Balanced families of perfect hash functions and their applications

Abstract

Abstract. The construction of perfect hash functions is a well-studied topic. In this paper, this concept is generalized with the following definition. We say that a family of functions from [n] to [k] is a δ-balanced (n, k)-family of perfect hash functions if for every S ⊆ [n], |S| = k, the number of functions that are 1-1 on S is between T /δ and δT for some constant T > 0. The standard definition of a family of perfect hash functions requires that there will be at least one function that is 1-1 on S, for each S of size k. In the new notion of balanced families, we require the number of 1-1 functions to be almost the same (taking δ to be close to 1) for every such S. Our main result is that for any constant δ > 1, a δ-balanced (n, k)-family of perfect hash functions of size 2 O(k log log k) log n can be constructed in time 2 O(k log log k) n log n. Using the technique of color-coding we can apply our explicit constructions to devise approximation algorithms for various counting problems in graphs. In particular, we exhibit a deterministic polynomial time algorithm for approximating both the number of simple paths of length k and the number of simple cycles of size k for any k ≤ O( log n log log log n ) in a graph with n vertices. The approximation is up to any fixed desirable relative error.

Approximating the Number of Network Motifs

“…Several existing algorithms for counting and detection non-induced motifs [6,4,2,1,16] used the color coding technique of Alon et al [3]. Color coding is an innovative combinatorial approach that was introduced by Alon et al [3] to detect simple paths, trees and bounded treewidth subgraphs in unlabeled graphs.…”

“…Dost et al [6] showed how to solve the subgraph detection problem for subgraphs of size O(log n), provided that the query subgraph is either a simple path, a tree, or a bounded treewidth subgraph. Arvind and Raman [4] counted the number of subgraphs in a given graph G which are isomorphic to a bounded treewidth graph H. They gave a randomized approximate counting algorithm with a running time of k O(k) · n b+O (1) , where n and k are the number of vertices in G and H, respectively, and b is the treewidth of H. Alon and Gutner [2] combined the color coding technique with a construction of Balanced Families of Perfect Hash Functions to obtain a deterministic algorithm to count the number of simple paths or cycles of size k in an input graph G. Alon et al [1] improved the algorithm of Alon and Gutner. They presented a polynomial time algorithm for approximating the number of non-induced occurrences of trees and bounded treewidth subgraphs with k = O(log n) vertices with a running time of 2 O(k log log k) · n O (1) .…”

Counting Thin Subgraphs via Packings Faster than Meet-in-the-Middle Time

“…The possibility to split into two controllable parts immediately suggests that one should pursue an algorithm that runs in no worse time than n k/2+O (1) ; such an algorithm was indeed discovered in 2009 by Vassilevska and Williams [26] for counting k-vertex subgraphs that admit an independent set of size k/2. This result was accompanied, within the same year, of two publications presenting the same runtime restricted to counting paths and matchings.…”

“…Fomin et al [13] generalized the latter result into an algorithm that counts occurrences of a k-vertex pattern graph with pathwidth p in n k/2+2p+O(1) time. Splitting into three parts enables faster listing of the parts in n k/3+O (1) time, but requires more elaborate control at the interface between parts. This strategy enables one to count also dense subgraphs such as k-cliques via an algorithm of Nešetřil and Poljak [24] (see also [11,18]) that uses fast matrix multiplication to achieve a pairwise join of the three parts, resulting in running time n ωk/3+O(1) , where 2 ≤ ω < 2.3728639 is the limiting exponent of square matrix multiplication [22,27].…”

“…This strategy enables one to count also dense subgraphs such as k-cliques via an algorithm of Nešetřil and Poljak [24] (see also [11,18]) that uses fast matrix multiplication to achieve a pairwise join of the three parts, resulting in running time n ωk/3+O(1) , where 2 ≤ ω < 2.3728639 is the limiting exponent of square matrix multiplication [22,27]. Even in the case ω = 2 this running time is, however, n 2k/3+O (1) , which is inferior to "meeting in the middle" by splitting into two parts.…”