Johnson: Computers and Intractability-A Guide to the Theory of NP-Completeness

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

An algorithm is presented which finds all occurrences of one given string within another, in running time proportional to the sum of the lengths of the strings. The constant of proportionality is low enough to make this algorithm of practical use, and the procedure can also be extended to deal with some more general pattern-matching problems. A theoretical application of the algorithm shows that the set of concatenations of even palindromes, i.e., the language $\{\alpha \alpha ^R\}^*$, can be recognized in linear time. Other algorithms which run even faster on the average are also considered.

Fast Pattern Matching in Strings

B-trees have become, de facto, a standard for file organization. File indexes of users, dedicated database systems, and general-purpose access methods have all been proposed and nnplemented using B-trees This paper reviews B-trees and shows why they have been so successful It discusses the major variations of the B-tree, especially the B+-tree, contrasting the relatwe merits and costs of each implementatmn. It illustrates a general purpose access method whmh uses a B-tree.

http://people.cs.aau.dk/~simas/aalg06/UbiquitBtree.pdf

Ubiquitous B-Tree

In 1970, Knuth, Pratt, and Morris [1] showed how to do basic pattern matching in linear time. Related problems, such as those discussed in [4], have previously been solved by efficient but sub-optimal algorithms. In this paper, we introduce an interesting data structure called a bi-tree. A linear time algorithm for obtaining a compacted version of a bi-tree associated with a given string is presented. With this construction as the basic tool, we indicate how to solve several pattern matching problems, including some from [4] in linear time.

Linear pattern matching algorithms

In this paper we look at a number of matching problems and devise general techniques for attacking such problems. In particular, we describe a strategy for constructing efficient algorithms for solving two types of matching problems. We use this strategy to develop explicit algorithms for these two problems applied to strings (where the patterns are substrings) and arrays (where the patterns are subarrays or blocks). We also develop algorithms for these and related problems for trees, where the patterns are subtrees. Certain special cases of these algorithms are also discussed. Although we do not claim that these algorithms are optimal, we analyze each algorithm to estimate its computational cost. This provides some basis for choosing which algorithm is most desirable in any given situation.

Rapid identification of repeated patterns in strings, trees and arrays

We study the graph-theoretic problem of embedding a graph in a book with its vertices in a line along the spine of the book and its edges on the pages in such a way that edges residing on the same page do not cross. This problem abstracts layout problems arising in the routing of multilayer printed circuit boards and in the design of fault-tolerant processor arrays. In devising an embedding, one strives to minimize both the number of pages used and the “cutwidth” of the edges on each page. Our main results (1) present optimal embeddings of a variety of families of graphs; (2) exhibit situations where one can achieve small pagenumber only at the expense of large cutwidth; and (3) establish bounds on the minimum pagenumber of a graph based on various structural properties of the graph. Notable in the last category are proofs that (a) every n-vertex d-valent graph can be embedded using $O( dn^{1/ 2} )$ pages, and (b) for every $d > 2$ and all large n, there are n-vertex d-valent graphs whose pagenumber is at...

/pdf/embedding-graphs-in-books-a-layout-problem-with-applications-46pzc5se35.pdf

Embedding graphs in books: a layout problem with applications to VLSI design

The authors develop an algebraic framework that exposes the structural kinship among the deBruijn, shujCfle-exchange, butterjy, and cube-connected cycles networks and illustrate algorithmic benefits that ensue from the exposed relationships. The framework builds on two alge- braically specified genres of graphs: A group action graph (GAG, for short) is given by a set V of vertices and a set H of permutations of V: For each v E V and each r EII, there is an arc labeled r from vertex v to vertex v. A Cayley graph is a GAG (V, H), where V is the group Gr(H) generated by H and where each r 6 H acts on each g 6 Gr(H) by right multiplication. The graphs (Gr(H), H) and (V, H) are called associated graphs. It is shown that every GAG is a quotient graph of its associated Cayley graph. By applying such general results, the authors determine the following: The butterfly network (a Cayley graph) and the deBruijn network (a GAG) are associated graphs. The cube-connected cycles network (a Cayley graph) and the shuffle-exchange network (a GAG) are associated graphs. The order-n instance of both the butterfly and the cube-connected cycles share the same underlying group, but have slightly different generator sets II. By analyzing these algebraic results, it is delimited, for any Cayley graph G and associated GAG 7-/, a family of "leveled" algorithms which run as efficiently on T/ as they do on (the much larger) G. Further analysis of the results yields new, surprisingly efficient simulations by the shuffle-oriented networks (the shuffle-exchange and deBruijn networks) of like-sized butterfly-oriented networks (the butterfly and cube-connected cycles networks): An N-vertex butterfly-oriented network can be simulated by the smallest shuffle-oriented network that is big enough to hold it with slowdown O(log log N). This simulation is exponentially faster than the anticipated logarithmic slowdown. The mappings that underlie the simulation can be computed in linear time; and they afford one an algorithmic tech- nique for translating any program developed for a butterfly-oriented architecture into an equivalent program for a shuffle-oriented architecture, the latter program incurring only the indicated slowdown factor.

Group action graphs and parallel architectures

The languages recognizable by time- and space-restricted multiple-counter machines are compared to the languages recognizable by similarly restricted multipletape Turing machines. Special emphasis is placed on languages definable by machines which operate in “real time”. Time and space requirements for counter machines and Turing machines are analyzed. A number of questions which remain open for time-restricted Turing machines are settled for their counter machine counterparts.

Arnold L. Rosenberg

Papers

Rapid identification of repeated patterns in strings, trees and arrays

Embedding graphs in books: a layout problem with applications to VLSI design

Group action graphs and parallel architectures

Counter machines and counter languages

Embedding graphs in books: a layout problem with applications to VLSI design