Balanced Hashing, Color Coding and Approximate Counting

Color Coding is an algorithmic technique for deciding efficiently if a given input graph contains a path of a given length (or another small subgraph of constant tree-width). Applications of the method in computational biology motivate the study of similar algorithms for counting the number of copies of a given subgraph. While it is unlikely that exact counting of this type can be performed efficiently, as the problem is #W [1]-complete even for paths, approximate counting is possible, and leads to the investigation of an intriguing variant of families of perfect hash functions. A family of functions from [n] to [k] is an ( , k)-balanced family of hash functions, if there exists a positive T so that for every K ⊂ [n] of size |K| = k, the number of functions in the family that are one-to-one on K is between (1− )T and (1 + )T . The family is perfectly k-balanced if it is (0, k)-balanced. We show that every such perfectly k-balanced family is of size at least c(k)nbk/2c, and that for every > 1 poly(k) there are explicit constructions of ( , k)-balanced families of hash functions from [n] to [k] of size e log n. This is tight up to the o(1)-term in the exponent, and supplies deterministic polynomial time algorithms for approximately counting the number of paths or cycles of a specified length k (or copies of any graph H with k vertices and bounded tree-width) in a given input graph of size n, up to relative error , for all k ≤ O(log n).