On the Structure of Sequential Search: Beyond “No Free Lunch”

Abstract

Novel results are obtained by investigating the search algorithms predicated in “no free lunch” (NFL) theorems rather than NFL itself. Finite functions are represented as strings of values. The set of functions is partitioned into maximal blocks of functions that are permutations of one another. A search algorithm effectively maps each test function to a permutation of the function. This mapping is partitioned into one-to-one correspondences on blocks. A distribution on the functions is referred to as block uniform if each function has the same probability as all its permutations. For any search algorithm, the distributions of test functions and permuted test functions have identical Kullback-Leibler distance to the nearest block uniform distribution. That is, divergence from block uniformity is conserved by search. There is NFL in the special case of no divergence, i.e., the distributions are block uniform.