M. D. Atkinson

Min-max heaps and generalized priority queues

A simple implementation of double-ended priority queues is presented. The proposed structure, called a min-max heap, can be built in linear time; in contrast to conventional heaps, it allows both</italic> FindMin <italic>and</italic> FindMax <italic>to be performed in constant time;</italic> Insert, DeleteMin, <italic>and</italic> DeleteMax <italic>operations can be performed in logarithmic time. Min-max heaps can be generalized to support other similar order-statistics operations efficiently (e.g., constant time</italic> FindMedian <italic>and logarithmic time</italic> DeleteMedian<italic>); furthermore, the notion of min-max ordering can be extended to other heap-ordered structures, such as leftist trees.

Simple permutations and pattern restricted permutations

A simple permutation is one that does not map any non-trivial interval onto an interval. It is shown that, if the number of simple permutations in a pattern restricted class of permutations is finite, the class has an algebraic generating function and is defined by a finite set of restrictions. Some partial results on classes with an infinite number of simple permutations are given. Examples of results obtainable by the same techniques are given; in particular it is shown that every pattern restricted class properly contained in the 132-avoiding permutations has a rational generating function.

Integer Sets with Distinct Sums and Differences and Carrier Frequency Assignments for Nonlinear Repeaters

The problem of assigning n carrier frequencies so as to avoid certain types (third and fifth order) of intermodulation interference is discussed. For the third-order case, close upper and lower bounds on the optimal solution are established; and close to optimal solutions are given for n (previously, suboptimal solutions were known only for n \leq 23 ). For the fifth-order case, it is shown that some existing results can be applied to this problem, and suboptimal solutions obtained by this construction are given for n \leq 17 (no solutions were known previously).

The cyclic towers of Hanoi

The famous Towers of Hanoi puzzle consists of 3 pegs (A, B, C) on one of which (A) are stacked n rings of different sizes, each ring resting on a larger ring. The objective is to move the n rings one by one until they are all stacked on another peg (B) in such a way that no ring is ever placed on a smaller ring; the other peg (C) can be used as workspace. The problem has tong been a favourite iir programming courses as one which admits a concise recursive solution. This solution hinges on the observation that, when the largest ring is moved from A to B, the n 1 remaining rings must all be on peg C. This immediately leads to the recursive procedure

An optimal algorithm for geometrical congruence

Abstract An algorithm is given which, in time O ( n log n ), determines all the Euclidean congruences (if any) between two n -point sets in 3-dimensional space. The algorithm is shown to be optimal to within a constant factor.

Partially Well-Ordered Closed Sets of Permutations

It is known that the “pattern containment” order on permutations is not a partial well-order. Nevertheless, many naturally defined subsets of permutations are partially well-ordered, in which case they have a strong finite basis property. Several classes are proved to be partially well-ordered under pattern containment. Conversely, a number of new antichains are exhibited that give some insight as to where the boundary between partially well-ordered and not partially well-ordered classes lies.

Restricted permutations and the wreath product

Restricted permutations are those constrained by having to avoid subsequences ordered in various prescribed ways. A closed set is a set of permutations all satisfying a given basis set of restrictions. A wreath product construction is introduced and it is shown that this construction gives rise to a number of useful techniques for deciding the finite basis question and solving the enumeration problem. Several applications of these techniques are given.

Algorithms for Pattern Involvement in Permutations

We consider the problem of developing algorithms for the recognition of a fixed pattern within a permutation. These methods are based upon using a carefully chosen chain or tree of subpatterns to build up the entire pattern. Generally, large improvements over brute force search can be obtained. Even using on-line versions of these methods allow for such improvements, though often not as great as for the full method. Furthermore, by using carefully chosen data structures to fine tune the methods, we establish that any pattern of length 4 can be detected in O(n log n) time. We also improve the complexity bound for detection of a separable pattern from O(n6) to O(n5 log n).

Regular closed sets of permutations

Machines whose main purpose is to permute and sort data are studied. The sets of permtutations that can arise are analysed by means of finite automata and avoided pattern techniques. Conditions are given for these sets to be enumerated by rational generating functions. As a consequence we give the first non-trivial examples of pattern closed sets of permutations all of whose closed subclasses have rational generating functions.

On the maximal multiplicative complexity of a family of bilinear forms

Abstract The number of nonscalar multiplications required to evaluate a general family of bilinear forms is investigated. An upper bound is obtained which is about half that obtained from naive arguments. In certain cases the best possible upper bound is obtained.