Paul Walton Purdom

Search rearrangement backtracking and polynomial average time

The average time required for simple search rearrangement backtracking is compared with that for ordinary backtracking when each algorithm is used to find all solutions for random conjunctive normal form predicates. The sets of random predicates are characterized by v: the number of variables, t(v): the number of clauses, and p(v): the probability that a literal appears in clause. For large v if vp(v) 1n 2, there is a difficult region where the average number of solutions per problem is exponentially small, but backtracking requires an exponentially large time. The difficult region for search rearrangement backtracking is only slightly smaller than the difficult region for ordinary backtracking. It is conjectured that search rearrangement backtracking is exponentially faster than ordinary backtracking for nearly all of the difficult region. It is proved that there is no major advantage in using search rearrangement backtracking outside of the difficult region.

A sentence generator for testing parsers

A fast algorithm is given to produce a small set of short sentences from a context free grammar such that each production of the grammar is used at least once. The sentences are useful for testing parsing programs and for debugging grammars (finding errors in a grammar which causes it to specify some language other than the one intended). Some experimental results from using the sentences to test some automatically generated simpleLR(1) parsers are also given.

Tree Size by Partial Backtracking

Knuth [1] recently showed how to estimate the size of a backtrack tree by repeatedly following random paths from the root. Often the efficiency of his method can be greatly improved by occasionally following more than one path from a node. This results in estimating the size of the backtrack tree by doing a very abbreviated partial backtrack search. An analysis shows that this modification results in an improvement which increases exponentially with the height of the tree. Experimental results for a particular tree of height 84 show an order of magnitude improvement. The measuring method is easy to add to a backtrack program.

The Pure Literal Rule and Polynomial Average Time

For a simple parameterized model of conjunctive normal form predicates, we show that a simplified version of the Davis–Putnam procedure can, for many values of the parameters, solve the satisfiability problem in polynomial average time. Let v be the number of variables,

Average-Case Performance of the Apriori Algorithm

t(v)

Polynomial-average-time satisfiability problems

the number of clauses in a predicate, and

Backtracking with multi-level dynamic search rearrangement

p(v)

Solving Satisfiability with Less Searching

the probability that a given literal appears in a clause (

A new base change algorithm for permutation groups

p(v)

An Analysis of Backtracking with Search Rearrangement

is the same for all literals). Let