Optimal Permutation Routing for Low-dimensional Hypercubes

We consider the offline problem of routing a permutation of tokens on the nodes of a d-dimensional hypercube, under a queueless MIMD communication model (under the constraints that each hypercube edge may only communicate one token per communication step, and each node may only be occupied by a single token between communication steps). For a d-dimensional hypercube, it is easy to see that d communication steps are necessary. We develop a theory of “separability ” which enables an analytical proof that d steps suffice for the case d = 3, and facilitates an experimental verification that d steps suffice for d = 4. This result improves the upper bound for the number of communication steps required to route an arbitrary permutation on arbitrarily large hypercubes to 2d − 4. We also find an interesting side-result, that the number of possible communication steps in a d-dimensional hypercube is the same as the number of perfect matchings in a (d + 1)-dimensional hypercube, a combinatorial quantity for which there is no closed-form expression. Finally we present some experimental observations which may lead to a proof of a more general result for arbitrarily large dimension d. 2

Diameter-preserving Orientations of the Torus

The diameter of a directed graph is the maximum of the lengths of the shortest paths between all pairs of vertices. A directed graph is said to be tightly oriented if it has the same diameter as its undirected image graph. Our main result is tight orientations for all sufficiently large toroids, except those whose sizes in both dimensions are odd. We also prove the impossibility of tightly orienting all the toroids for which we do not present tight orientations, and we give partial results for dimensionality higher than two. Key words: Torus, toroidal grids, diameter, combinatorial problem, gossiping 1 Introduction An orientation of an undirected graph is an assignment of directions to the edges so as to produce a directed graph. Among the properties of orientations that are of interest are preserving strong connectivity [1] and minimizing distances [2]. We address the problem of finding tight orientations : those that preserve the graph's diameter (the maximum distance between any pa..

Gossiping in Minimal Time

The gossip problem involves communicating a unique item from each node in a graph to every other node. We study the minimum time required to do this under the weakest model of parallel communication which allows each node to participate in just one communication at a time as either sender or receiver. We study a number of topologies including the complete graph, grids, hypercubes and rings. Definitive new optimal time algorithms are derived for complete graphs, rings, regular grids and toroidal grids that significantly extend existing results. In particular, we settle an open problem about minimum time gossiping in complete graphs. Specifically, for a graph with N nodes, at least log ae N communication steps, where the logarithm is in the base of the golden ratio ae, are required by any algorithm under the weakest model of communication. This bound, which is approximately 1:44 log 2 N , can be realized for some networks and so the result is optimal. KEYWORDS: Gossiping, broadcasting. ..

Minimum eccentricity multicast trees

We consider the problem of constructing a multicast tree that connects a group of source nodes to a group of sink nodes (receivers) and minimizes the maximum end-to-end delay between any pair of source/sink nodes. This is known as the minimum eccentricity multicast tree problem, and is directly related to the quality of service requirements of real multipoint applications. We deal directly with the problem in its general form, meaning that the sets of source and sink nodes need not be overlapping nor disjoint. The main contribution of this work is a polynomial algorithm for this problem on general networks which is inspired by an innovative method that uses geometric relationships on the xy-plane

North-Holland FIXED HYPERCUBE EMBEDDING *

The problem of embedding a graph into a fixed-size hypercube is shown to be NP-complete. This work complements recent work of the present authors showing that deciding whether a graph is embeddable into any size hypercube is NP-complete as well. The reduction is from 3-partition. Keywords: Hypercube, graph embedding, NP-complete, complexity 1

Tight bounds for connecting sites across barrieres

Given m points (sites) and n obstacles (barriers) in the plane, we address the problem of finding a straight-line minimum cost spanning tree on the sites, where the cost is proportional to the number of intersections (crossings) between tree edges and barriers. If the barriers are infinite lines then there is a spanning tree where every barrier is crossed by O ( √ m) tree edges (connectors), and this bound is asymptotically optimal (spanning tree with low stabbing number). Asano et al. showed that if the barriers are pairwise disjoint line segments, then there is a spanning tree such that every barrier crosses at most 4 tree edges and so the total cost is at most 4n. Constructionswith3crossings per barrier and 2n total cost provide a lower bound. We obtain tight bounds on the minimum cost spanning tree in the most exciting special case where the barriers are interior disjoint line segments that form a convex subdivision and there is a point in every cell. In particular, we show that there is a spanning tree such that every barrier is crossed by at most 2 tree edges, and there is a spanning tree of total cost 5n/3. Both bounds are tight

Consistent models for electrical networks with distributed parameters

summary:A system of one-dimensional linear parabolic equations coupled by boundary conditions which include additional state variables, is considered. This system describes an electric circuit with distributed parameter lines and lumped capacitors all connected through a resistive multiport. By using the monotony in a space of the form $L^2(0,T;H^1)$, one proves the existence and uniqueness of a variational solution, if reasonable engineering hypotheses are fulfilled