Parameterized Complexity of Broadcasting in Graphs

The task of the broadcast problem is, given a graph G and a source vertex s, to compute the minimum number of rounds required to disseminate a piece of information from s to all vertices in the graph. It is assumed that, at each round, an informed vertex can transmit the information to at most one of its neighbors. The broadcast problem is known to NP-hard. We show that the problem is FPT when parametrized by the size k of a feedback edge-set, or by the size k of a vertex-cover, or by k=n-t where t is the input deadline for the broadcast protocol to complete.Comment: Full version of WG 2023 pape

Near-linear Time Algorithm for Approximate Minimum Degree Spanning Trees

Given a graph $G = (V, E)$, we wish to compute a spanning tree whose maximum vertex degree, i.e. tree degree, is as small as possible. Computing the exact optimal solution is known to be NP-hard, since it generalizes the Hamiltonian path problem. For the approximation version of this problem, a $\tilde{O}(mn)$ time algorithm that computes a spanning tree of degree at most $\Delta^* +1$ is previously known [F\"urer \& Raghavachari 1994]; here $\Delta^*$ denotes the minimum tree degree of all the spanning trees. In this paper we give the first near-linear time approximation algorithm for this problem. Specifically speaking, we propose an $\tilde{O}(\frac{1}{\epsilon^7}m)$ time algorithm that computes a spanning tree with tree degree $(1+\epsilon)\Delta^* + O(\frac{1}{\epsilon^2}\log n)$ for any constant $\epsilon \in (0,\frac{1}{6})$. Thus, when $\Delta^*=\omega(\log n)$, we can achieve approximate solutions with constant approximate ratio arbitrarily close to 1 in near-linear time.Comment: 17 page

Four Algorithms on the Swapped Dragonfly

The Swapped Dragonfly with M routers per group and K global ports per router is denoted D3(K;M) [1]. It has n=KMM routers and is a partially populated Dragonfly. A Swapped Dragonfly with K and M restricted is studied in this paper. There are four cases. matrix product: If K is a perfect square, a matrix product of size n can be performed in squareroot n rounds. all-to-all exchange: If K and M have a common factor s, an all-to-all exchange can be performed in n/s rounds. broadcast: If D3(K,M) is equipped with a synchronized source-vector header it can perform x broadcast in 3x/M rounds. ascend-descend: If K and M are powers of 2 an ascend-descend algorithm can be performed at twice the cost of the algorithm on a Boolean hypercube of size n. In each case the algorithm on the Swapped Dragonfly is free of link conflicts and is compared with algorithms on a hypercube as well as on the fully populated Dragonfly. The results on the Swapped Dragonfly are more applicable than the special cases because D3(K,M) contains emulations of every Swapped Dragonfly with J less than equal to K and/or L less than or equal to M. Keywords: Swapped Interconnection Network, Matrix Product, All-to-all, Universal Exchange, Boolean Hypercube, Ascend-descend algorithm, Broad- cast, Edge-disjoint spanning tree. References [1] R. Draper. The Swapped Dragonfly , ArXiv for Computer Science:2202.01843. 1Comment: 8 page

Power assignment problems in wireless communication

A fundamental class of problems in wireless communication is concerned with the assignment of suitable transmission powers to wireless devices/stations such that the resulting communication graph satisfies certain desired properties and the overall energy consumed is minimized. Many concrete communication tasks in a wireless network like broadcast, multicast, point-to-point routing, creation of a communication backbone, etc. can be regarded as such a power assignment problem. This paper considers several problems of that kind; for example one problem studied before in (Vittorio Bil{\`o} et al: Geometric Clustering to Minimize the Sum of Cluster Sizes, ESA 2005) and (Helmut Alt et al.: Minimum-cost coverage of point sets by disks, SCG 2006) aims to select and assign powers to $k$ of the stations such that all other stations are within reach of at least one of the selected stations. We improve the running time for obtaining a $(1+\epsilon)$-approximate solution for this problem from $n^{((\alpha/\epsilon)^{O(d)})}$ as reported by Bil{\`o} et al. (see Vittorio Bil{\`o} et al: Geometric Clustering to Minimize the Sum of Cluster Sizes, ESA 2005) to $O\left( n+ {\left(\frac{k^{2d+1}}{\epsilon^d}\right)}^{ \min{\{\; 2k,\;\; (\alpha/\epsilon)^{O(d)} \;\}} } \right)$ that is, we obtain a running time that is \emph{linear} in the network size. Further results include a constant approximation algorithm for the TSP problem under squared (non-metric!) edge costs, which can be employed to implement a novel data aggregation protocol, as well as efficient schemes to perform $k$-hop multicasts

On the design and implementation of broadcast and global combine operations using the postal model

There are a number of models that were proposed in recent years for message passing parallel systems. Examples are the postal model and its generalization the LogP model. In the postal model a parameter λ is used to model the communication latency of the message-passing system. Each node during each round can send a fixed-size message and, simultaneously, receive a message of the same size. Furthermore, a message sent out during round r will incur a latency of hand will arrive at the receiving node at round r + λ - 1. Our goal in this paper is to bridge the gap between the theoretical modeling and the practical implementation. In particular, we investigate a number of practical issues related to the design and implementation of two collective communication operations, namely, the broadcast operation and the global combine operation. Those practical issues include, for example, 1) techniques for measurement of the value of λ on a given machine, 2) creating efficient broadcast algorithms that get the latency hand the number of nodes n as parameters and 3) creating efficient global combine algorithms for parallel machines with λ which is not an integer. We propose solutions that address those practical issues and present results of an experimental study of the new algorithms on the Intel Delta machine. Our main conclusion is that the postal model can help in performance prediction and tuning, for example, a properly tuned broadcast improves the known implementation by more than 20%