I. Hal Sudborough

Combinatorial Optimization of Group Key Management

Given the growing number of group applications in many existing and evolving domains recent attention has been focused on secure multicasting over the Internet. When such systems are required to manage large groups that undergo frequent fluctuations in group membership, the need for efficient encryption key management becomes critical. This paper presents a new key management framework based on a combinatorial formulation of the group multicast key management problem that is applicable to the general problem of managing keys for any type of trusted group communication, regardless of the underlying transmission method between group participants. Specifically, we describe Exclusion Basis Systems and show exactly when they exist. In addition, the framework separates key management from encrypted message transmission, resulting in a more efficient implementation of key management.

On the Diameter of the Pancake Network

Then-dimensionalpancake network,Pn, has processors labeled with each of then! distinct permutations of lengthnand a connection between two processors when the label of one is obtained from the other by some prefix reversal. Each permutation is considered as a stack of different size pancakes. The well knownpancake problemconcerns the number,f(n), of prefix reversals required to sortnpancakes.We describe a (9/8)n+2 step sorting sequence for Gates and Papadimitrious stack ofnpancakes, ?n, used for establishing their lower bound, thus disproving the conjecture that (19/16)nsteps are required. Furthermore, we improve their lower bound by showingf(n)?(15/14)n. In fact, we define for eachn?0 (mod14) a stack of pancakes, ?n, and show that (15/14)n?f(?n)?(8/7)n?1.We show that ?In, the conjectured hardest stack of burnt pancakes, can be sorted in (3(n+1))/2 steps, for alln?3 (mod4) andn?23. If ?Inis indeed hardest, this implies that both the “burnt” and “unburnt” pancake networks of dimensionnhave diameter at most (3(n+1))/2.Values off(n), forn?11, were given previously. We note thatf(12)=14,f(13)=15, andf(19)?22.

Embedding grids into hypercubes

Abstract We consider efficient simulations of mesh connected networks (or good representations of array structures) by hypercube machines. In particular, we consider embedding a mesh or grid G into the smallest hypercube that at least as many points as G, called the optimal hypercube for G. In order to minimize simulation time we derive embeddings which minimize dilation, i.e., the maximum distance in the hypercube between images of adjacent points of G. Our results are: (1) There is a dilation 2 embedding of the [m × k] grid into its optimal hypercube, provided that ⌈ log m⌉+ log mk 2 ⌈ log m⌉ + log m 2 ⩽⌈ log mk⌉ and (2) For any k B k = a d ∏ k i=1 a i ∏ k i=1 2 ⌈ log a i ⌉ + ∑ i=1 k ⌊ log a 1 ⌋ 2 ⌉

Note: Short proofs for cut-and-paste sorting of permutations

We consider the problem of determining the maximum number of moves required to sort a permutation of [n] using cut-and-paste operations, in which a segment is cut out and then pasted into the remaining string, possibly reversed. We give short proofs that every permutation of [n] can be transformed to the identity in at most @?2n/3@? such moves and that some permutations require at least @?n/2@? moves.

Bounding prefix transposition distance for strings and permutations

A transposition is an operation that exchanges two adjacent substrings. When it is restricted so that one of the substrings is a prefix, it is called a prefix transposition. The prefix transposition distance between a pair of strings (permutations) is the shortest sequence of prefix transpositions required to transform a given string (permutation) into another given string (permutation). This problem is a variation of the transposition distance problem, related to genome rearrangements. An upper bound of n-1 and a lower bound of n/2 are known. We improve the bounds to n-log8 n and 2n/3 respectively. We also give upper and lower bounds for the prefix transposition distance on strings. For example, n/2 prefix transpositions are always sufficient for binary strings. We also prove that the exact prefix transposition distance problem on strings is NP complete.

Comparing star and pancake networks

Low dilation embedding are used to compare the computational capabilities of star and pancake networks.

A faster and simpler 2-approximation algorithm for block sorting

Block sorting is used in connection with optical character recognition (OCR). Recent work has focused on finding good strategies which perform well in practice. Block sorting is

On the complexity of tree embedding problems

\mathcal{NP}

Calibrating embedded protocols on asynchronous systems

-hard and all of the previously known heuristics lack proof of any approximation ratio. We present here an approximation algorithm for the block sorting problem with approximation ratio of 2 and run time O(n2). The approximation algorithm is based on finding an optimal sequence of absolute block deletions.

A quadratic time 2-approximation algorithm for block sorting

The papier is relative to how a tree can be embedded into another while minimizing various costs. It considers the general complexity of the problem and shows that there exist polynomial time algorithms hich determine whether a graph can be embedded into a complete binary tree with fixed dilation k, or fixed congestion cost k