Douglas S. Stones

Efficient parallel lists intersection and index compression algorithms using graphics processing units

Major web search engines answer thousands of queries per second requesting information about billions of web pages. The data sizes and query loads are growing at an exponential rate. To manage the heavy workload, we consider techniques for utilizing a Graphics Processing Unit (GPU). We investigate new approaches to improve two important operations of search engines -- lists intersection and index compression. For lists intersection, we develop techniques for efficient implementation of the binary search algorithm for parallel computation. We inspect some representative real-world datasets and find that a sufficiently long inverted list has an overall linear rate of increase. Based on this observation, we propose Linear Regression and Hash Segmentation techniques for contracting the search range. For index compression, the traditional d-gap based compression schemata are not well-suited for parallel computation, so we propose a Linear Regression Compression schema which has an inherent parallel structure. We further discuss how to efficiently intersect the compressed lists on a GPU. Our experimental results show significant improvements in the query processing throughput on several datasets.

NetMODE: network motif detection without Nauty.

A motif in a network is a connected graph that occurs significantly more frequently as an induced subgraph than would be expected in a similar randomized network. By virtue of being atypical, it is thought that motifs might play a more important role than arbitrary subgraphs. Recently, a flurry of advances in the study of network motifs has created demand for faster computational means for identifying motifs in increasingly larger networks. Motif detection is typically performed by enumerating subgraphs in an input network and in an ensemble of comparison networks; this poses a significant computational problem. Classifying the subgraphs encountered, for instance, is typically performed using a graph canonical labeling package, such as Nauty, and will typically be called billions of times. In this article, we describe an implementation of a network motif detection package, which we call NetMODE. NetMODE can only perform motif detection for [Formula: see text]-node subgraphs when [Formula: see text], but does so without the use of Nauty. To avoid using Nauty, NetMODE has an initial pretreatment phase, where [Formula: see text]-node graph data is stored in memory ([Formula: see text]). For [Formula: see text] we take a novel approach, which relates to the Reconstruction Conjecture for directed graphs. We find that NetMODE can perform up to around [Formula: see text] times faster than its predecessors when [Formula: see text] and up to around [Formula: see text] times faster when [Formula: see text] (the exact improvement varies considerably). NetMODE also (a) includes a method for generating comparison graphs uniformly at random, (b) can interface with external packages (e.g. R), and (c) can utilize multi-core architectures. NetMODE is available from netmode.sf.net.

On the number of Latin rectangles

This thesis primarily investigates the number Rk,n of reduced k X n Latin rectangles. Specifically, we find many congruences that involve Rk,n with the aim of improving our understanding of Rk,n. In general, the problem of finding Rk,n is difficult and furthermore, the literature contains many published errors. Modern enumeration algorithms, such as that of McKay and Wanless, require lengthy computations and storage of a large amount of data. Consequently, even into the future, the possibility of obtaining an erroneous result remains, for example, through a hardware or bookkeeping error. In this thesis we find many congruences satisfied by Rk,n so that future researchers will be able to check that their purported value of Rk,n satisfies these congruences. We extend the methodology developed in this thesis to encompass the number of certain graph factorisations, the number of orthomorphisms and partial orthomorphisms and the size of certain subsets of Latin hypercuboids. Consequently we find new congruences satisfied by all these numbers. Additionally, we give new sufficient conditions for when a partial orthomorphism admits a completion to an orthomorphism. In a 1997 paper, Drisko suggested some ideas for future research in the study of the Alon-Tarsi Conjecture, which we show to be futile. We find a new bound on the maximum size of an autotopism group of a Latin square which enables us to find new divisors of Rn,n for large n. A similar method gives a bound on the maximum number of k X k subsquares in a Latin square, for general k. Finally, we find new strong necessary conditions for when an isotopism can be an autotopism of some Latin square.

How not to prove the Alon-Tarsi conjecture

The sign of a Latin square is −1 if it has an odd number of rows and columns that are odd permutations; otherwise, it is +1. Let L E n and L o n be, respectively, the number of Latin squares of order n with sign +1 and −1. The Alon-Tarsi conjecture asserts that L E n ≠ L o n when n is even. Drisko showed that L E p+1 ≢ L o p+1 (mod p 3 ) for prime p ≥ 3 and asked if similar congruences hold for orders of the form p k + 1, p + 3, or pq + 1. In this article we show that if t ≤ n , then L E n+1 ≢ L 0 n+1 (mod t 3 ) only if t = n and n is an odd prime, thereby showing that Drisko’s method cannot be extended to encompass any of the three suggested cases. We also extend exact computation to n ≤ 9, discuss asymptotics for L o /L E , and propose a generalization of the Alon-Tarsi conjecture.

Bounds on the number of autotopisms and subsquares of a Latin square

A subsquare of a Latin square L is a submatrix that is also a Latin square. An autotopism of L is a triplet of permutations (α, β, γ) such that L is unchanged after the rows are permuted by α, the columns are permuted by β and the symbols are permuted by γ. Let n!(n−1)!Rn be the number of n×n Latin squares. We show that an n×n Latin square has at most nO(log k) subsquares of order k and admits at most nO(log n) autotopisms. This enables us to show that {ie11-1} divides Rn for all primes p. We also extend a theorem by McKay and Wanless that gave a factorial divisor of Rn, and give a new proof that Rp≠1 (mod p) for prime p.

Symmetries of partial Latin squares

Abstract In this paper, we study symmetries (autoparatopisms) of partial Latin squares. Let s ( n ) be the minimum number of non-empty cells in a partial Latin square of order n with a trivial autoparatopism group. We show 1 5 ( 6 n − 7 ) ≤ s ( n ) ≤ 1 2 ( 3 n − 3 ) for all n ≥ 5 . We also show that, if G is a finite group, then there exists a partial Latin square whose autoparatopism group is isomorphic to G (as are its autotopism and automorphism groups). Computational methods are also introduced, and are used to study symmetries of partial Latin squares of small orders; the source code has been made available as supplementary material.

Exploiting query term correlation for list caching in web search engines

Caching technologies have been widely employed to boost the performance of Web search engines. Motivated by the correlation between terms in query logs from a commercial search engine, we explore the idea of a caching scheme based on pairs of terms, rather than individual terms (which is the typical approach used by search engines today). We propose an inverted list caching policy, based on the Least Recently Used method, in which the co-occurring correlation between terms in the query stream is accounted for when deciding on which terms to keep in the cache. We consider not only the term co-occurrence within the same query but also the co-occurrence between separate queries. Experimental results show that the proposed approach can improve not only the cache hit ratio but also the overall throughput of the system when compared to existing list caching algorithms.