Raymond Greenlaw

A compendium of problems complete for symmetric logarithmic space

Abstract. The papers main contributions are a compendium of problems that are complete for symmetric logarithmic space (SL), a collection of material relating to SL, a list of open problems, and an extension to the number of problems known to be SL-complete. Complete problems are one method of studying SL, a class for which programming is non-intuitive. Our exposition helps make the class SL less mysterious and more accessible to other researchers.

Computing Prüfer codes efficiently in parallel

Abstract A Prufer code of a labeled free tree with n nodes is a sequence of length n−2 constructed by the following sequential process: for i ranging from 1 to n−2 insert the label of the neighbor of the smallest remaining leaf into the ith position of the sequence, and then delete the leaf. Prufer codes provide an alternative to the usual representation of trees. We present an optimal O ( log n) time, n/ log n processor EREW-PRAM algorithm for determining the Prufer code of an n-node labeled chain and an O ( log n) time, n processor EREW-PRAM algorithm for constructing the Prufer code of an n-node labeled free tree. This resolves an open question posed by Wang et al. (IEEE Trans. Parallel Distributed Systems 8 (12) (1997) 1236–1240).

The graph relabeling problem and its variants

Graph labeling is a classic problem in mathematics and computing. In this paper we study an interesting set of graph labeling problems which were first introduced by Kantabutra (2007). The general problem, here called the graph relabeling problem, is to take an undirected graph G=(V, E), two labelings l1 and l2 of G, and a label switching function f and then to determine the complexity of transforming the labeling l1 into l2 using f. We define several variants of the problem and discuss their complexity. We give tight bounds for one version of the problem on chains, and show another version is NP-complete. These problems have applications in areas such as bioinformatics, networks, and VLSI.

The Structure of Rooted Weighted Trees Modeling Layered Cyber-security Systems

In this paper we consider the structure and topology of a layered-security model in which the containers and their nestings are given in the form of a rooted tree T. A cyber-security model is an ordered three-tuple M = (T, C, P) where C and P are multisets of penetration costs for the containers and target-acquisition values for the prizes that are located within the containers, respectively, both of the same cardinality as the set of the non-root vertices of T. The problem that we study is to assign the penetration costs to the edges and the target-acquisition values to the vertices of the tree T in such a way that minimizes the total prize that an attacker can acquire given a limited budget. The attacker breaks into containers starting at the root of T and once a vertex has been broken into, its children can be broken into by paying the associated penetration costs. The attacker must deduct the corresponding penetration cost from the budget, as each new container is broken into. For a given assignment of costs and target values we obtain a security system. We show that in general it is not possible to develop an optimal security system for a given cyber-security model M. We define P- and C-models where the penetration costs and prizes, respectively, all have unit value. We show that if T is a rooted tree such that any P- or C-model M = (T, C, P) has an optimal security system, then T is one of the following types: (i) a rooted path, (ii) a rooted star, (iii) a rooted 3-caterpillar, or (iv) a rooted 4-spider. Conversely, if T is one of these four types of trees, then we show that any P- or C-model M = (T, C, P) does have an optimal security system. Finally, we study a duality between P- and C-models that allows us to translate results for P-models into corresponding results for C-models and vice versa. The results obtained give us some mathematical insights into how layered-security defenses should be organized.

Fast sequential and parallel algorithms for label selection to obtain space efficient implementations in a software configuration management system

Software configuration management (SCM) systems require the ability to maintain collections of a large number of specific versions of files. One technique for managing these configurations is to maintain a single time-stamp identifying which versions of the files belong to the configuration along with an exception list to that time-stamp. Given the individual time-stamps of n files represented by half-open intervals, we develop an O(n log n) tone sequential algorithm that finds a time-stamp leading to the smallest possible exception list. The technique is used in a commercial SCM system and works well in practice. We also develop a parallel version of the algorithm that runs in O(log n) time using n processors on an EREW-PRAM. If the intervals are sorted, the sequential algorithm runs optimally in O(n) time and the parallel algorithm runs optimally in O(log n) time using n/log n processors on an EREW-PRAM.