On Efficient Parallel Algorithms for Solving Set Recurrence Equations

We present an O((log log N)/sup 2/) -time algorithm for computing the distance transform of an N /spl times/ N binary image. Our algorithm is designed for the common concurrent read concurrent write parallel random access machine (CRCW PRAM) and requires O(N/sup 2+/spl epsi///log log N) processors, for any /spl epsi/ such that 0 < /spl epsi/ < 1. Our algorithm is based on a novel deterministic sampling scheme and can be used for computing distance transforms for a very general class of distance functions. We also present a scalable version of our algorithm when the number of processors is available p/sup 2+/spl epsi///log log p for some p < N. In this case, our algorithm runs in O((N/sup 2//p/sup 2/)+(N/p) log log p + (log log p)/sup 2/) time. This scalable algorithm is more practical since usually the number of available processors is much less than the size of the image.

In this paper we prove lower bounds on the computational complexity of symmetric functions on parallel machines. The parallel models considered are PRIORITY and ARBITRARY, both widely used versions of the Parallel Random Access Machine (PRAM). We first consider PRIORITY. We prove that a PRIORITY PRAM with one shared memory cell and O(n) processors requires Ω(g(n)) time in order to decide if a binary vector of length n has at least g(n) 1's, for the case g(n)=o(n1/4). This is the decision problem for the threshold language Lg. Our lower bound is tight. For PRIORITY with m shared memory cells we prove an Ω(g(n)/m) lower bound. The limitation on the number of processors is essential, since with enough processors Lg can be decided by the same model in O(√g(n)) time. The limitation on g is important: for instance, for g(n)=n/2 Lg can be decided in time O(√g(n)), which is optimal [VW].

Simulating the CRCW PRAM on reconfigurable networks

Observations on log ( n) time parallel recognition of unambiguous cfl's

Frequently found implementations of the parallel random access machine (PRAM) include the common-bus multiprocessor as well as shared-memory machines based on several types of switching networks. The performance of the common-bus multiprocessor decreases as the total number of required data exchanges among processors, for a given computation, grows. The same applies to switching networks but, because of the higher number of available data paths, to a lesser degree. In this sense, the advantages of the crossbar network as compared to other switching networks are well known. However, the high number of switching elements required by this network has been a limitation on its use in practical machines. In this paper, a variation of the crossbar network based on m(m-l) switching elements, where m is the number of processors, is discussed. The architecture of the machine is described, and the design of the chip set of the machine, based on field-programmable gate arrays (FPGA's), is presented. A brief description of the functions implemented in each device is given, followed by a detailed discussion of the switching elements, and, in particular, of the access priority arbitration function implemented in the device. Performance estimations of an 8-processor machine are presented and compared with a common-bus multiprocessor design using similar technology. It is concluded that the crossbar network represents a technologically viable alternative to other implementations of limited parallelism.

Parallel time O(log n) recognition of unambiguous context-free languages

The authors describe a novel parallel algorithm for discrete-event simulation on an exclusive-read exclusive-write parallel random-access machine (EREW PRAM). This algorithm is based on a recently developed parallel data structure, namely the parallel heap, which, when used as a parallel event-queue, allows deletion of O(p) earliest messages and insertion of O(p) new messages each in O(log n) time, using p processors, where n is the number of messages scheduled. The proposed algorithm can simulate up to p independent messages in O(log n) time, thus achieving O(p) speedup. The number of processors, p, can be optimally varied in the range 1 >

This paper introduces a novel parallel data structure called the recursive star-tree (denoted “${}^ * $-tree”). For its definition a generalization of the $ * $ functional is used (where for a function $f * f(n) = \min \{ {i|f^{(i)} (n) \leqslant 1} \}$ and $f^{(i)} $ is the ith iterate of f). Recursive ${}^ * $-trees are derived by using recursion in the spirit of the inverse Ackermann function. The recursive ${}^ * $-tree data structure leads to a new design paradigm for parallel algorithms. This paradigm allows for extremely fast parallel computations, specifically, $O(\alpha (n))$ time (where $\alpha (n)$ is the inverse of the Ackermann function), using an optimal number of processors on the (weakest) concurrent-read, concurrent-write parallel random-access machine (CRCW PRAM). These computations need only constant time, and use an optimal number of processors if the following nonstandard assumption about the model of parallel computation is added to the CRCW PRAM: an extremely small number of processors each can write simultaneously into different bits of the same word. Applications include finding lowest common ancestors in trees by a new algorithm that is considerably simpler than the known algorithms for the problem, restricted domain merging, parentheses matching, and a new parallel reducibility.

Article Free Access Share on Limits on the power of concurrent-write parallel machines Author: P Beame Dept. of Computer Science, University of Toronto, Toronto, Canada, M5S 1A4 Dept. of Computer Science, University of Toronto, Toronto, Canada, M5S 1A4View Profile Authors Info & Claims STOC '86: Proceedings of the eighteenth annual ACM symposium on Theory of computingNovember 1986 Pages 169–176https://doi.org/10.1145/12130.12147Online:01 November 1986Publication History 11citation229DownloadsMetricsTotal Citations11Total Downloads229Last 12 Months5Last 6 weeks2 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

An Asynchronous PRAM allows processors to run at different and unpredictable speeds. Thus a fundamental problem in designing asynchronous PRAM algorithms is constructing a synchronization primitive which determines that a set of tasks has been completed. The Certified Write All Problem (CWA) is: given an array A[1..n] and a flag f which are both initialized to zero, set all elements of A to one, and then set f to one. A solution to the Certified Write All problem can be used as a synchronization primitive in a wide variety of settings. This paper investigates the complexity of CWA algorithms by presenting several new algorithms and lower bound proofs.We present a new randomized CWA algorithm which uses expected O(n) work using up to n/log n processors. We show that this algorithm is optimal in both work and processor utilization by proving an Ω(n + p log n) lower bound on the expected work done by a p processor randomized CWA algorithm. Our CWA algorithm uses concurrent reads and concurrent writes. We show that this is necessary by proving that no concurrent read exclusive write (CREW) asynchronous PRAM can solve the CWA problem. However, for a fail-stop PRAM, where processors operate synchronously until they fail, we present a randomized CREW CWA algorithm. This algorithm also uses O(n) expected work using up to n/log n CREW fail-stop processors.KeywordsLeaf NodeProcessor UtilizationRandom IntegerSpeed FunctionComplete Binary TreeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The fastest deterministic algorithms for connected components take logarithmic time and perform superlinear work on a Parallel Random Access Machine (PRAM). These algorithms maintain a spanning forest by merging and compressing trees, which requires pointer-chasing operations that increase memory access latency and are limited to shared-memory systems. Many of these PRAM algorithms are also very complicated to implement. Another popular method is “leader-contraction” where the challenge is to select a constant fraction of leaders that are adjacent to a constant fraction of non-leaders with high probability, but this can require adding more edges than were in the original graph. Instead we investigate label propagation because it is deterministic, easy to implement, and does not rely on pointer-chasing. Label propagation exchanges representative labels within a component using simple graph traversal, but it is inherently difficult to complete in a sublinear number of steps. We are able to overcome the problems with label propagation for graph connectivity. We introduce a surprisingly simple framework for deterministic, undirected graph connectivity using label propagation that is easily adaptable to many computational models. It achieves logarithmic convergence independently of the number of processors and without increasing the edge count. We employ a novel method of propagating directed edges in alternating direction while performing minimum reduction on vertex labels. We present new algorithms in PRAM, Stream, and MapReduce. Given a simple, undirected graph [Formula: see text] with [Formula: see text] vertices, [Formula: see text] edges, our approach takes O(m) work each step, but we can only prove logarithmic convergence on a path graph. It was conjectured by Liu and Tarjan (2019) to take [Formula: see text] steps or possibly [Formula: see text] steps. Our experiments on a range of difficult graphs also suggest logarithmic convergence. We leave the proof of convergence as an open problem.

In a 2-dimensional (2D) pattern matching problem, the text is arranged as a matrix 𝖬[1..n, 1..n] and consists of N = n × n symbols drawn from alphabet set Σ of size σ. The query consists of a m × m square matrix 𝖯[1..m, 1..m] drawn from the same alphabet set Σ and the task is to find all the locations in 𝖬 where 𝖯 appears as a (contiguous) submatrix. The patterns can be of any size, but as long as they are square in shape data structures like suffix trees and suffix array exist [Raffaele Giancarlo, 1995; Dong Kyue Kim et al., 1998] for the task of efficient pattern matching. These are essentially 2D counterparts of classic suffix trees and arrays known for traditional 1-dimensional (1D) pattern matching. They work based on linearization of 2D suffixes which would preserve the prefix match property (i.e., every pattern match is a prefix of some suffix). The main limitation of the suffix trees and the suffix arrays (in 1D) was their space utilization of O(N log N) bits, where N is the size of the text. This was suboptimal compared to Nlog σ bits of space, which is information theoretic optimal for the text. With the advent of the field of succinct/compressed data structures, it was possible to develop compressed variants of suffix trees and array based on Burrows-Wheeler Tansform and LF-mapping (or Φ function) [Roberto Grossi and Jeffrey Scott Vitter, 2005; Paolo Ferragina and Giovanni Manzini, 2005; Kunihiko Sadakane, 2007]. These data structures indeed achieve O(N log σ) bits of space or better. This gives rise to the question: analogous to 1D case, can we design a succinct or compressed index for 2D pattern matching? Can there be a 2D compressed suffix tree? Are there analogues of Burrows-Wheeler Transform or LF-mapping? The problem has been acknowledged for over a decade now and there have been a few attempts at applying Φ function [Ankur Gupta, 2004] and achieving entropy based compression [Veli Mäkinen and Gonzalo Navarro, 2008]. However, achieving the complexity breakthrough akin to 1D case has yet to be found. In this paper, we still do not know how to answer suffix array queries in O(N log σ) bits of space - which would have led to efficient pattern matching. However, for the first time, we show an interesting result that it is indeed possible to compute inverse suffix array (ISA) queries in near compact space in O(polylog n) time. Our 2D succinct text index design is based on two 1D compressed suffix trees and it takes O(N log log N + N logσ) bits of space which is much smaller than its naive design that takes O(N log N) bits. Although the main problem is still evasive, this index gives a hope on the existence of a full 2D succinct index with all functionalities similar to that of 1D case.

A fast algorithm for order-preserving pattern matching

Efficient protocols for private wildcards pattern matching

We propose a protocol for the problem of secure two-party pattern matching, where Alice holds a text t?{0,1}? of length n, while Bob has a pattern p?{0,1}? of length m. The goal is for Bob to (only) learn where his pattern occurs in Alice's text, while Alice learns nothing. Private pattern matching is an important problem that has many applications in the area of DNA search, computational biology and more. Our construction guarantees full simulation in the presence of malicious, polynomial-time adversaries (assuming the hardness of DDH assumption) and exhibits computation and communication costs of O(n+m) group elements in a constant round complexity. This improves over previous work by Gennaro et al. (Public Key Cryptography, pp. 145---160, 2010) whose solution requires overhead of O(nm) group elements and exponentiations in O(m) rounds. In addition to the above, we propose a collection of protocols for important variations of the secure pattern matching problem that are significantly more efficient than the current state of art solutions: First, we deal with secure pattern matching with wildcards. In this variant the pattern may contain wildcards that match both 0 and 1. Our protocol requires O(n+m) communication and O(1) rounds using O(nm) computation. Then we treat secure approximate pattern matching. In this variant the matches may be approximated, i.e., have Hamming distance less than some threshold, ?. Our protocol requires O(n?) communication in O(1) rounds using O(nm) computation. Third, we have secure pattern matching with hidden pattern length. Here, the length, m, of Bob's pattern remains a secret. Our protocol requires O(n+M) communication in O(1) rounds using O(n+M) computation, where M is an upper bound on m. Finally, we have secure pattern matching with hidden text length. Finally, in this variant the length, n, of Alice's text remains a secret. Our protocol requires O(N+m) communication in O(1) rounds using O(N+m) computation, where N is an upper bound on n.