We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

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.

Probabilistic approximation of metric spaces and its algorithmic applications

Text algorithms

Preface 1. Basic Concepts of Molecular Biology 2. Strings, Graphs, and Algorithms 3. Sequence Comparison and Database Search 4. Fragment Assembly of DNA 5. Physical Mapping of DNA 6. Phylogenetic Trees 7. Genome Rearrangements 8. Molecular Structure Prediction 9. Epilogue: Computing with DNA Answers to Selected Exercises / References / Index

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Introduction to Computational Molecular Biology

1. Programs. 2. Functions and Categories. 3. Applications. 4. Relationships and Allegories. 5. Datatypes in Allegories. 6. Optimisation Problems. 7. Thinning Algorithms. 8. Dynamic Programming. 9. Greedy Algorithms. Appendices.

Algebra of programming

The k-server problem is investigated when the metric space is a tree. For this case an on-line k-competitive algorithm for k-servers is presented. The competitiveness ratio k is optimal. The algorithm is memoryless, in the sense that it does not use any information from the past.

An optimal on-line algorithm for K-servers on trees

The string editing problem for input strings x and y consists of transforming x into y by performing a series of weighted edit operations on x of overall minimum cost. An edit operation on x can be the deletion of a symbol from x, the insertion of a symbol in x or the substitution of a symbol x with another symbol. This problem has a well known O((absolute value of x)(absolute value of y)) time sequential solution (25). The efficient Program Requirements Analysis Methods (PRAM) parallel algorithms for the string editing problem are given. If m = ((absolute value of x),(absolute value of y)) and n = max((absolute value of x),(absolute value of y)), then the CREW bound is O (log m log n) time with O (mn/log m) processors. In all algorithms, space is O (mn).

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Efficient parallel algorithms for string editing and related problems

An O(nL)-time algorithm is introduced for constructing an optimal Huffman code for a weighted alphabet of size n, where each code string must have length no greater than L. The algorithm uses O(n) space.

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A fast algorithm for optimal length-limited Huffman codes

2 The Server Problem 5 2.1 Convex Hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 The Equipoise Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 The Work Function Algorithm for k Servers . . . . . . . . . . . . . . . . . . . . 17 2.3.1 A Conjectured Outline of the Proof of Optimality . . . . . . . . . . . . 20 2.3.2 WFA is 2-Competitive for 2 Servers . . . . . . . . . . . . . . . . . . . . 22

The Server Problem and On-Line Games.

The least weight subsequence (LWS) problem is introduced, and is shown to be equivalent to the classic minimum path problem for directed graphs. A special case of the LWS problem is shown to be solvable in $O(n\log n)$ time generally and, for certain weight functions, in linear time. A number of applications are given, including an optimum paragraph formation problem and the problem of finding a minimum height B-tree, whose solutions realize improvement in asymptotic time complexity.

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Lawrence L. Larmore

Papers

An optimal on-line algorithm for K-servers on trees

Efficient parallel algorithms for string editing and related problems

A fast algorithm for optimal length-limited Huffman codes

The Server Problem and On-Line Games.

The least weight subsequence problem