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

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

Energy Complexity of Distance Computation in Multi-hop Networks

Energy efficiency is a critical issue for wireless devices operated under stringent power constraint (e.g., battery). Following prior works, we measure the energy cost of a device by its transceiver usage, and define the energy complexity of an algorithm as the maximum number of time slots a device transmits or listens, over all devices. In a recent paper of Chang et al. (PODC 2018), it was shown that broadcasting in a multi-hop network of unknown topology can be done in $\text{poly} \log n$ energy. In this paper, we continue this line of research, and investigate the energy complexity of other fundamental graph problems in multi-hop networks. Our results are summarized as follows. 1. To avoid spending $\Omega(D)$ energy, the broadcasting protocols of Chang et al. (PODC 2018) do not send the message along a BFS tree, and it is open whether BFS could be computed in $o(D)$ energy, for sufficiently large $D$. In this paper we devise an algorithm that attains $\tilde{O}(\sqrt{n})$ energy cost. 2. We show that the framework of the ${\Omega}(n)$ round lower bound proof for computing diameter in CONGEST of Abboud et al. (DISC 2017) can be adapted to give an $\tilde{\Omega}(n)$ energy lower bound in the wireless network model (with no message size constraint), and this lower bound applies to $O(\log n)$-arboricity graphs. From the upper bound side, we show that the energy complexity of $\tilde{O}(\sqrt{n})$ can be attained for bounded-genus graphs (which includes planar graphs). 3. Our upper bounds for computing diameter can be extended to other graph problems. We show that exact global minimum cut or approximate $s$--$t$ minimum cut can be computed in $\tilde{O}(\sqrt{n})$ energy for bounded-genus graphs

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

Resilient Wireless Sensor Networks Using Topology Control: A Review

Wireless sensor networks (WSNs) may be deployed in failure-prone environments, and WSNs nodes easily fail due to unreliable wireless connections, malicious attacks and resource-constrained features. Nevertheless, if WSNs can tolerate at most losing k − 1 nodes while the rest of nodes remain connected, the network is called k − connected. k is one of the most important indicators for WSNs’ self-healing capability. Following a WSN design flow, this paper surveys resilience issues from the topology control and multi-path routing point of view. This paper provides a discussion on transmission and failure models, which have an important impact on research results. Afterwards, this paper reviews theoretical results and representative topology control approaches to guarantee WSNs to be k − connected at three different network deployment stages: pre-deployment, post-deployment and re-deployment. Multi-path routing protocols are discussed, and many NP-complete or NP-hard problems regarding topology control are identified. The challenging open issues are discussed at the end. This paper can serve as a guideline to design resilient WSNs

Minimal contention-free matrices with application to multicasting

In this paper, we show that the multicast problem in trees can be expressed in term of arranging rows and columns of boolean matrices. Given a $p \times q$ matrix $M$ with 0-1 entries, the {\em shadow} of $M$ is defined as a boolean vector $x$ of $q$ entries such that $x_i=0$ if and only if there is no 1-entry in the $i$th column of $M$, and $x_i=1$ otherwise. (The shadow $x$ can also be seen as the binary expression of the integer $x=\sum_{i=1}^{q}x_i 2^{q-i}$. Similarly, every row of $M$ can be seen as the binary expression of an integer.) According to this formalism, the key for solving a multicast problem in trees is shown to be the following. Given a $p \times q$ matrix $M$ with 0-1 entries, finding a matrix $M^*$ such that: 1- $M^*$ has at most one 1-entry per column; 2- every row $r$ of $M^*$ (viewed as the binary expression of an integer) is larger than the corresponding row $r$ of $M$, $1 \leq r \leq p$; and 3- the shadow of $M^*$ (viewed as an integer) is minimum. We show that there is an $O(q(p+q))$ algorithm that returns $M^*$ for any $p \times q$ boolean matrix $M$. The application of this result is the following: Given a {\em directed} tree $T$ whose arcs are oriented from the root toward the leaves, and a subset of nodes $D$, there exists a polynomial-time algorithm that computes an optimal multicast protocol from the root to all nodes of $D$ in the all-port line model.Peer Reviewe

Approximation Algorithms for Broadcasting in Simple Graphs with Intersecting Cycles

Broadcasting is an information dissemination problem in a connected network in which one node, called the originator, must distribute a message to all other nodes by placing a series of calls along the communication lines of the network. Every time the informed nodes aid the originator in distributing the message. Finding the minimum broadcast time of any vertex in an arbitrary graph is NP-Complete. The problem remains NP-Complete even for planar graphs of degree 3 and for a graph whose vertex set can be partitioned into a clique and an independent set. The best theoretical upper bound gives logarithmic approximation. It has been shown that the broadcasting problem is NP-Hard to approximate within a factor of 3-ɛ. The polynomial time solvability is shown only for tree-like graphs; trees, unicyclic graphs, tree of cycles, necklace graphs and some graphs where the underlying graph is a clique; such as fully connected trees and tree of cliques. In this thesis we study the broadcast problem in different classes of graphs where cycles intersect in at least one vertex. First we consider broadcasting in a simple graph where several cycles have common paths and two intersecting vertices, called a k-path graph. We present a constant approximation algorithm to find the broadcast time of an arbitrary k-path graph. We also study the broadcast problem in a simple cactus graph called k-cycle graph where several cycles of arbitrary lengths are connected by a central vertex on one end. We design a constant approximation algorithm to find the broadcast time of an arbitrary k-cycle graph. Next we study the broadcast problem in a hypercube of trees for which we present a 2-approximation algorithm for any originator. We provide a linear algorithm to find the broadcast time in hypercube of trees with one tree. We extend the result for any arbitrary graph whose nodes contain trees and design a linear time constant approximation algorithm where the broadcast scheme in the arbitrary graph is already known. In Chapter 6 we study broadcasting in Harary graph for which we present an additive approximation which gives 2-approximation in the worst case to find the broadcast time in an arbitrary Harary graph. Next for even values of n, we introduce a new graph, called modified-Harary graph and present a 1-additive approximation algorithm to find the broadcast time. We also show that a modified-Harary graph is a broadcast graph when k is logarithmic of n. Finally we consider a diameter broadcast problem where we obtain a lower bound on the broadcast time of the graph which has at least (d+k-1 choose d) + 1 vertices that are at a distance d from the originator, where k >= 1

Book of Abstracts of the Sixth SIAM Workshop on Combinatorial Scientific Computing

Book of Abstracts of CSC14 edited by Bora UçarInternational audienceThe Sixth SIAM Workshop on Combinatorial Scientific Computing, CSC14, was organized at the Ecole Normale Supérieure de Lyon, France on 21st to 23rd July, 2014. This two and a half day event marked the sixth in a series that started ten years ago in San Francisco, USA. The CSC14 Workshop's focus was on combinatorial mathematics and algorithms in high performance computing, broadly interpreted. The workshop featured three invited talks, 27 contributed talks and eight poster presentations. All three invited talks were focused on two interesting fields of research specifically: randomized algorithms for numerical linear algebra and network analysis. The contributed talks and the posters targeted modeling, analysis, bisection, clustering, and partitioning of graphs, applied in the context of networks, sparse matrix factorizations, iterative solvers, fast multi-pole methods, automatic differentiation, high-performance computing, and linear programming. The workshop was held at the premises of the LIP laboratory of ENS Lyon and was generously supported by the LABEX MILYON (ANR-10-LABX-0070, Université de Lyon, within the program ''Investissements d'Avenir'' ANR-11-IDEX-0007 operated by the French National Research Agency), and by SIAM