4,648 research outputs found
On the Complexity of Polytope Isomorphism Problems
We show that the problem to decide whether two (convex) polytopes, given by
their vertex-facet incidences, are combinatorially isomorphic is graph
isomorphism complete, even for simple or simplicial polytopes. On the other
hand, we give a polynomial time algorithm for the combinatorial polytope
isomorphism problem in bounded dimensions. Furthermore, we derive that the
problems to decide whether two polytopes, given either by vertex or by facet
descriptions, are projectively or affinely isomorphic are graph isomorphism
hard.
The original version of the paper (June 2001, 11 pages) had the title ``On
the Complexity of Isomorphism Problems Related to Polytopes''. The main
difference between the current and the former version is a new polynomial time
algorithm for polytope isomorphism in bounded dimension that does not rely on
Luks polynomial time algorithm for checking two graphs of bounded valence for
isomorphism. Furthermore, the treatment of geometric isomorphism problems was
extended.Comment: 16 pages; to appear in: Graphs and Comb.; replaces our paper ``On the
Complexity of Isomorphism Problems Related to Polytopes'' (June 2001
Delay-Doppler Channel Estimation with Almost Linear Complexity
A fundamental task in wireless communication is Channel Estimation: Compute
the channel parameters a signal undergoes while traveling from a transmitter to
a receiver. In the case of delay-Doppler channel, a widely used method is the
Matched Filter algorithm. It uses a pseudo-random sequence of length N, and, in
case of non-trivial relative velocity between transmitter and receiver, its
computational complexity is O(N^{2}log(N)). In this paper we introduce a novel
approach of designing sequences that allow faster channel estimation. Using
group representation techniques we construct sequences, which enable us to
introduce a new algorithm, called the flag method, that significantly improves
the matched filter algorithm. The flag method finds the channel parameters in
O(mNlog(N)) operations, for channel of sparsity m. We discuss applications of
the flag method to GPS, radar system, and mobile communication as well.Comment: 11 page
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