The cost of decompressing (decoding) data can be prohibitive for certain real-time applications. In many scenarios, it is acceptable to sacrifice (to some extent) on compression in the interest of fast decoding. We study anovel problem of finding a prefix tree having the best decode time under the constraint that the code length does not exceed a certain threshold for a natural class of memory access cost functions that use blocking (also referred to as lookup tables). We present exact and approximation algorithms for this problem that are based on dynamic programming and capitalize on interesting structures of the optimal solutions. The full version of this paper is available at [1]

In this paper, we study the problem of designing prefix-free encoding schemes having minimum average code length that can be decoded efficiently under a decode cost model that captures memory hierarchy induced cost functions. We also study a special case of this problem that is closely related to the length limited Huffman coding (LLHC) problem; we call this the soft-length limited Huffman coding problem. In this version, there is a penalty associated with each of the n characters of the alphabet whose encodings exceed a specified bound D(≤ n) where the penalty increases linearly with the length of the encoding beyond D. The goal of the problem is to find a prefix-free encoding having minimum average code length and total penalty within a pre-specified bound P. This generalizes the LLHC problem. We present an algorithm to solve this problem that runs in time O(nD). We study a further generalization in which the penalty function and the objective function can both be arbitrary monotonically non-decreasing functions of the codeword length. We provide dynamic programming based exact and PTAS algorithms for this setting.

The cost of uncompressing (decoding) data can be prohibitive in certain real-time applications, for example when predicting using compressed deep learning models. In many scenarios, it is acceptable to sacrifice to some extent on compression in the interest of fast decoding. In this work, we are interested in finding the prefix tree having the best decode time under the constraint that the code length does not exceed a certain threshold for a natural class of algorithms under the hierarchical memory model. We present an efficient optimal algorithm for this problem based on a dynamic program that capitalizes on an interesting structure of the optimal solution.

By combining several interesting applications of random sampling in geometric algorithms like point location, linear programming, segment intersections, binary space partitioning, Clarkson and Shor [Kenneth L. Clarkson and Peter W. Shor, 1989] developed a general framework of randomized incremental construction (RIC ). The basic idea is to add objects in a random order and show that this approach yields efficient/optimal bounds on expected running time. Even quicksort can be viewed as a special case of this paradigm. However, unlike quicksort, for most of these problems, sharper tail estimates on their running times are not known. Barring some promising attempts in [Kurt Mehlhorn et al., 1993; Kenneth L. Clarkson et al., 1992; Raimund Seidel, 1991], the general question remains unresolved. In this paper we present a general technique to obtain tail estimates for RIC and and provide applications to some fundamental problems like Delaunay triangulations and construction of Visibility maps of intersecting line segments. The main result of the paper is derived from a new and careful application of Freedman’s [David Freedman, 1975] inequality for Martingale concentration that overcomes the bottleneck of the better known Azuma-Hoeffding inequality. Further, we explore instances, where an RIC based algorithm may not have inverse polynomial tail estimates. In particular, we show that the RIC time bounds for trapezoidal map can encounter a running time of Omega (n log n log log n) with probability exceeding 1/(sqrt{n)}. This rules out inverse polynomial concentration bounds within a constant factor of the O(n log n) expected running time.

We present an algorithm for maintaining a maximal matching in a graph under addition and deletion of edges. Our algorithm is randomized and it takes expected amortized $O(\log n)$ time for each edge update, where $n$ is the number of vertices in the graph. Moreover, for any sequence of $t$ edge updates, the total time taken by the algorithm is $O(t\log n + n \log^2 n)$ with high probability. (Original article at https://doi.org/10.1137/130914140.)

Scratchpad memory (SPM) is considered a useful component in the memory hierarchy, solely or along with caches, for meeting the power and energy constraints as performance ceases to be the sole criteria for processor design. Although the efficiency of SPM is well known, its use has been restricted owing to difficulties in programmability. Real applications usually have regions that are amenable to exploitation by either SPM or cache and hence can benefit if the two are used in conjunction. Dynamically adjusting the local memory resources to suit application demand can significantly improve the efficiency of the overall system. In this article, we propose a compiler technique to map application data objects to the SPM-cache and also partition the local memory between the SPM and cache depending on the dynamic requirement of the application. First, we introduce a novel graph-based structure to tackle data allocation in an application. Second, we use this to present a data allocation heuristic to map program objects for a fixed-size SPM-cache hybrid system that targets whole program optimization. We finally extend this formulation to adapt the SPM and cache sizes, as well as the data allocation as per the requirement of different application regions. We study the applicability of the technique on various workloads targeted at both SPM-only and hardware reconfigurable memory systems, observing an average of 18% energy-delay improvement over state-of-the-art techniques.

In this paper, we study the maximum density, threshold and emptiness queries for intervals in the streaming model. The input is a stream S of n points in the real line R and a floating closed interval W of width alpha. The specific problems we consider in this paper are as follows. - Maximum density: find a placement of W in R containing the maximum number of points of S. - Threshold query: find a placement of W in R, if it exists, that contains at least Delta elements of S. - Emptiness query: find, if possible, a placement of W within the extent of S so that the interior of W does not contain any element of S. The stream S, being huge, does not fit into main memory and can be read sequentially at most a constant number of times, usually once. The problems studied here in the geometric setting have relations to frequency estimation and heavy hitter identification in a stream of data. We provide lower bounds and results on trade-off between extra space and quality of solution. We also discuss generalizations for the higher dimensional variants for a few cases.

We present an algorithm for maintaining maximal matching in a graph under addition and deletion of edges. Our algorithm is randomized and it takes expected amortized $O(\log n)$ time for each edge update, where $n$ is the number of vertices in the graph. While there exists a trivial $O(n)$ time algorithm for each edge update, the previous best known result for this problem is due to Ivković and Lloyd [ Lecture Notes in Comput. Sci. 790, Springer-Verlag, London, 1994, pp. 99--111]. For a graph with $n$ vertices and $m$ edges, they gave an $O( {(n+ m)}^{0.7072})$ update time algorithm which is sublinear only for a sparse graph. For the related problem of maximum matching, Onak and Rubinfeld [ Proceedings of STOC'10, Cambridge, MA, 2010, pp. 457--464] designed a randomized algorithm that achieves expected amortized $O(\log^2 n)$ time for each update for maintaining a $c$-approximate maximum matching for some unspecified large constant $c$. In contrast, we can maintain a factor 2 approximate maximum matching in expected amortized $O(\log n )$ time per update as a direct corollary of the maximal matching scheme. This in turn also implies a 2-approximate vertex cover maintenance scheme that takes expected amortized $O(\log n )$ time per update. (A corrected version is at https://epubs.siam.org/doi/abs/10.1137/16M1106158.)

We present a fully dynamic algorithm for maintaining approximate maximum weight matching in general weighted graphs. The algorithm maintains a matching M whose weight is at least 1/8 M^{*} where M^{*} is the weight of the maximum weight matching. The algorithm achieves an expected amortized O(log n log C) time per edge insertion or deletion, where C is the ratio of the weights of the highest weight edge to the smallest weight edge in the given graph.

In this paper we present a cache-oblivious framework for randomized divide and conquer algorithms on the multicore model with private cache. We first derive an O(n/p log n + log n log log n) expected parallel depth algorithm for sorting n numbers with expected O(n/B logM n) cache misses where p,M and B respectively denote the number of processors, the size of an individual cache memory and the block size respectively. Although similar results have been obtained recently for sorting, we feel that our approach is simpler and general and we apply it to obtain an algorithm for 3D convex hulls with similar bounds.

We present an algorithm for maintaining maximal matching in a graph under addition and deletion of edges. Our data structure is randomized that takes $O( \log n)$ expected amortized time for each edge update where $n$ is the number of vertices in the graph. While there is a trivial $O(n)$ algorithm for edge update, the previous best known result for this problem was due to Ivkovi\'c and Llyod\cite{llyod}. For a graph with $n$ vertices and $m$ edges, they give an $O( {(n+ m)}^{0.7072})$ update time algorithm which is sub linear only for a sparse graph. %To the best of our knowledge this %is the first polylog update time for maximal matching that implies an % exponential improvement from the previous results. For the related problem of maximum matching, Onak and Rubinfeld \cite{onak} designed a randomized data structure that achieves $O(\log^2 n)$ expected amortized time for each update for maintaining a $c$-approximate maximum matching for some large constant $c$. In contrast, we can maintain a factor two approximate maximum matching in $O(\log n )$ expected amortized time per update as a direct corollary of the maximal matching scheme. This in turn also implies a two approximate vertex cover maintenance scheme that takes $O(\log n )$expected amortized time per update.

We present a framework for computing with input data specified by intervals, representing uncertainty in the values of the input parameters. To compute a solution, the algorithm can query the input parameters that yield more refined estimates in form of sub-intervals and the objective is to minimize the number of queries.The previous approaches address the scenario where every query returns an exact value. Our framework is more general as it can deal with a wider variety of inputs and query responses and we establish interesting relationships between them that have not been investigated previously. Although some of the approaches of the previous restricted models can be adapted to the more general model, we require more sophisticated techniques for the analysis and we also obtain improved algorithms for the previous model. We address selection problems in the generalized model and show that there exist 2-update competitive algorithms that do not depend on the lengths or distribution of the sub-intervals and hold against the worst case adversary. We also obtain similar bounds on the competitive ratio for the MST problem in graphs.

We present a general approach for designing approximation algorithms for a fundamental class of geometric clustering problems in arbitrary dimensions. More specifically, our approach leads to simple randomized algorithms for the k -means, k -median and discrete k -means problems that yield (1+ε) approximations with probability ≥ 1/2 and running times of O (2 ( k /ε) O (1) dn ). These are the first algorithms for these problems whose running times are linear in the size of the input ( nd for n points in d dimensions) assuming k and ε are fixed. Our method is general enough to be applicable to clustering problems satisfying certain simple properties and is likely to have further applications.

The parallel disks model (PDM) has been proposed to alleviate the I/O bottleneck that arises in the processing of massive data sets. Sorting has been extensively studied on the PDM model due to the fundamental nature of the problem - several asymptotically optimal algorithms are known for sorting. Although randomization has been frequently exploited, most of the prior algorithms suffer from complications in memory layouts, implementation, restrictions in range of parameters, and laborious analysis. In this paper, we present a randomized mergesort algorithm based on a simple idea that sorts using an asymptotically optimal number of I/O operations with high probability and has all of the desirable features for practical implementation. In the second part of the paper, we also present several novel algorithms for sorting on the PDM that take only a small number of passes through the data. Recently, considerable interest has been shown by researchers in developing algorithms for problem sizes of practical interest and we are able to obtain several improvements and simplification, in particular for random input.

Let G = ( V , E ) be an undirected graph on n vertices, and let δ( u , v ) denote the distance in G between two vertices u and v . Thorup and Zwick showed that for any positive integer t , the graph G can be preprocessed to build a data structure that can efficiently report t -approximate distance between any pair of vertices. That is, for any u , v ∈ V , the distance reported is at least δ( u , v ) and at most t δ( u , v ). The remarkable feature of this data structure is that, for t ≥3, it occupies subquadratic space, that is, it does not store all-pairs distances explicitly, and still it can answer any t -approximate distance query in constant time. They named the data structure “approximate distance oracle” because of this feature. Furthermore, the trade-off between the stretch t and the size of the data structure is essentially optimal.In this article, we show that we can actually construct approximate distance oracles in expected O ( n 2 ) time if the graph is unweighted. One of the new ideas used in the improved algorithm also leads to the first expected linear-time algorithm for computing an optimal size (2, 1)-spanner of an unweighted graph. A (2, 1) spanner of an undirected unweighted graph G = ( V , E ) is a subgraph ( V , Ê), Ê ⊆ E , such that for any two vertices u and v in the graph, their distance in the subgraph is at most 2δ( u , v ) + 1.

We present the first linear time (1 + /spl epsiv/)-approximation algorithm for the k-means problem for fixed k and /spl epsiv/. Our algorithm runs in O(nd) time, which is linear in the size of the input. Another feature of our algorithm is its simplicity - the only technique involved is random sampling.

We present improved algorithms for maintaining transitive closure and all-pairs shortest paths/distances in a digraph under deletion of edges.(MATH) For the problem of transitive closure, the previous best known algorithms, for achieving O(1) query time, require O(\min(m, \frac{n^3}{m}))$ amortized update time, implying an upper bound of O(n^{\frac{3}{2}})$ on update time per edge-deletion. We present an algorithm that achieves $O(1)$ query time and O(n \log^2n + \frac{n^2}{\sqrt{m}}{\sqrt{\log n}})$ update time per edge-deletion, thus improving the upper bound to O(n^{\frac{4}{3}}\sqrt[3]{\log n})$.(MATH) For the problem of maintaining all-pairs shortest distances in unweighted digraph under deletion of edges, we present an algorithm that requires O(\frac{n^3}{m} \log^2 n)$ amortized update time and answers a distance query in O(1) time. This improves the previous best known update bound by a factor of log n. For maintaining all-pairs shortest paths, we present an algorithm that achieves O(\min(n^{\frac{3}{2}} \sqrt{\log n}, \frac{n^3}{m} \log ^2n))$ amortized update time and reports a shortest path in optimal time (proportional to the length of the path). For the latter problem we improve the worst amortized update time bound by a factor of O(\sqrt{\frac{n}{\log n}})$.(MATH) We also present the first decremental algorithm for maintaining all-pairs (1+ε) approximate shortest paths/distances, for any ε > 0, that achieves a sub-quadratic update time of O(n log2n + \frac{n^2}{\sqrt{\epsilon m}}\sqrt{\log n})$ and optimal query time.Our algorithms are randomized and have one-sided error for query (with probability O(1/nc) for any constant c).

We present a model that enables us to analyze the running time of an algorithm on a computer with a memory hierarchy with limited associativity, in terms of various cache parameters. Our cache model, an extension of Aggarwal and Vitter's I/O model, enables us to establish useful relationships between the cache complexity and the I/O complexity of computations. As a corollary, we obtain cache-efficient algorithms in the single-level cache model for fundamental problems like sorting, FFT, and an important subclass of permutations. We also analyze the average-case cache behavior of mergesort, show that ignoring associativity concerns could lead to inferior performance, and present supporting experimental evidence.We further extend our model to multiple levels of cache with limited associativity and present optimal algorithms for matrix transpose and sorting. Our techniques may be used for systematic exploitation of the memory hierarchy starting from the algorithm design stage, and for dealing with the hitherto unresolved problem of limited associativity.

We investigate the memory system performance of several algorithms for transposing an N/spl times/N matrix in-place, where N is large. Specifically, we investigate the relative contributions of the data cache, the translation lookaside buffer, register tiling, and the array layout function to the overall running time of the algorithms. We use various memory models to capture and analyze the effect of various facets of cache memory architecture that guide the choice of a particular algorithm, and attempt to experimentally validate the predictions of the model. Our major conclusions are as follows: limited associativity in the mapping from main memory addresses to cache sets can significantly degrade running time; the limited number of TLB entries can easily lead to thrashing; the fanciest optimal algorithms are not competitive on real machines even at fairly large problem sizes unless cache miss penalties are quite high: low-level performance tuning "hacks", such as register tiling and array alignment, can significantly distort the effects of improved algorithms; and hierarchical non-linear layouts are inherently superior to the standard canonical layouts (such as row- or column-major) for this problem.

We describe a deterministic parallel algorithm for linear programming in fixed dimension d that takes poly(log log n) time in the common concurrent read concurrent write (CRCW) PRAM model and does optimal O(n) work. In the exclusive read exclusive write (EREW) model, the algorithm runs in O(log n · log log d-1 n ) time. Our algorithm is based on multidimensional search and effective use of approximation algorithms to speed up the basic search in the CRCW model. Our method also yields very fast poly(log log n) algorithms for smallest enclosing sphere and approximate ham-sandwich cuts and an O(log n) time work-optimal algorithm for exact ham-sandwich cuts of separable point sets. For these problems, in particular for fixed-dimensional linear programming, o(log n) time efficient deterministic PRAM algorithms were not known until very recently.

In this paper we focus on the problem of designing very fast parallel algorithms for the convex hull problem in two and three dimensions in the arbitrary CRCW model whose running times are output-size sensitive.

We describe a very simple deterministic parallel algorithm for linear programming in fixed dimension d that takes poly(log log n) time in the common CRCW PRAM model and does optimal O(n) work.Our algorithm is based on multidimensional search and an effective use of approximation algorithms to speed-up the basic search in the CRCW model.Our method also yields very fast poly(log log n) algorithm for smallest enclosing sphere and approximate ham-sandwich cuts as well as an O (log n) time work-optimal algorithm for exact ham-sandwich cuts of separable point sets.For all these problems, particularly for the fixed-dimensional linear programming, o(log n) time efficient deterministic PRAM algorithms were not known until very recently.

Further applications of random sampling techniques which have been used for deriving efficient parallel algorithms are presented by J. H. Reif and S. Sen [Proc. 16th International Conference on Parallel Processing, 1987]. This paper presents an optimal parallel randomized algorithm for computing intersection of half spaces in three dimensions. Because of well-known reductions, these methods also yield equally efficient algorithms for fundamental problems like the convex hull in three dimensions, Voronoi diagram of point sites on a plane, and Euclidean minimal spanning tree. The algorithms run in time $T = O(\log n)$ for worst-case inputs and use $P = O(n)$ processors in a CREW PRAM model where n is the input size. They are randomized in the sense that they use a total of only polylogarithmic number of random bits and terminate in the claimed time bound with probability $1 - n^{ - \alpha } $ for any fixed $\alpha > 0$. They are also optimal in $P\cdot T$ product since the sequential time bound for all thes...

There are now a number of fundamental problems in computational geometry that have optimal algorithms on PRAM models. This paper presents randomized parallel algorithms that execute on an n-processor butterfly interconnection network in $O(\log n)$ time for the following problems of input size n: trapezoidal decomposition, visibility, triangulation, and two-dimensional convex hull. These algorithms involve tackling some of the very basic problems, like binary search and load balancing, that are taken for granted in PRAM models. Apart from a two-dimensional convex hull algorithm, these are the first nontrivial geometric algorithms that attain this performance on fixed connection networks. These techniques use a number of ideas from Flashsort that have to be modified to handle more difficult situations; it seems likely that they will have wider applications.

Further applications of random sampling techniques which have been used for deriving efficient parallel algorithms are presented by J. H. Reif and S. Sen [Proc. 16th International Conference on Parallel Processing, 1987]. This paper presents an optimal parallel randomized algorithm for computing intersection of half spaces in three dimensions. Because of well-known reductions, these methods also yield equally efficient algorithms for fundamental problems like the convex hull in three dimensions, Voronoi diagram of point sites on a plane, and Euclidean minimal spanning tree. The algorithms run in time $T = O(\log n)$ for worst-case inputs and use $P = O(n)$ processors in a CREW PRAM model where n is the input size. They are randomized in the sense that they use a total of only polylogarithmic number of random bits and terminate in the claimed time bound with probability $1 - n^{ - \alpha } $ for any fixed $\alpha > 0$. They are also optimal in $P\cdot T$ product since the sequential time bound for all these problems is $\Omega (n\log n)$. The best known deterministic parallel algorithms for two-dimensional Voronoi-diagram and three-dimensional convex hull run in $O(\log ^2 n)$ and $O(\log ^2 n\log ^ * n)$ time, respectively, while using $O(n/\log n)$ and $O(n)$ processors, respectively.

We present algorithms for maintaining data structures supporting fast point location queries in arrangements of hyperplanes with dimension less than or equal to four.This data structure allows for ity which is likely to have further applications to other dynamic algorithms.

Article Parallel searching in generalized Monge arrays with applications Share on Authors: A. Aggarwal IBM Research Division, T. J. Watson Research Center, Yorktown Heights, NY IBM Research Division, T. J. Watson Research Center, Yorktown Heights, NYView Profile , D. Kravets Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, MA Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, MAView Profile , J. Park Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, MA Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, MAView Profile , S. Sen Department of Computer Science, Duke University, Durham, NC Department of Computer Science, Duke University, Durham, NCView Profile Authors Info & Claims SPAA '90: Proceedings of the second annual ACM symposium on Parallel algorithms and architecturesMay 1990 Pages 259–268https://doi.org/10.1145/97444.97693Online:01 May 1990Publication History 11citation348DownloadsMetricsTotal Citations11Total Downloads348Last 12 Months4Last 6 weeks0 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 SiteGet Access

There are now a number of fundamental problems in computational geometry that have optimal algorithms on PRAM models.We present randomized parallel algorithms which execute on an n-processor butterfly inter-connection network in O(log n) time for the following problems of input size n: trapezoidal decomposition, visibility, triangulation and Z-D convex hull.These are based on some previous work of the authors on PRAM algorithms and a new algorithm for doing binary search on fixed connection network.Apart from a 2-D convex hull algorithm, these are the first non-trivial geometric algorithms which attain this performance on fixed connection networks.The techniques developed in this paper rely on random sampling methods to do loadbalancing on fixed-connection networks; it seems likely that they will have wider applications.

We introduce a new randomized sampling technique, called Polling which has applications to deriving efficient parallel algorithms. As an example of its use in computational geometry, we present an optimal parallel randomized algorithm for intersection of half-spaces in three dimensions. Because of well-known reductions, our methods also yield equally efficient algorithms for fundamental problems like t,he convex hull in three dimensions, Voronoi diagram of point sites on a plane and Euclidean minimal spanning tree. Our algorithms run in time T = O(logn) for worst-case inputs and uses P = O(n) processors in a CREW PRAM model where n is the input size. They are randomized in the sense that they use a total of only O(log2 n) random bits and terminate in the claimed time bound with probability 1- n--(y for any o> 0. They are also optimal in P. T product since the sequential time bound for all these problems is Sl(nlogn). The best known deterministic parallel algorithms for 2-D Voronoi-diagram and 3-D Convex hull run in O(log2 n) and O(log2 nlog * n) time respectively while using O(n) processors.

The gradual refinement of a general approach to two-dimensional sorting, the shear-sort algorithm, to more sophisticated and specialized sorting algorithms on mesh-connected computers is described. The analysis of the shear-sort algorithm gives rise to a novel perspective of two-dimensional sorting, which seems to be a very powerful tool for developing efficient algorithms. The same methods can be extended for sorting in higher dimensions, for example, in the three-dimensional mesh. The concept of clean and dirty rows can be modified to clean and dirty planes (or hyperplanes for dimensions greater than three). Although only two schemes (purely recursive and iterative) are explicitly described, the reader may construct his own algorithm using similar technique and slight modifications. Designing an O(n) algorithm for sorting on a mesh becomes much simpler using the techniques developed.>

In this paper we present an algorithm for hidden surface removal for a class of polyhedral surfaces which have a property that they can be ordered relatively quickly like the terrain maps. A distinguishing feature of this algorithm is that its running time is sensitive to the actual size of the visible image rather than the total number of intersections in the image plane which can be much larger than the visible image. The time complexity of this algorithm is Ο((k +n)lognloglogn) where n and k are respectively the input and the output sizes. Thus, in a significant number of situations this will be faster than the worst case optimal algorithms which have running time Ω(n2) irrespective of the output size (where as the output size k is Ο(n2) only in the worst case). We also present a parallel algorithm based on a similar approach which runs in time Ο(log4(n+k)) using Ο((n + k)/log(n+k)) processors in a CREW PRAM model. All our bounds are obtained using ammortized analysis.

In this paper, We consider the problem of sorting n2 numbers, initially distributed randomly in an n × n mesh-connected processor array with one element per processor. We show a lower bound, based on distance arguments, of 4n routing steps on mesh-connected processors operating in an SIMD mode with no wraparounds in rows or columns, We present an algorithm using a novel approach, which is optimal upto the conslant of the leading term, and hence, succeed in proving the tightness of the lower bound based on distance. Keeping in mind the practical difficulties in implementation of this algorithm, we also present an extremely practical O(n) algorithm amenable for VLSI implementation and for existing mesh- connected computers. All the results in this paper were derived by using a new method of analysis inspired by the discovery of shear-sort or row-column sort.