Recognizing Partial Cubes in Quadratic Time

We show how to test whether a graph with n vertices and m edges is a partial cube, and if so how to find a distance-preserving embedding of the graph into a hypercube, in the near-optimal time bound O(n^2), improving previous O(nm)-time solutions.Comment: 25 pages, five figures. This version significantly expands previous versions, including a new report on an implementation of the algorithm and experiments with i

Optimal simulation of full binary trees on faulty hypercubes

The problem of operating full binary tree based algorithms on a hypercube with faulty nodes was investigated. Developing a method for embedding a full binary tree into the faulty hypercube is the solution to this problem. Two outcomes for embedding an (n-1)-tree into an n-cube with unit dilation and load, that were based on a new embedding technique, were presented. For the problem where the root can be mapped to any nonfaulty hypercube node, the optimum toleration of faults was shown. Moreover, it was demonstrated that the algorithm for the variable root embedding problem is maximal within a class algorithms called recursive embedding algorithms as far as the number of tolerable faults is concerned. Lastly, it was demonstrated that when an O(1/√n) fraction of nodes in the hypercube are faulty, a O(1)-load variable root embedding is not always possible regardless of the significance of the dilation.published_or_final_versio

Lecture notes: Semidefinite programs and harmonic analysis

Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th International Workshop on High Performance Optimization Techniques (Algebraic Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg University, The Netherlands.Comment: 31 page

Embedding cube-connected cycles graphs into faulty hypercubes

We consider the problem of embedding a cube-connected cycles graph (CCC) into a hypercube with edge faults. Our main result is an algorithm that, given a list of faulty edges, computes an embedding of the CCC that spans all of the nodes and avoids all of the faulty edges. The algorithm has optimal running time and tolerates the maximum number of faults (in a worst-case setting). Because ascend-descend algorithms can be implemented efficiently on a CCC, this embedding enables the implementation of ascend-descend algorithms, such as bitonic sort, on hypercubes with edge faults. We also present a number of related results, including an algorithm for embedding a CCC into a hypercube with edge and node faults and an algorithm for embedding a spanning torus into a hypercube with edge faults