By Daniel I.A. Cohen

ISBN-10: 0471035351

ISBN-13: 9780471035350

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The inductive definitions have the following form: If p is a parameter value of S such that \p\ < c then S(p) is defined directly. Otherwise, we define the structures in S(p) by choosing a shorter parameter value p\ and defining truncations and augmentations in such a way that S(p) is equal to the set of augmentations (with respect to p) of structures in S{pi). In general, there is no reason why we should have to limit ourselves to a single shorter parameter value pi. Suppose that p is an arbitrary parameter value of 5 and that \p\ > c.

Since there are many interesting families of structures for which efficient random sampling algorithms are known there are many cases in which we can use the reducer to obtain probabilistic polynomial delay listing algorithms which have exponentially small failure probability. The reducer itself is straightforward. It repeatedly calls S-Sample to generate random structures. It maintains a queue which it uses to store structures that have not yet been output. In addition, it maintains a dictionary which it uses to keep track of the structures that have been put on the queue.

S(P) = I t S(p) . otherwise Let S'-Sample be the following random sampling algorithm for 5': Function S'-S ample (p) s <— S-Sample{p) If (p = p') If (s = -') s <— 5 — 1 (mod2") Return s 34 2. Techniques for Listing Combinatorial Structures We assumed earlier that the function g which is associated with S-Sample is sufficiently large that for every fixed parameter value p the expected running time of S'-Sample is bounded from above by g\. It is easy to see that the bias factor of S'-Sample is bounded from above by 6.

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