Finding succinct ordered minimal perfect hashing functions

An ordered minimal perfect hash table is one in which no collisions occur among a predefined set of keys, no space is unused, and the data are placed in the table in order. A new method for creating ordered minimal perfect hashing functions is presented. The method presented is based on a method developed by Fox, Heath, Daoud, and Chen, but it creates hash functions with representation space requirements closer to the theoretical lower bound. The method presented requires approximately 10% less space to represent generated hash functions, and is easier to implement than Fox et al's. However, a higher time complexity makes it practical for small sets only (< 1000)

GPERF : a perfect hash function generator

gperf is a widely available perfect hash function generator written in C++. It automates a common system software operation: keyword recognition. gperf translates an n element user-specified keyword list keyfile into source code containing a k element lookup table and a pair of functions, phash and in_word_set. phash uniquely maps keywords in keyfile onto the range 0 .. k - 1, where k >/= n. If k = n, then phash is considered a minimal perfect hash function. in_word_set uses phash to determine whether a particular string of characters str occurs in the keyfile, using at most one string comparison.This paper describes the user-interface, options, features, algorithm design and implementation strategies incorporated in gperf. It also presents the results from an empirical comparison between gperf-generated recognizers and other popular techniques for reserved word lookup

Simple, compact and robust approximate string dictionary

This paper is concerned with practical implementations of approximate string dictionaries that allow edit errors. In this problem, we have as input a dictionary $D$ of $d$ strings of total length $n$ over an alphabet of size $\sigma$. Given a bound $k$ and a pattern $x$ of length $m$, a query has to return all the strings of the dictionary which are at edit distance at most $k$ from $x$, where the edit distance between two strings $x$ and $y$ is defined as the minimum-cost sequence of edit operations that transform $x$ into $y$. The cost of a sequence of operations is defined as the sum of the costs of the operations involved in the sequence. In this paper, we assume that each of these operations has unit cost and consider only three operations: deletion of one character, insertion of one character and substitution of a character by another. We present a practical implementation of the data structure we recently proposed and which works only for one error. We extend the scheme to $2\leq k<m$. Our implementation has many desirable properties: it has a very fast and space-efficient building algorithm. The dictionary data structure is compact and has fast and robust query time. Finally our data structure is simple to implement as it only uses basic techniques from the literature, mainly hashing (linear probing and hash signatures) and succinct data structures (bitvectors supporting rank queries).Comment: Accepted to a journal (19 pages, 2 figures

Fast Scalable Construction of (Minimal Perfect Hash) Functions

Recent advances in random linear systems on finite fields have paved the way for the construction of constant-time data structures representing static functions and minimal perfect hash functions using less space with respect to existing techniques. The main obstruction for any practical application of these results is the cubic-time Gaussian elimination required to solve these linear systems: despite they can be made very small, the computation is still too slow to be feasible. In this paper we describe in detail a number of heuristics and programming techniques to speed up the resolution of these systems by several orders of magnitude, making the overall construction competitive with the standard and widely used MWHC technique, which is based on hypergraph peeling. In particular, we introduce broadword programming techniques for fast equation manipulation and a lazy Gaussian elimination algorithm. We also describe a number of technical improvements to the data structure which further reduce space usage and improve lookup speed. Our implementation of these techniques yields a minimal perfect hash function data structure occupying 2.24 bits per element, compared to 2.68 for MWHC-based ones, and a static function data structure which reduces the multiplicative overhead from 1.23 to 1.03

Efficient Gauss Elimination for Near-Quadratic Matrices with One Short Random Block per Row, with Applications

In this paper we identify a new class of sparse near-quadratic random Boolean matrices that have full row rank over F_2 = {0,1} with high probability and can be transformed into echelon form in almost linear time by a simple version of Gauss elimination. The random matrix with dimensions n(1-epsilon) x n is generated as follows: In each row, identify a block of length L = O((log n)/epsilon) at a random position. The entries outside the block are 0, the entries inside the block are given by fair coin tosses. Sorting the rows according to the positions of the blocks transforms the matrix into a kind of band matrix, on which, as it turns out, Gauss elimination works very efficiently with high probability. For the proof, the effects of Gauss elimination are interpreted as a ("coin-flipping") variant of Robin Hood hashing, whose behaviour can be captured in terms of a simple Markov model from queuing theory. Bounds for expected construction time and high success probability follow from results in this area. They readily extend to larger finite fields in place of F_2. By employing hashing, this matrix family leads to a new implementation of a retrieval data structure, which represents an arbitrary function f: S -> {0,1} for some set S of m = (1-epsilon)n keys. It requires m/(1-epsilon) bits of space, construction takes O(m/epsilon^2) expected time on a word RAM, while queries take O(1/epsilon) time and access only one contiguous segment of O((log m)/epsilon) bits in the representation (O(1/epsilon) consecutive words on a word RAM). The method is readily implemented and highly practical, and it is competitive with state-of-the-art methods. In a more theoretical variant, which works only for unrealistically large S, we can even achieve construction time O(m/epsilon) and query time O(1), accessing O(1) contiguous memory words for a query. By well-established methods the retrieval data structure leads to efficient constructions of (static) perfect hash functions and (static) Bloom filters with almost optimal space and very local storage access patterns for queries

Multiparticle entanglement purification for two-colorable graph states

We investigate multiparticle entanglement purification schemes which allow one to purify all two colorable graph states, a class of states which includes e.g. cluster states, GHZ states and codewords of various error correction codes. The schemes include both recurrence protocols and hashing protocols. We analyze these schemes under realistic conditions and observe for a generic error model that the threshold value for imperfect local operations depends on the structure of the corresponding interaction graph, but is otherwise independent of the number of parties. The qualitative behavior can be understood from an analytically solvable model which deals only with a restricted class of errors. We compare direct multiparticle entanglement purification protocols with schemes based on bipartite entanglement purification and show that the direct multiparticle entanglement purification is more efficient and the achievable fidelity of the purified states is larger. We also show that the purification protocol allows one to produce private entanglement, an important aspect when using the produced entangled states for secure applications. Finally we discuss an experimental realization of a multiparty purification protocol in optical lattices which is issued to improve the fidelity of cluster states created in such systems.Comment: 22 pages, 8 figures; replaced with published versio