A powerful heuristic for telephone gossiping

A refined heuristic for computing schedules for gossiping in the telephone model is presented. The heuristic is fast: for a network with n nodes and m edges, requiring R rounds for gossiping, the running time is O(R n log(n) m) for all tested classes of graphs. This moderate time consumption allows to compute gossiping schedules for networks with more than 10,000 PUs and 100,000 connections. The heuristic is good: in practice the computed schedules never exceed the optimum by more than a few rounds. The heuristic is versatile: it can also be used for broadcasting and more general information dispersion patterns. It can handle both the unit-cost and the linear-cost model. Actually, the heuristic is so good, that for CCC, shuffle-exchange, butterfly de Bruijn, star and pancake networks the constructed gossiping schedules are better than the best theoretically derived ones. For example, for gossiping on a shuffle-exchange network with 2^{13} PUs, the former upper bound was 49 rounds, while our heuristic finds a schedule requiring 31 rounds. Also for broadcasting the heuristic improves on many formerly known results. A second heuristic, works even better for CCC, butterfly, star and pancake networks. For example, with this heuristic we found that gossiping on a pancake network with 7! PUs can be performed in 15 rounds, 2 fewer than achieved by the best theoretical construction. This second heuristic is less versatile than the first, but by refined search techniques it can tackle even larger problems, the main limitation being the storage capacity. Another advantage is that the constructed schedules can be represented concisely

Dynamic Monopolies in Colored Tori

The {\em information diffusion} has been modeled as the spread of an information within a group through a process of social influence, where the diffusion is driven by the so called {\em influential network}. Such a process, which has been intensively studied under the name of {\em viral marketing}, has the goal to select an initial good set of individuals that will promote a new idea (or message) by spreading the "rumor" within the entire social network through the word-of-mouth. Several studies used the {\em linear threshold model} where the group is represented by a graph, nodes have two possible states (active, non-active), and the threshold triggering the adoption (activation) of a new idea to a node is given by the number of the active neighbors. The problem of detecting in a graph the presence of the minimal number of nodes that will be able to activate the entire network is called {\em target set selection} (TSS). In this paper we extend TSS by allowing nodes to have more than two colors. The multicolored version of the TSS can be described as follows: let $G$ be a torus where every node is assigned a color from a finite set of colors. At each local time step, each node can recolor itself, depending on the local configurations, with the color held by the majority of its neighbors. We study the initial distributions of colors leading the system to a monochromatic configuration of color $k$, focusing on the minimum number of initial $k$-colored nodes. We conclude the paper by providing the time complexity to achieve the monochromatic configuration

Circuit-Switched Gossiping in the 3-Dimensional Torus Networks

In this paper we describe, in the case of short messages, an efficient gossiping algorithm for 3-dimensional torus networks (wrap-around or toroidal meshes) that uses synchronous circuit-switched routing. The algorithm is based on a recursive decomposition of a torus. The algorithm requires an optimal number of rounds and a quasi-optimal number of intermediate switch settings to gossip in an $7^i \times 7^i \times 7^i$ torus

Partial multinode broadcast and partial exchange algorithms for d-dimensional meshes

Caption title. "Revision of January 1992."Includes bibliographical references (p. 24-26).Supported by NSF. NSF-ECS-8519058 Supported by ARO. DAAL03-86-K-0171by Emmanouel A. Varvarigos and Dimitri P. Bertsekas

Gossiping in chordal rings under the line model

The line model assumes long distance calls between non neighboring processors. In this sense, the line model is strongly related to circuit-switched networks, wormhole routing, optical networks supporting wavelength division multiplexing, ATM switching, and networks supporting connected mode routing protocols. Since the chordal rings are competitors of networks as meshes or tori because of theirs short diameter and bounded degree, it is of interest to ask whether they can support intensive communications (typically all-to-all) as efficiently as these networks. We propose polynomial algorithms to derive optimal or near optimal gossip protocols in the chordal ring

Deterministic 1-k routing on meshes with applications to worm-hole routing

In $1$-$k$ routing each of the $n^2$ processing units of an $n \times n$ mesh connected computer initially holds $1$ packet which must be routed such that any processor is the destination of at most $k$ packets. This problem reflects practical desire for routing better than the popular routing of permutations. $1$-$k$ routing also has implications for hot-potato worm-hole routing, which is of great importance for real world systems. We present a near-optimal deterministic algorithm running in \sqrt{k} \cdot n / 2 + \go{n} steps. We give a second algorithm with slightly worse routing time but working queue size three. Applying this algorithm considerably reduces the routing time of hot-potato worm-hole routing. Non-trivial extensions are given to the general $l$-$k$ routing problem and for routing on higher dimensional meshes. Finally we show that $k$-$k$ routing can be performed in \go{k \cdot n} steps with working queue size four. Hereby the hot-potato worm-hole routing problem can be solved in \go{k^{3/2} \cdot n} steps

Multicolored Dynamos on Toroidal Meshes

Detecting on a graph the presence of the minimum number of nodes (target set) that will be able to "activate" a prescribed number of vertices in the graph is called the target set selection problem (TSS) proposed by Kempe, Kleinberg, and Tardos. In TSS's settings, nodes have two possible states (active or non-active) and the threshold triggering the activation of a node is given by the number of its active neighbors. Dealing with fault tolerance in a majority based system the two possible states are used to denote faulty or non-faulty nodes, and the threshold is given by the state of the majority of neighbors. Here, the major effort was in determining the distribution of initial faults leading the entire system to a faulty behavior. Such an activation pattern, also known as dynamic monopoly (or shortly dynamo), was introduced by Peleg in 1996. In this paper we extend the TSS problem's settings by representing nodes' states with a "multicolored" set. The extended version of the problem can be described as follows: let G be a simple connected graph where every node is assigned a color from a finite ordered set C = {1, . . ., k} of colors. At each local time step, each node can recolor itself, depending on the local configurations, with the color held by the majority of its neighbors. Given G, we study the initial distributions of colors leading the system to a k monochromatic configuration in toroidal meshes, focusing on the minimum number of initial k-colored nodes. We find upper and lower bounds to the size of a dynamo, and then special classes of dynamos, outlined by means of a new approach based on recoloring patterns, are characterized