution for all values .
In the second part, we develop and compare two different approaches for
synthesising synchronous counting algorithms. Both approaches are based on
casting the synthesis problem as a propositional satisfiability (SAT) problem
and employing modern SAT-solvers. The difference lies in how to solve the SAT
problem: either in a direct fashion, or incrementally within a counter-example
guided abstraction refinement loop. Empirical results suggest that the former
technique is more efficient if we want to synthesise time-optimal algorithms,
while the latter technique discovers non-optimal algorithms more quickly.Comment: 35 pages, extended and revised versio
Hardware Based Projection onto The Parity Polytope and Probability Simplex
This paper is concerned with the adaptation to hardware of methods for
Euclidean norm projections onto the parity polytope and probability simplex. We
first refine recent efforts to develop efficient methods of projection onto the
parity polytope. Our resulting algorithm can be configured to have either
average computational complexity or worst case
complexity on a serial processor where
is the dimension of projection space. We show how to adapt our projection
routine to hardware. Our projection method uses a sub-routine that involves
another Euclidean projection; onto the probability simplex. We therefore
explain how to adapt to hardware a well know simplex projection algorithm. The
hardware implementations of both projection algorithms achieve area scalings of
at a delay of
. Finally, we present numerical results in
which we evaluate the fixed-point accuracy and resource scaling of these
algorithms when targeting a modern FPGA
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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