Narrow Results

This paper introduces a calculus of weakest specification for supporting reuse of established components in deriving a design (in the sense of formal methods). The weakest specifunction generalizes the notions of weakest pre-specification and weakest parallel environment; but instead of calculating the weakest required component of a target specification, it calculates the weakest specification function whose value refines the target when applied to an established component. In particular it overcomes the restriction of those other calculi to taking merely one required component at a time.

The theory of specifunctions is applied to a new weakest-design calculus in the context of BSP. The calculus is based on the par-seq specifunction which involves two required components: it places one established component in parallel with one required component and the result in sequence with another required component to meet a given specification. A calculus is provided for the par-seq specifunction and it is applied to the derivation of a distributed BSP algorithm for greatest common divisor.

We present simple and efficient parallel algorithms for obtaining a chordless cycle of length greater than or equal to k in a graph whenever such a cycle exists. Our results generalize and simplify existing results for detecting chordless cycles.

Two techniques for managing memory on a parallel random access machine (PRAM) are presented. One is a scheme for an n/log n processor EREW PRAM that dynamically allocates and deallocates up to n records of the same size in O(log n) time. The other is a simulation of a PRAM with initialized memory by one with uninitialized memory. A CREW PRAM variant of the technique justifies the assumption that memory can be assumed to be appropriately initialized with no asymptotic increase in time but a factor of n increase in space. An EREW PRAM solution incurs a factor of O(log n) increase in time but only a constant factor increase in space.

We consider the problem of copying information stored initially in a single memory cell by Exclusive Read Exclusive Write Parallel Random Access Machines (EREW PRAMs). We prove lower bounds for this problem and present algorithms matching them tightly (in many cases up to an additive constant). The bounds presented depend on the number of cells used and size of information copied. The lower bounds apply also to functions where a change of a single argument influences the output in many memory locations.

We describe a randomized parallel algorithm to solve the single function coarsest partition problem. The algorithm runs in O(log n) time using O(n) operations with high probability on the Priority CRCW PRAM. The previous best known algorithms run in O(log² n) time using O(n log² n) operations on the CREW PRAM and O(log n) time using O(n log log n) operations on the Arbitrary CRCW PRAM. The technique presented can be used to generate the Euler tour of a rooted tree optimally from the parent representation.

Distance hereditary graphs are graphs in which every two vertices have the same distance in every connected induced subgraph containing them. In this paper, we study properties of distance hereditary graphs from the view point of parallel computations. We present efficient parallel algorithms for finding a minimum weighted connected dominating set, a minimum weighted Steiner tree, which take O(log n) time using O(n + m) processors on CRCW PRAM, where n and m are the number of vertices and edges of a given graph, respectively. We also find a maximum weighted clique of a distance hereditary graph in O(log² n) time using O(n + m) processors on a CREW PRAM.