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.

Quantum Overlapping Tomography

“…Finding (n, k) families with this T -property is difficult, but there is a useful approximate notion. A (δ, T )-balanced (n, k) family of perfect hash functions f 1 , ..., f N has the property that for every S in [n] where |S| = k, there are between T /δ and δT functions which are 1-to-1 on S [25]. For any given δ > 1, [25] provides a construction of a (δ, T )-balanced (n, k) family of size e O(k log log(k)) log(n), where T is determined by the construction.…”

“…A (δ, T )-balanced (n, k) family of perfect hash functions f 1 , ..., f N has the property that for every S in [n] where |S| = k, there are between T /δ and δT functions which are 1-to-1 on S [25]. For any given δ > 1, [25] provides a construction of a (δ, T )-balanced (n, k) family of size e O(k log log(k)) log(n), where T is determined by the construction. This would allow for QOT with (M/T ) e O(k log log(k)) log(n) measurements, which in certain parameter regimes would require less measurements than the non-balanced case.…”

The challenges of unbounded treewidth in parameterised subgraph counting problems

“…Section 4 then contains the proof of our new complexity results, and finally in Section 5 we review the limitations of the existing methods and suggest directions for future research. 3…”

Waring Rank, Parameterized and Exact Algorithms

“…The next fact essentially appears in [AG07]; we reproduce the proof for completeness. Here (n) k := n(n − 1) · · · (n − k + 1) denotes the falling factorial.…”