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Journal Article•DOI•

Competitive paging algorithms

Amos Fiat1, Richard M. Karp2, Michael Luby3, Lyle A. McGeoch4, Daniel D. Sleator5, Neal E. Young6 •
Tel Aviv University1, University of California, Berkeley2, University of Toronto3, Amherst College4, Carnegie Mellon University5, Princeton University6
01 Dec 1991-Journal of Algorithms (Academic Press, Inc.)-Vol. 12, Iss: 4, pp 685-699
TL;DR: The marking algorithm is developed, a randomized on-line algorithm for the paging problem, which it is proved that its expected cost on any sequence of requests is within a factor of 2Hk of optimum.
About: This article is published in Journal of Algorithms.The article was published on 1991-12-01 and is currently open access. It has received 489 citations till now. The article focuses on the topics: Page replacement algorithm & K-server problem.

Summary (2 min read)

Jump to: [Introduction] – [686 FIAT ET AL.] – [688 FIAT ET AL.] – [690 FIAT ET AL.] – [692 FIAT ET AL.] – [694 FIAT ET AL.] – [6. ALGORITHMS THAT ARE COMPETITIVE AGAINST SEVERAL OTHERS] – [696 FIAT ET AL.] and [7. EXTENSIONS]

Introduction

  • The authors develop the marking algorithm, a randomized on-line algorithm for the paging problem.
  • The best such factor that can be achieved is Hk. 3Support was provided by the International Computer Science Institute and operating grant A8092 of the Natural Sciences and Engineering Research Council of Canada.

686 FIAT ET AL.

  • And a set of appropriate constants, the authors describe a way of constructing another on-line algorithm whose performance is within the appropriate constant factor of each algorithm in the set.
  • They showed that two strategies for paging (ejecting the least recently used page, or LRU, and first-in-first-out, or FIFO) could be worse than the optimum off-line algorithm by a factor of k, but not more, and that no on-line algorithm could achieve a factor less than k.
  • This function is closely approximated by the natural logarithms: ln(k +.
  • These three groups collaborated in the writing of this paper.

688 FIAT ET AL.

  • Recent extensions to this work are described in Section 7, along with several open problems.
  • If the requested vertex is not covered, then a server is chosen uniformly at random from among the unmarked vertices, and this server is moved to cover the requested vertex.
  • THEOREM 1. The marking algorithm is a 2H,-competitive algorithm for the uniform k-server problem on n vertices.
  • The marking algorithm (denoted M) implicitly divides u (excluding some requests at the beginning) into phases.
  • A vertex is called clean if it was not requested in the previous phase and has not yet been requested in this phase.

690 FIAT ET AL.

  • [lo, 111 showed that for any given algorithm, there is always a lazy one that incurs no more cost.
  • The authors shall first argue that the amortized cost incurred by A over the phase is at least l/2.
  • The expected cost of the request is c/s because there are c unserved vertices distributed uniformly among s stale vertices.
  • Since the cost incurred by A during the phase is l/2, this proves that the marking algorithm is 2H,-competitive.
  • The above proof can be modified slightly to give this theorem.

692 FIAT ET AL.

  • ALGORITHM EATR Algorithm EATR is a randomized algorithm for the uniform 2-server problem.
  • The algorithm maintains one server on the most recently requested vertex, and the other uniformly at random among the set of stale vertices.
  • After the first request of the phase, one vertex is marked.
  • Armed with these tools (the marking and the probability vector), the adversary can generate a sequence such that the expected cost of each phase to A is H,,-l, and the cost to the optimum off-line algorithm is 1.
  • The expected cost of this request is at least l/u. ‘Raghavan [13, pp. 118-1191 presents a different proof of this theorem based on a generalization of the minimax principle due to Andy Yao 1161.

694 FIAT ET AL.

  • Next, a set of requests are generated by the following loop (P denotes the current total probability of the marked vertices):.
  • If the total expected cost ends up exceeding l/u, then an arbitrary request is made to an unmarked vertex, and the subphase is over.

6. ALGORITHMS THAT ARE COMPETITIVE AGAINST SEVERAL OTHERS

  • In many applications of the k-server model, the following situation arises: one is given several on-line algorithms with desirable characteristics and would like to construct a single on-line algorithm that has the advantages of all the given ones.
  • In the case of the paging problem (the uniform-cost k-server problem), the least-recently-used page replacement algorithm (LRU) is believed to work well in practice, but, in the worst case, it can be k times as costly as the optimal off-line algorithm.
  • On the other hand, the authors have exhibited a randomized on-line algorithm that is 2H,-competitive, and thus it has theoretical advantages over LRU.
  • The ordered pair (k, n) is called the type of the algorithm.
  • Thus, it suffices to show that A can punish each B(i) at least lC,(u>/c(i)J times.

696 FIAT ET AL.

  • This completes the proof of the sufficiency of (1).
  • Then CA(r(N)) = N, and CCBci)(7(N)) I N. If A is to be c(i)-competitive with each of the on-line algorithms B(i) then there must exist constants a(i) such that, for all i and all N, But these inequalities, together with the fact that Cl/c(i) > 1, lead to a contradiction for sufficiently large N. q COMPETITIVE PAGING ALGORITHMS 697.
  • The authors now extend their definitions to the case of randomized algorithms.

7. EXTENSIONS

  • The problem of devising a strongly competitive algorithm for any k and n was solved by McGeoch and Sleator [12].
  • Most notable of these is to extend the technique of constructing an algorithm competitive with several others to other problems besides the uniform server problem.
  • The authors thank Jorge Stolfi and an anonymous referee for many helpful suggestions.
  • A competitive 3-server algorithm, in “First Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, January 1990,” pp. 280-290. 14. P. RAGHAVAN AND M. SNIR, Memory versus randomization in on-line algorithms, in “Automata, Languages, and Programming,” Lecture Notes in Computer Science, Vol. 372, pp. 687-703, Springer-Verlag, New York/Berlin, 1989; revised version available as an IBM research report.

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Citations
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Proceedings Article•DOI•
Probabilistic approximation of metric spaces and its algorithmic applications

[...]

Yair Bartal
14 Oct 1996
TL;DR: It is proved that any metric space can be probabilistically-approximated by hierarchically well-separated trees (HST) with a polylogarithmic distortion.
Abstract: This paper provides a novel technique for the analysis of randomized algorithms for optimization problems on metric spaces, by relating the randomized performance ratio for any, metric space to the randomized performance ratio for a set of "simple" metric spaces. We define a notion of a set of metric spaces that probabilistically-approximates another metric space. We prove that any metric space can be probabilistically-approximated by hierarchically well-separated trees (HST) with a polylogarithmic distortion. These metric spaces are "simple" as being: (1) tree metrics; (2) natural for applying a divide-and-conquer algorithmic approach. The technique presented is of particular interest in the context of on-line computation. A large number of on-line algorithmic problems, including metrical task systems, server problems, distributed paging, and dynamic storage rearrangement are defined in terms of some metric space. Typically for these problems, there are linear lower bounds on the competitive ratio of deterministic algorithms. Although randomization against an oblivious adversary has the potential of overcoming these high ratios, very little progress has been made in the analysis. We demonstrate the use of our technique by obtaining substantially improved results for two different on-line problems.

797 citations

Cites background or methods from "Competitive paging algorithms"

  • ...Both of our algoi ithms run recursively copies of the randomized marking algorithm of [ FKLMSY88 ] for the K-...

    [...]

  • ...The most obvious example is the paging problem with a cache of size Ii, where the deterministic competitive ratio is Ii [ST851 while the randomized competitive ratio (against oblivious adversaries) is O( log K) [ FKLMSY88 ]....

    [...]

Randomized Algorithms

[...]

Rajeev Motwani1, Prabhakar Raghavan2•
Stanford University1, IBM2
05 Mar 2013
TL;DR: For many applications, a randomized algorithm is either the simplest or the fastest algorithm available, and sometimes both. as discussed by the authors introduces the basic concepts in the design and analysis of randomized algorithms and provides a comprehensive and representative selection of the algorithms that might be used in each of these areas.
Abstract: For many applications, a randomized algorithm is either the simplest or the fastest algorithm available, and sometimes both. This book introduces the basic concepts in the design and analysis of randomized algorithms. The first part of the text presents basic tools such as probability theory and probabilistic analysis that are frequently used in algorithmic applications. Algorithmic examples are also given to illustrate the use of each tool in a concrete setting. In the second part of the book, each chapter focuses on an important area to which randomized algorithms can be applied, providing a comprehensive and representative selection of the algorithms that might be used in each of these areas. Although written primarily as a text for advanced undergraduates and graduate students, this book should also prove invaluable as a reference for professionals and researchers.

785 citations

Journal Article•DOI•
How to use expert advice

[...]

Nicolò Cesa-Bianchi1, Yoav Freund2, David Haussler3, David P. Helmbold3, Robert E. Schapire2, Manfred K. Warmuth3 •
University of Milan1, AT&T Labs2, University of California, Santa Cruz3
01 May 1997-Journal of the ACM
TL;DR: This work analyzes algorithms that predict a binary value by combining the predictions of several prediction strategies, called experts, and shows how this leads to certain kinds of pattern recognition/learning algorithms with performance bounds that improve on the best results currently know in this context.
Abstract: We analyze algorithms that predict a binary value by combining the predictions of several prediction strategies, called experts. Our analysis is for worst-case situations, i.e., we make no assumptions about the way the sequence of bits to be predicted is generated. We measure the performance of the algorithm by the difference between the expected number of mistakes it makes on the bit sequence and the expected number of mistakes made by the best expert on this sequence, where the expectation is taken with respect to the randomization in the predictins. We show that the minimum achievable difference is on the order of the square root of the number of mistakes of the best expert, and we give efficient algorithms that achieve this. Our upper and lower bounds have matching leading constants in most cases. We then show how this leads to certain kinds of pattern recognition/learning algorithms with performance bounds that improve on the best results currently know in this context. We also compare our analysis to the case in which log loss is used instead of the expected number of mistakes.

629 citations

Cites background from "Competitive paging algorithms"

  • ...…also related to that taken in recent work on the competitive ratio of on-line algorithms, and in particular to work on combining on-line algorithms to obtain the best competitive ratio [Fiat et al. 1991a; 1991b; 1994], except that we look at the difference in performance rather than the ratio....

    [...]

Proceedings Article•DOI•
How to use expert advice

[...]

Nicolò Cesa-Bianchi1, Yoav Freund2, David P. Helmbold2, David Haussler2, Robert E. Schapire3, Manfred K. Warmuth2 •
University of Milan1, University of California, Santa Cruz2, Alcatel-Lucent3
01 Jun 1993
TL;DR: This work analyzes algorithms that predict a binary value by combining the predictions of several prediction strategies, called `experts', and shows how this leads to certain kinds of pattern recognition/learning algorithms with performance bounds that improve on the best results currently known in this context.
Abstract: We analyze algorithms that predict a binary value by combining the predictions of several prediction strategies, called `experts''. Our analysis is for worst-case situations, i.e., we make no assumptions about the way the sequence of bits to be predicted is generated. We measure the performance of the algorithm by the difference between the expected number of mistakes it makes on the bit sequence and the expected number of mistakes made by the best expert on this sequence, where the expectation is taken with respect to the randomization in the predictions. We show that the minimum achievable difference is on the order of the square root of the number of mistakes of the best expert, and we give efficient algorithms that achieve this. Our upper and lower bounds have matching leading constants in most cases. We then show how this leads to certain kinds of pattern recognition/learning algorithms with performance bounds that improve on the best results currently known in this context. We also extend our analysis to the case in which log loss is used instead of the expected number of mistakes.

541 citations

Journal Article•DOI•
Competitive algorithms for server problems

[...]

Mark S. Manasse, Lyle A. McGeoch1, Daniel D. Sleator2•
Amherst College1, Carnegie Mellon University2
01 May 1990-Journal of Algorithms
TL;DR: This paper seeks to develop on-line algorithms whose performance on any sequence of requests is as close as possible to the performance of the optimum off-line algorithm.

475 citations

References
More filters
Journal Article•DOI•
Amortized efficiency of list update and paging rules

[...]

Daniel D. Sleator1, Robert E. Tarjan1•
Bell Labs1
01 Feb 1985-Communications of The ACM
TL;DR: This article shows that move-to-front is within a constant factor of optimum among a wide class of list maintenance rules, and analyzes the amortized complexity of LRU, showing that its efficiency differs from that of the off-line paging rule by a factor that depends on the size of fast memory.
Abstract: In this article we study the amortized efficiency of the “move-to-front” and similar rules for dynamically maintaining a linear list. Under the assumption that accessing the ith element from the front of the list takes t(i) time, we show that move-to-front is within a constant factor of optimum among a wide class of list maintenance rules. Other natural heuristics, such as the transpose and frequency count rules, do not share this property. We generalize our results to show that move-to-front is within a constant factor of optimum as long as the access cost is a convex function. We also study paging, a setting in which the access cost is not convex. The paging rule corresponding to move-to-front is the “least recently used” (LRU) replacement rule. We analyze the amortized complexity of LRU, showing that its efficiency differs from that of the off-line paging rule (Belady's MIN algorithm) by a factor that depends on the size of fast memory. No on-line paging algorithm has better amortized performance.

2,378 citations

Proceedings Article•DOI•
Probabilistic computations: Toward a unified measure of complexity

[...]

Andrew Chi-Chin Yao
30 Sep 1977
TL;DR: Two approaches to the study of expected running time of algoritruns lead naturally to two different definitions of intrinsic complexity of a problem, which are the distributional complexity and the randomized complexity, respectively.
Abstract: 1. Introduction The study of expected running time of algoritruns is an interesting subject from both a theoretical and a practical point of view. Basically there exist two approaches to this study. In the first approach (we shall call it the distributional approach), some "natural" distribution is assumed for the input of a problem, and one looks for fast algorithms under this assumption (see Knuth [8J). For example, in sorting n numbers, it is usually assumed that all n! initial orderings of the numbers are equally likely. A common criticism of this approach is that distributions vary a great deal in real life situations; fu.rthermore, very often the true distribution of the input is simply not known. An alternative approach which attempts to overcome this shortcoming by allowing stochastic moves in the computation has recently been proposed. This is the randomized approach made popular by Habin [lOJ(also see Gill[3J, Solovay and Strassen [13J), although the concept was familiar to statisticians (for exa'1lple, see Luce and Raiffa [9J). Note that by allowing stochastic moves in an algorithm, the input is effectively being randomized. We shall refer to such an algoritlvn as a randomized algorithm. These two approaches lead naturally to two different definitions of intrinsic complexity of a problem, which we term the distributional complexity and the randomized complexity, respectively. (Precise definitions and examples will be given in Sections 2 and 3.) To solidify the ideas, we look at familiar combinatorial problems that can be modeled by decision trees. In particular, we consider (a) the testing of an arbitrary graph property from an adjacency matrix (Section 2), and (b) partial order problems on n We will show that for these two classes of problems, the two complexity measures always agree by virtue of a famous theorem, the Minimax Theorem of Von Neumann [14J. The connection between the two approaches lends itself to applications. With two different views (and in a sense complementary to each other) on the complexity of a problem, it is frequently easier to derive upper and lower bounds. For example, using adjacency matrix representation for a graph, it can be shown that no randomized algorithm can determine 2 the existence of a perfect matching in less than O(n) probes. Such lower bounds to the randomized approach were lacking previously. As another example of application , we can prove that for the partial order problems in (b), assuming uniform …

1,188 citations

"Competitive paging algorithms" refers methods in this paper

  • ...If the total expected cost ends up exceeding 7Raghavan ([13], pages 118–9) presents a different proof of this theorem based on a generalization of the minimax principle due to Andy Yao [16]....

    [...]

  • ...Raghavan ([13], pages 118–9) presents a different proof of this theorem based on a generalization of the minimax principle due to Andy Yao [16]....

    [...]

Journal Article•DOI•
Competitive snoopy caching

[...]

Anna R. Karlin1, Mark S. Manasse, Larry Rudolph2, Daniel D. Sleator3•
Stanford University1, Hebrew University of Jerusalem2, Carnegie Mellon University3
01 Nov 1988-Algorithmica
TL;DR: This work presents new on-line algorithms to be used by the caches of snoopy cache multiprocessor systems to decide which blocks to retain and which to drop in order to minimize communication over the bus.
Abstract: In a snoopy cache multiprocessor system, each processor has a cache in which it stores blocks of data. Each cache is connected to a bus used to communicate with the other caches and with main memory. Each cache monitors the activity on the bus and in its own processor and decides which blocks of data to keep and which to discard. For several of the proposed architectures for snoopy caching systems, we present new on-line algorithms to be used by the caches to decide which blocks to retain and which to drop in order to minimize communication over the bus. We prove that, for any sequence of operations, our algorithms' communication costs are within a constant factor of the minimum required for that sequence; for some of our algorithms we prove that no on-line algorithm has this property with a smaller constant.

593 citations

Journal Article•DOI•
Competitive algorithms for server problems

[...]

Mark S. Manasse, Lyle A. McGeoch1, Daniel D. Sleator2•
Amherst College1, Carnegie Mellon University2
01 May 1990-Journal of Algorithms
TL;DR: This paper seeks to develop on-line algorithms whose performance on any sequence of requests is as close as possible to the performance of the optimum off-line algorithm.

475 citations

Proceedings Article•DOI•
Competitive algorithms for on-line problems

[...]

Mark S. Manasse, Lyle A. McGeoch1, Daniel D. Sleator2•
Amherst College1, Carnegie Mellon University2
01 Jan 1988
TL;DR: This paper presents several general results concerning competitive algorithms, as well as results on specific on-line problems.
Abstract: An on-line problem is one in which an algorithm must handle a sequence of requests, satisfying each request without knowledge of the future requests. Examples of on-line problems include scheduling the motion of elevators, finding routes in networks, allocating cache memory, and maintaining dynamic data structures. A competitive algorithm for an on-line problem has the property that its performance on any sequence of requests is within a constant factor of the performance of any other algorithm on the same sequence. This paper presents several general results concerning competitive algorithms, as well as results on specific on-line problems.

412 citations

"Competitive paging algorithms" refers background or methods in this paper

  • ...Server problems were introduced by Manasse, McGeoch and Sleator [10, 11]....

    [...]

  • ...The problem of devising a strongly competitive algorithm for any k and n was solved by McGeoch and Sleator [12]....

    [...]

  • ...[10, 11] showed that for any given algorithm, there is always a lazy one that incurs no more cost....

    [...]

  • ...Sleator and Tarjan [15] analyzed on-line paging algorithms by comparing their performance on any sequence of requests to that of the optimum off-line algorithm (that is, one that has knowledge of the entire sequence of requests in advance)....

    [...]

  • ...Sleator and Tarjan [15] used a slightly different framework to study competitiveness in paging problems....

    [...]

Related Papers (5)
Amortized efficiency of list update and paging rules

[...]

01 Feb 1985-Communications of The ACM
Daniel D. Sleator, Robert E. Tarjan
Online Computation and Competitive Analysis

[...]

01 Jan 1998
Allan Borodin, Ran El-Yaniv
A study of replacement algorithms for a virtual-storage computer

[...]

01 Jun 1966-Ibm Systems Journal
Laszlo A. Belady
Competitive snoopy caching

[...]

01 Nov 1988-Algorithmica
Anna R. Karlin, Mark S. Manasse, Larry Rudolph, Daniel D. Sleator 
Competitive algorithms for server problems

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01 May 1990-Journal of Algorithms
Mark S. Manasse, Lyle A. McGeoch, Daniel D. Sleator
Frequently Asked Questions (11)
Q1. What are the contributions mentioned in the paper "Competitive paging algorithms" ?

In this paper, the authors proposed a method for the analysis of the relationship between computer science degrees and their application in the field of artificial intelligence. 

Karlin et al. [8] have shown that for two servers in a graph that is an isosceles triangle the best competitive factor that can be achieved is a constant that approaches e/(e - 1) z 1.582 as the length of the similar sides go to infinity. 

A randomized on-line algorithm may be viewed as basing its actions on the request sequence (T presented to it and on an infinite sequence p of independent unbiased random bits. 

The marking algorithm is strongly competitive (its competitive factor is Hk) if k = n - 1, but it is not strongly competitive if k < n - 1. 

They showed that LRU running with k servers performs within a factor of k/(k - h + 1) of any off-line algorithm with h 5 k servers and that this is the minimum competitive factor that can be achieved. 

They showed that no deterministic algorithm for the k-server problem can be better than k-competitive, they gave k-competitive algorithms for the case when k = 2 and k = II - 1, and they conjectured that there exists a k-competitive k-server algorithm for any graph. 

The adversary is, however, able to maintain a vector p = (pl, p2,. . . , p,) of probabilities, where pi is the probability that vertex i is not covered by a server. 

In that proof, deterministic on-line algorithms B(l), B(2), . . . , B(m) of type (k, n) were given, and the deterministic on-line algorithm A of type (k, n) was constructed to be &)-competitive against B(i) for each i. 

If the total expected cost ends up exceeding l/u, then an arbitrary request is made to an unmarked vertex, and the subphase is over. 

During this phase exactly the vertices of S were requested, so since A is lazy, the authors know that at least d’ of A’s servers were outside of S during the entire phase. 

Armed with these tools (the marking and the probability vector), the adversary can generate a sequence such that the expected cost of each phase to A is H,,-l, and the cost to the optimum off-line algorithm is 1.