Description bttroductian. A maximal complete subgraph (clique) is a complete subgraph that is not contained in any other complete subgraph. A recent paper [1] describes a number of techniques to find maximal complete subgraphs of a given undirected graph. In this paper, we present two backtracking algorithms, using a branchand-bound technique [4] to cut off branches that cannot lead to a clique. The first version is a straightforward implementation of the basic algorithm. It is mainly presented to illustrate the method used. This version generates cliques in alphabetic (lexicographic) order. The second version is derived from the first and generates cliques in a rather unpredictable order in an attempt to minimize the number of branches to be traversed. This version tends to produce the larger cliques first and to generate sequentially cliques having a large common intersection. The detailed algorithm for version 2 is presented here. Description o f the algorithm--Version 1. Three sets play an important role in the algorithm. (1) The set compsub is the set to be extended by a new point or shrunk by one point on traveling along a branch of the backtracking tree. The points that are eligible to extend compsub, i.e. that are connected to all points in compsub, are collected recursively in the remaining two sets. (2) The set candidates is the set of all points that will in due time serve as an extension to the present configuration of compsub. (3) The set not is the set of all points that have at an earlier stage already served as an extension of the present configuration of compsub and are now explicitly excluded. The reason for maintaining this set trot will soon be made clear. The core of the algorithm consists of a recursively defined extension operator that will be applied to the three sets Just described. It has the duty to generate all extensions of the given configuration of compsub that it can make with the given set of candidates and that do not contain any of the points in not. To put it differently: all extensions of compsub containing any point in not have already been generated. The basic mechanism now consists of the following five steps:

Algorithm 457: finding all cliques of an undirected graph

Association rule discovery has emerged as an important problem in knowledge discovery and data mining. The association mining task consists of identifying the frequent itemsets, and then forming conditional implication rules among them. We present efficient algorithms for the discovery of frequent itemsets which forms the compute intensive phase of the task. The algorithms utilize the structural properties of frequent itemsets to facilitate fast discovery. The items are organized into a subset lattice search space, which is decomposed into small independent chunks or sublattices, which can be solved in memory. Efficient lattice traversal techniques are presented which quickly identify all the long frequent itemsets and their subsets if required. We also present the effect of using different database layout schemes combined with the proposed decomposition and traversal techniques. We experimentally compare the new algorithms against the previous approaches, obtaining improvements of more than an order of magnitude for our test databases.

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Scalable algorithms for association mining

Discovery of association rules is an important problem in database mining. In this paper we present new algorithms for fast association mining, which scan the database only once, addressing the open question whether all the rules can be efficiently extracted in a single database pass. The algorithms use novel itemset clustering techniques to approximate the set of potentially maximal frequent itemsets. The algorithms then make use of efficient lattice traversal techniques to generate the frequent itemsets contained in each cluster. We propose two clustering schemes based on equivalence classes and maximal hypergraph cliques, and study two traversal techniques based on bottom-up and hybrid search. We also use a vertical database layout to cluster related transactions together. Experimental results show improvements of over an order of magnitude compared to previous algorithms.

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New algorithms for fast discovery of association rules

The maximum clique problem is a classical problem in combinatorial optimization which finds important applications in different domains. In this paper we try to give a survey of results concerning algorithms, complexity, and applications of this problem, and also provide an updated bibliography. Of course, we build upon precursory works with similar goals [39, 232, 266].

The maximum clique problem

This paper presents a unifying procedure, called Facet, for the automated synthesis of data paths at the register-transfer level. The procedure minimizes the number of storage elements, data operators, and interconnection units. A design generator named Emerald, based on Facet, was developed and implemented to facilitate extensive experiments with the methodology. The input to the design generator is a behavioral description which is viewed as a code sequence. Emerald provides mechanisms for interactively manipulating the code sequence. Different forms of the code sequence are mapped into data paths of different cost and speed. Data paths for the behavioral descriptions of the AM2910, the AM2901, and the IBM System/370 were produced and analyzed. Designs for the AM2910 and the AM2901 are compared with commercial designs. Overall, the total number of gates required for Emerald's designs is about 15 percent more than the commercial designs. The design space spanned by the behavioral specification of the AM2901 is extensively explored.

Automated Synthesis of Data Paths in Digital Systems

Recently Augustson and Minker presented a version of the Bierstone algorithm for finding the set of cliques of a finite undirected linear graph. Their version contains two errors. In this paper the counterexamples to their version and the modified version of the Bierstone algorithm are presented.

Gordon D. Mulligan

Papers

Corrections to Bierstone's Algorithm for Generating Cliques