A. Aggarwal and J. S. Vitter. The input/output complexity of sorting and related problems. C. ACM, 31(8):1116--27, 1988.  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

....search (BFS) and an improved external algorithm for directed depth first search. We also demonstrate the equivalence of various formulations of external undirected BFS, and we use these to give the first I O optimal BFS algorithm for undirected trees. 1 Introduction We use the standard I O model [1], which counts disk accesses incurred by an algorithm, using the following parameters: M is the memory size, B is the block size, and we assume that B M=2. Define sort(N ) Theta( N B log M=B N B ) the number of I Os needed to sort N items, and scan(N ) dN=Be, the number of I Os needed ....

A. Aggarwal and J. S. Vitter. The input/output complexity of sorting and related problems. C. ACM, 31(8):1116--27, 1988.

....of lines in S that have different relative orderings at the two boundaries of W . It is well known that the number of such pairs can be counted by modifying any sorting algorithm, e.g. merge sort. Hence, we can compute oe = j Sigmaj using O( N=B) log B N) I Os by modifying external mergesort [5]. Set u = doe=Ne. We choose u random points of Sigma and sort them by their t coordinates. Let p 1 ; p 2 ; p u be these points in the sorted order, and let W i be the vertical slab whose boundary lines pass through p i and p i 1 . As shown in [22] one can modify the merge sort algorithm ....

A. Aggarwal and J. S. Vitter. The Input/Output complexity of sorting and related problems. Communications of the ACM, 31(9):1116--1127, 1988.

....database applications, the set S is too large to fit in the main memory, therefore it is stored on disk. In that case, the main goal is to minimize the number of disk accesses needed to answer a window query, and the performance of an algorithm is analyzed under the standard external memory model [2]. This model assumes that each disk access transmits a contiguous block of t units of data in a single input output operation (or I O) The efficiency of a data structure is measured in terms of the amount of disk space it uses (measured in units of disk blocks) the number of I Os required to ....

A. Aggarwal and J. S. Vitter. The Input/Output complexity of sorting and related problems. Communications of the ACM, 31(9):1116--1127, 1988.

....not assume that I O is performed in blocks. They also proved lower bounds for other problems that are discussed in this survey, such as FFTs. Their bounds were later extended to more complex models that assume that I O is performed in blocks on multiple independent disks by Aggarwal and Vitter [1] and by Vitter and Shriver [58] A SURVEY OF OUT OF CORE ALGORITHMS IN NUMERICAL LINEAR ALGEBRA 7 The misperception that partitioned schedules require a data layout by blocks caused partitioned schedules to go out of fashion in the 1970s. Virtual memory was becoming a standard way to solve ....

....for data reuse so that they can be scheduled. This idea is explored in Section 3.2. 2.5. Fast Fourier Transforms (FFTs) The FFT can be scheduled for outof core execution fairly e#ciently. Theoretically, both lower and upper bounds for the number of words transferred is #(n log n log M)[33, 1]. In practice, the 10 SIVAN TOLEDO log n log M factor is a small constant, rarely larger than 2, so asymptotic bounds are not particularly useful. Soon after the introduction of the FFT algorithm by Cooley and Tukey, Gentleman and Sande [27] presented an FFT algorithm which is suitable for ....

Alok Aggarwal and Je#rey S. Vitter. The input/output complexity of sorting and related problems. Communications of the ACM, 31(9):1116--1127, August 1988.

....manner in which the arrays are accessed. Conflict misses pose an additional challenge in designing efficient algorithms in the cache. This class of misses is not present in the I O models, where the mapping between internal and external memory is fully associative. Existing memory hierarchy models [4, 2, 3, 5] do not model certain salient features of caches, notably the lack of full associativity in address mapping and the lack of explicit control over data movement and replacement. Unfortunately, these small differences are malign in the effect. 1 The conflict misses that they introduce make ....

....important to the performance of the remaining steps of the FFT algorithm. In the first part of this paper, we develop a two level memory hierarchy model to capture the interaction between cache and main memory. Our model is a simple extension of the two level I O model that Aggarwal and Vitter [4] proposed for analyzing external memory algorithms. However, it captures three additional constraints of caches: lower miss penalties; lack of full associativity in address mapping; and lack of explicit program control over data movement. The work in this paper shows that the constraint imposed by ....

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A. Aggarwal and J. Vitter. The input/output complexity of sorting and related problems. Commun. ACM, 31(5):1118--1127, 1988.

....i.e. the total elapsed execution time when the algorithms are run on a real computer, depends heavily on the realization of the computer. A real computer may have multiple processors (see, e.g. 18] and or the I O subsystem can transfer data between several disks at the same time (cf. [2, 25, 30]) the processor operations (see, e.g. 27] and or the I Os (cf. 19] might be pipelined, but the effect of these factors is not considered here. It has been observed that in many large scale computations the increasing bottleneck of the computation is the performance of the I O subsystem (see, ....

....structure prior to the ith operation. The number of comparisons required when handling the sequence is O( P S i=1 log 2 N i ) When this data structure is used for sorting N elements, both the processor and I O performance match the well known lower bounds Omega ( N B log M=B N M ) I Os [2] and Omega (N log 2 N ) comparisons (see, e.g. 20] which are valid for all comparison based algorithms. To achieve the above bounds as well as our bounds the following facilities must be provided: 1. we should know the capacity of a block and the internal memory beforehand, 2. we must ....

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A. Aggarwal and J. S. Vitter. The input/output complexity of sorting and related problems. Communications of the ACM, volume 31, pages 1116--1127, 1988.

....Science, Rutgers University, Piscataway, NJ 08855, USA. martin google.com, http: www.cs.rutgers.edu farach) Supported by NSF grant CCR9820879. The distinguishing feature of multilevel memory hierarchies is that memory transfers are done in blocks in order to amortize the cost of a transfer [4]. This amortization only works when each transfer contains many pieces of data to be used by the CPU. Thus, our objective is to maintain locality of reference, meaning that memory accesses are clustered in time and space. Maintaining Locality in Irregular and Dynamic Data. Data locality is easier ....

....the relative speeds and block sizes at each memory level. While this leads to accurate time predictions, it makes it difficult to design and analyze optimal algorithms in these models. A second body of work concentrates on two level memory hierarchies, either in the context of memory and disk [4, 9, 18, 37, 38], or cache and memory [30, 20] In such a model there are only a few parameters, making it relatively easy to design efficient algorithms. The motivation is that it is common for one level of the memory hierarchy to dominate the running time. The difficulty with this approach is that the ....

A. Aggarwal and J. S. Vitter. The input/output complexity of sorting and related problems. Comm. ACM, 31(9):1116--1127, Sept. 1988.

....dependencies form a grid graph, the tradeo between the computation time and memory access time was quanti ed by Papadimitriou and Ullman [PU87] The I O complexity of an algorithm is the cost of inputs and outputs between faster internal memory and slower secondary memory. Aggarwal and Vitter [AV88] proved tight upper and lower bounds for the I O complexity of sorting, computing the FFT, permuting, and matrix transposition. Hong and Kung [HK81] introduced an abstract model of pebbling a computation graph to analyze the I O complexity of algorithms. The vertices of the graph that hold ....

A. Aggarwal and J. S. Vitter. The input/output complexity of sorting and related problems. Communications of the ACM, 31(9):1116-1127, September 1988.

....feature of disks that we want to model is their extremely long access time relative to that of internal memory. In order to amortize the access time over a large amount of data, typical disks read or write large blocks of contiguous data at once. Therefore we use a theoretical model defined in [5] with the following parameters: N = number of elements in the problem instance; M = number of elements that can fit into internal memory; B = number of elements per disk block; where M N and 1 B M=2. In order to study the performance of external memory algorithms, we use the standard ....

....parameters: N = number of elements in the problem instance; M = number of elements that can fit into internal memory; B = number of elements per disk block; where M N and 1 B M=2. In order to study the performance of external memory algorithms, we use the standard notion of I O complexity [5]. We define an Input Output (I O) operation to be the process of simultaneously reading or writing a block of B contiguous data elements to or from the disk. As I O communication is our primary concern, we define the I O complexity of an algorithm simply to be the number of I Os it performs. Thus ....

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A. Aggarwal and J. S. Vitter. The Input/Output complexity of sorting and related problems. Communications of the ACM, 31(9):1116--1127, 1988.

....both in sequential and parallel settings. The notions of block transfer and hierarchy are developed further in a parallel model in which memory consists of a tree of modules, where computation takes place at the leaves [6] I O complexity models start with a single disk and CPU with block transfer [18, 4] and continue through parallel disks with flat memory and hierarchical memory [35, 36] Our analytical model draws on ideas from several of these papers, though our intent is not to characterize the complexity of problems, but rather to predict performance on many real architectures. 2.3 Related ....

Aggarwal, A. and Vitter, J. S. The Input/Output Complexity of Sorting and Related Problems. CACM, 31(9):1116--1127, Sept. 1988.

....area of external memory algorithms. In this paper we study lower bound for external memory computational geometry problems. 1. 1 The I O model and lower bounds for fundamental problems We will be working in variations of the external memory model of computation introduced by Aggarwal and Vitter [AV88] The model has the following parameters: Supported in part by U.S. Army Research Office MURI grant DAAH04 96 1 0013. Part of this work was done while at BRICS, Department of Computer Science, University of Aarhus, Denmark. Email: large cs.duke.edu. y Supported in part by the ESPRIT Long ....

....of many of the I Osystems in use today, and depending on the size of the data records, typical values for workstations and file servers are on the order of M = 10 6 or 10 7 and B = 10 3 . Large scale problem instances can be in the range N = 10 10 to N = 10 12 . Aggarwal and Vitter [AV88] defined an I O operation in the model to be a swap of B records from internal memory with B consecutive records from external memory. The measure of performance is then the number of such I Os needed to solve a given problem. Internal computation is free. Aggarwal and Vitter proved lower bounds ....

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A. Aggarwal and J. S. Vitter, The Input/Output complexity of sorting and related problems, Communications of the ACM 31 (1988), no. 9, 1116--1127.

....search (BFS) and an improved external algorithm for directed depth first search. We also demonstrate the equivalence of various formulations of external undirected BFS, and we use these to give the first I O optimal BFS algorithm for undirected trees. 1 Introduction We use the standard I O model [1], which counts disk accesses incurred by an algorithm, using the following parameters: M is the memory size, B is the block size, and we assume that B M=2. Define sort(N) N B log M=B N B ) the number of I Os needed to sort N items, and scan(N) dN=Be, the number of I Os needed to ....

A. Aggarwal and J. S. Vitter. The input/output complexity of sorting and related problems. C. ACM, 31(8):1116--27, 1988.

....presented in this paper, a choice of # = 3## suf ces. Of this, ## amount of memory is used to prefetch data in order to overlap computation and data access. From hereon, we use # to denote ##. The problem of disk sorting was rst studied by Aggarwal and Vitter in their foundational paper [5]. In the model they considered, each I O operation results in the transfer of # blocks each block having # records. A more realistic model was envisioned in [20] Several asymptotically optimal algorithms have been given for sorting on this model. Nodine and Vitter s optimal algorithm [13] ....

A. Aggarwal and J. S. Vitter, The Input /Output Complexity of Sorting and Related Problems, Communications of the ACM, 1988, 31(9):1116-1127.

....Work Because of its importance, compiler writers have spent considerable effort on generating code with good locality [7] however substantial additional improvements can be gained by proper algorithm design. There has also been prior work on algorithm design and analysis for memory hierarchies [1, 2, 3, 4] but most of this prior work assumes greater control over movement between the memory levels than exists in most current systems. Moret and Shapiro discuss cache effects on graph algorithms in their minimum spanning tree paper [21] They comment that data caching and performance is affected by ....

A. Aggrawal, and J. Vitter. The input/output complexity of sorting and related problems. Communications of the ACM, 31(9):1116-1127, 1988.

....BMSTs, and maximal matchings. In Section 5, we consider semi external graph problems and give improved I O bounds for the semi external case of connected components. We conclude in Section 6. 1. 1 The Functional I O Model We adapt the I O model of complexity as defined by Aggarwal and Vitter [1]. For some problem instance, we define N to be the number of items in the instance, M to be the number of items that can fit in main memory, B to be the number of items per disk block, and b = bM=Bc. A typical compute server might have M 10 9 and B 10 3 . We assume that the input graph is ....

....output is the file of C i s catenated together, with a separator record between adjacent components. For the MST and BMST problems the output is a delineated list of edges in each tree in the spanning forest. For matching, the output is the list of edges in the matching. Following Chiang et al. [1] we define scan(N) dN=Be to be the number of disk I Os required to transfer N contiguous items between disk and memory, and we define sort(N) Theta(scan(N ) log b N B ) to be the number of I Os required to sort N items. The I O model stresses the importance of disk accesses over ....

A. Aggarwal and J. S. Vitter. The input/output complexity of sorting and related problems. C. ACM, 31(8):1116--27, 1988.

.... Sort(n B ) I Os and O(n B ) external space for an arbitrary parameter 0 1=2. The result carries over to BFS, depth rst search (DFS) and single source shortest paths (SSSP) on undirected planar graphs with arbitrary node degrees. 1 Introduction We use the standard I O model of [1], which counts accesses to a disk of potentially in nite size using the parameters M for the memory size and B for the block size where B M=2. Let Sort(x) x B log M=B x B ) denote the number of I Os needed to sort x items, and Scan(x) x B the number of I Os required ....

A. Aggarwal and J.S. Vitter. The input/output complexity of sorting and related problems. Communications of the ACM, pages 1116-1127, 1988.

....If nbytes is positive, then both the mtime and ctime of the file are updated. 4 Applications of zero( 4.1 I O Efficient Computation In recent years, I O efficient algorithms for a wide variety of problems have been developed. These include algorithms from the domains of sorting [AV88, Arg94, NV90, NV93, VS94] permuting [CSW94, Cor93, Cor92] computational geometry [GTVV93] line segment intersection [AVV95] graph algorithms [CGG 95] and scientific computation [VV95] The TPIE library [Ven95, Ven94] supports efficient implementation of many of these algorithms. A ....

Alok Aggarwal and Jeffrey S. Vitter. The input/output complexity of sorting and related problems. Communications of the ACM, 31(9):1116--1127, 1988.

....consideration that data must often reside on secondary storage rather than main memory; in a parallel setting this may involve the parallel access of multiple disks. Therefore it is necessary to design parallel algorithms which consider the possible data movement between main and secondary memory [5], and more generally which consider multiple levels of memory including register and cache. In the P HMM model, each processor has a memory hierarchy organized into discrete levels, much like the memory organization in the HMM, and all of P separate memories are connected together at the base ....

A. Aggarwal and J. S. Vitter. The input/output complexity of sorting and related problems. Comm. of the ACM, 31(9):1116--1127, Sept. 1988.

....bound Theta i N BD lg(N=B) lg(M=B) j , which was shown by Vitter and Shriver for randomized sorting and by Nodine and Vitter [NV93] and by Arge [Arg95] for deterministic sorting. These bounds are asymptotically tight, for they match the lower bounds proven earlier by Aggarwal and Vitter [AV88] using a model with one disk and D independent read write heads, which is at least as powerful as the Vitter Shriver model. Specific classes of structured permutations sometimes require fewer parallel I Os than general permutations. Vitter and Shriver showed how to transpose an R Theta S matrix ....

Alok Aggarwal and Jeffrey Scott Vitter. The input/output complexity of sorting and related problems. Communications of the ACM, 31(9):1116--1127, September 1988.

....memory model, optimality means that the total number of I Os required to compute the K intersections or the trapezoidal decomposition of S, is Theta(n log m n k) where n = N=B, m = M=B and k = K=B. In fact, the first term derives from the optimal I O cost for sorting the N segments of S [1], whereas the latter term derives from the cost due to the storage of the produced output on the disk. No I O optimal algorithm is known. The best known algorithms are due to Arge et al. 3] They solve the segment intersection problem (their algorithm does not compute the trapezoidal ....

A. Aggarwal and J. S. Vitter. The Input/Output complexity of sorting and related problems. Communications of the ACM, 31(9):1116--1127, 1988.

....given the current motion information of all objects . We call this type of queries the one dimensional MOR query for objects moving in one dimension and the two dimensional MOR query for objects moving in two dimensions. We consider the problem in the standard external memory model of computation[3]. In this model each disk access (an I O) transmits in a single operation B units of data. We call B the page capacity. We measure the efficiency of an algorithm in terms of the number of I O s to perform an operation. If N is the number of the mobile objects and K is the number of objects ....

A. Aggarwal and J. S. Vitter. The Input/Output complexity of sorting and related problems. In Communications of the ACM, 31(9):1116-1127, 1988.

....segment hit by that ray. In this paper, weareinterested in the problem of locating points in avery large planar subdivision that is stored on disk while reducing the overall time spent on I O operations. Our model of computation is the standard twolevel I O model proposed by Aggarwal and Vitter [3]. In this model, N denotes the number of elements in the problem instance, M is the numberofelements fitting in internal memory, and B is the number of elements per disk block, where M Nand 2 B M=2. An I O is the operation of reading (or writing) a disk block from (into) external memory. ....

....An I O is the operation of reading (or writing) a disk block from (into) external memory. Computations can only be done on elements presentininternal memory. Our measures of performance are the number of I Os used to solve a problem and the amount of space (disk blocks) used. Aggarwal and Vitter [3] considered sorting and related problems in the I O model and proved that sorting requires Theta i (N=B) log M=B (N=B) j I Os. When dealing with massive data sets, it is likely that there are K point location queries to be answered at a time also called batchedpoint location. Most ....

A. Aggarwal and J. Vitter. The input/output complexity of sorting and related problems. Comm. ACM, 31(9):1116--1127, 1988.

....be computed in O(N) time. If we apply the same approach to our problem, however, we expend O(N) I Os in the worst case, or I O time of c Theta 10 7 sec. for some constant c. However, it has been shown that permutation in external memory requires only Theta( N B log (M=B) N B ) I Os [4], giving a time of c 0 Theta 10 4 log 5 Theta10 4 10 7 seconds for some constant c 0 . So, assuming similar constants c and c 0 we have the EM algorithm for permutation running about B = 1000 times faster than the internal memory approach. This illustrates the most important factor ....

....in subsequent sections which deal with specific external memory topics. Floyd [49] is credited with some of the earliest work towards optimal EM algorithms. He studied matrix transposition and permutation algorithms in a paged memory, with a limited set of basic operations. Aggarwal and Vitter [4] studied sorting on a model which involved a single disk which could read or write D blocks in one operation. They showed matching upper and lower bounds for the number of I O operations required for sorting, FFT, matrix transpose, and permutation under this model. Their model did not constrain ....

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A. Aggarwal and J. S. Vitter. The Input/Output complexity of sorting and related problems. Communications of the ACM, 31(9):1116--1127, 1988.

....one of the major theoretical open problems in the area, and in [23] it is even called the most elegant open question. In this paper we develop such an optimal structure. 1. 1 Memory model and previous results We will be working in the standard model for external memory with one (logical) disk [24, 1]. We assume that each external memory access transmits one page of B units of data, which we count as one I O. We measure the efficiency of an algorithm in terms of the number of I O operations it performs. Often one also makes the assumption that the main memory is capable of holding O(B 2 ) ....

A. Aggarwal and J. S. Vitter. The Input/Output complexity of sorting and related problems. Communications of the ACM, 31(9):1116--1127, 1988.

....to fit in internal memory and so have to be stored on disk, thus necessitating I O as a fundamental, frequently used operation during sorting. One approach to alleviate the effects of the I O bottleneck is to use parallel disk systems [HGK 94, PGK88, Uni89, GS84, Mag87] Aggarwal and Vitter [AV88] generalizing initial work done by Floyd [Flo72] and Hong and Kung [HK81] laid the foundation for I O algorithms by studying the I O complexity of sorting and related problems. The model they studied [AV88] considers an internal memory of size M and I O reads or writes that each result in a ....

....to use parallel disk systems [HGK 94, PGK88, Uni89, GS84, Mag87] Aggarwal and Vitter [AV88] generalizing initial work done by Floyd [Flo72] and Hong and Kung [HK81] laid the foundation for I O algorithms by studying the I O complexity of sorting and related problems. The model they studied [AV88] considers an internal memory of size M and I O reads or writes that each result in a transfer of D blocks, where each block is comprised of B contiguous records, from or to disks. Subsequently, Vitter and Shriver considered a realistic D disk two level memory model in which secondary memory is ....

Alok Aggarwal and Jeffrey S. Vitter. The input/output complexity of sorting and related problems. Communications of the ACM, 31(9):1116--1127, 1988.

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