Engineering Parallel String Sorting

We discuss how string sorting algorithms can be parallelized on modern multi-core shared memory machines. As a synthesis of the best sequential string sorting algorithms and successful parallel sorting algorithms for atomic objects, we first propose string sample sort. The algorithm makes effective use of the memory hierarchy, uses additional word level parallelism, and largely avoids branch mispredictions. Then we focus on NUMA architectures, and develop parallel multiway LCP-merge and -mergesort to reduce the number of random memory accesses to remote nodes. Additionally, we parallelize variants of multikey quicksort and radix sort that are also useful in certain situations. Comprehensive experiments on five current multi-core platforms are then reported and discussed. The experiments show that our implementations scale very well on real-world inputs and modern machines.Comment: 46 pages, extension of "Parallel String Sample Sort" arXiv:1305.115

Online Data Structures in External Memory

The original publication is available at www.springerlink.comThe data sets for many of today's computer applications are too large to t within the computer's internal memory and must instead be stored on external storage devices such as disks. A major performance bottleneck can be the input/output communication (or I/O) between the external and internal memories. In this paper we discuss a variety of online data structures for external memory, some very old and some very new, such as hashing (for dictionaries), B-trees (for dictionaries and 1-D range search), bu er trees (for batched dynamic problems), interval trees with weight-balanced B-trees (for stabbing queries), priority search trees (for 3-sided 2-D range search), and R-trees and other spatial structures. We also discuss several open problems along the way

Algorithm Libraries for Multi-Core Processors

By providing parallelized versions of established algorithm libraries, we ease the exploitation of the multiple cores on modern processors for the programmer. The Multi-Core STL provides basic algorithms for internal memory, while the parallelized STXXL enables multi-core acceleration for algorithms on large data sets stored on disk. Some parallelized geometric algorithms are introduced into CGAL. Further, we design and implement sorting algorithms for huge data in distributed external memory

Parallel String Sample Sort

We discuss how string sorting algorithms can be parallelized on modern multi-core shared memory machines. As a synthesis of the best sequential string sorting algorithms and successful parallel sorting algorithms for atomic objects, we propose string sample sort. The algorithm makes effective use of the memory hierarchy, uses additional word level parallelism, and largely avoids branch mispredictions. Additionally, we parallelize variants of multikey quicksort and radix sort that are also useful in certain situations.Comment: 34 pages, 7 figures and 12 table

Approximately Optimum Search Trees in External Memory Models

We examine optimal and near optimal solutions to the classic binary search tree problem of Knuth. We are given a set of n keys (originally known as words), B_1, B_2, ..., B_n and 2n+1 frequencies. {p_1, p_2, ..., p_n} represent the probabilities of searching for each given key, and {q_0, q_1, ..., q_n} represent the probabilities of searching in the gaps between and outside of these keys. We have that Σ_{i=0}^n q_i + Σ_{i=1}^n p_i = 1. We also assume without loss of generality that q_{i-1}+p_i+q_i != 0 for any i ϵ {1,...,n}. The keys must make up the internal nodes of the tree while the gaps make up the leaves. Our goal is to construct a binary search tree such that expected cost of search is minimized. First, we re-examine an approximate solution of Guttler, Mehlhorn and Schneider which was shown to have a worst case bound of c * H + 2 where c >= 1/(H(1/3,2/3)) ~ 1.08, and H = Σ_{i=1}^{n} p_i * log_2(1/p_i) + Σ_{j=0}^{n} q_i * log_2(1/q_j) is the entropy of the distribution. We give an improved worst case bound on the heuristic of H+4. Next, we examine the optimum binary search tree problem under a model of external memory. We use the Hierarchical Memory Model of Aggarwal et al. The model has an unlimited number of registers, R_1, R_2, ... each with its own location in memory (a positive integer). We have a set of memory sizes m_1, m_2, ..., m_l which are monotonically increasing. Each memory level has a finite size except m_l which we assume has infinite size. Each memory level has an associated cost of access c_1, c_2, ..., c_l. We assume that c_1 < c_2 < ... < c_l. We propose two approximate solutions which run in O(n) time where n is the number of words in our data set. Using these methods, we improve upon a bound given in Thite's 2001 thesis under the related HMM_2 model in the approximate setting. We also examine the related problem of binary trees on multisets of probabilities where keys are unordered and we do not differentiate between which probabilities must be leaves, and which must be internal nodes. We provide a simple O(n log_2(n)) algorithm that is within an additive (n+1)(2n) of optimal on a multiset of n keys

Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal

The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most $k$ of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a \BigOh(4^kkmn) time algorithm for it, the first algorithm with polynomial runtime of uniform degree for every fixed $k$. It is known that this implies a polynomial-time compression algorithm that turns OCT instances into equivalent instances of size at most \BigOh(4^k), a so-called kernelization. Since then the existence of a polynomial kernel for OCT, i.e., a kernelization with size bounded polynomially in $k$, has turned into one of the main open questions in the study of kernelization. This work provides the first (randomized) polynomial kernelization for OCT. We introduce a novel kernelization approach based on matroid theory, where we encode all relevant information about a problem instance into a matroid with a representation of size polynomial in $k$. For OCT, the matroid is built to allow us to simulate the computation of the iterative compression step of the algorithm of Reed, Smith, and Vetta, applied (for only one round) to an approximate odd cycle transversal which it is aiming to shrink to size $k$. The process is randomized with one-sided error exponentially small in $k$, where the result can contain false positives but no false negatives, and the size guarantee is cubic in the size of the approximate solution. Combined with an \BigOh(\sqrt{\log n})-approximation (Agarwal et al., STOC 2005), we get a reduction of the instance to size \BigOh(k^{4.5}), implying a randomized polynomial kernelization.Comment: Minor changes to agree with SODA 2012 version of the pape