By Leslie Ann Goldberg
This thesis is anxious with the layout of effective algorithms for directory combinatorial constructions. The learn defined right here supplies a few solutions to the subsequent questions: which households of combinatorial buildings have quickly computing device algorithms for directory their contributors, What normal equipment are necessary for directory combinatorial buildings, How can those be utilized to these households which are of curiosity to theoretical machine scientists and combinatorialists? between these households thought of are unlabeled graphs, first-order one houses, Hamiltonian graphs, graphs with cliques of targeted order, and k-colorable graphs. a few similar paintings is usually incorporated that compares the directory challenge with the trouble of fixing the lifestyles challenge, the development challenge, the random sampling challenge, and the counting challenge. particularly, the trouble of comparing Polya's cycle polynomial is confirmed.
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Extra info for Efficient Algorithms for Listing Combinatorial Structures (Distinguished Dissertations in Computer Science)
Let A be the listing algorithm that we get by combining Uniform Reducer 2 with S-Sample. Let d denote the delay of A and let r denote the space complexity of A. We know from the proof of theorem 2 that d is bounded from above by a polynomial in \p\ and that there is a polynomial q such that r(p) = 0(^(|p|) x |S(p)|). We know from theorem 3 that there is a polynomial q such that d(p)r(p) = £l(\S(p)\ / q'(\p\)). Therefore, the space complexity of algorithm A could only be improved by a polynomial factor without increasing the delay.
The functions g and b are large enough that they can be associated with a modified random sampling algorithm, S'-Sample. 2. S-Sample has bias factor 1. (We call a random sampling algorithm unbiased if its bias factor is 1. ) Using the same arguments that we used to prove theorem 3 we could prove the following stronger theorem. Theorem 3 (strengthened). Let U be any uniform reducer from probabilistic listing to efficient random sampling. Let S-Sample be any unbiased efficient random sampling algorithm for any simple family S.
The restriction that we have placed on the size of integers that can be stored in random access machine registers ensures that there is a polynomial q such that 4 |JD(p)| < 2q"p"r(p' for every parameter value p. Let m be a function satisfying 1 < m(p) < \S(p)\. We will construct the set \I>(p) as follows. An ra(p)-element set U C S(p) is a member of \I>(p) if and only if the probability that a given run of A with input p lists all of the structures in S(p) conditioned on the fact that U is the ra(p)-element starting set of the run is at least 1/4.