225 research outputs found
Throughput-Optimal Multihop Broadcast on Directed Acyclic Wireless Networks
We study the problem of efficiently broadcasting packets in multi-hop
wireless networks. At each time slot the network controller activates a set of
non-interfering links and forwards selected copies of packets on each activated
link. A packet is considered jointly received only when all nodes in the
network have obtained a copy of it. The maximum rate of jointly received
packets is referred to as the broadcast capacity of the network. Existing
policies achieve the broadcast capacity by balancing traffic over a set of
spanning trees, which are difficult to maintain in a large and time-varying
wireless network. We propose a new dynamic algorithm that achieves the
broadcast capacity when the underlying network topology is a directed acyclic
graph (DAG). This algorithm is decentralized, utilizes local queue-length
information only and does not require the use of global topological structures
such as spanning trees. The principal technical challenge inherent in the
problem is the absence of work-conservation principle due to the duplication of
packets, which renders traditional queuing modelling inapplicable. We overcome
this difficulty by studying relative packet deficits and imposing in-order
delivery constraints to every node in the network. Although in-order packet
delivery, in general, leads to degraded throughput in graphs with cycles, we
show that it is throughput optimal in DAGs and can be exploited to simplify the
design and analysis of optimal algorithms. Our characterization leads to a
polynomial time algorithm for computing the broadcast capacity of any wireless
DAG under the primary interference constraints. Additionally, we propose an
extension of our algorithm which can be effectively used for broadcasting in
any network with arbitrary topology
Graph Theory
Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures
Bounds relating the weakly connected domination number to the total domination number and the matching number
AbstractLet G=(V,E) be a connected graph. A dominating set S of G is a weakly connected dominating set of G if the subgraph (V,E∩(S×V)) of G with vertex set V that consists of all edges of G incident with at least one vertex of S is connected. The minimum cardinality of a weakly connected dominating set of G is the weakly connected domination number, denoted γwc(G). A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γt(G) of G. In this paper, we show that 12(γt(G)+1)≤γwc(G)≤32γt(G)−1. Properties of connected graphs that achieve equality in these bounds are presented. We characterize bipartite graphs as well as the family of graphs of large girth that achieve equality in the lower bound, and we characterize the trees achieving equality in the upper bound. The number of edges in a maximum matching of G is called the matching number of G, denoted α′(G). We also establish that γwc(G)≤α′(G), and show that γwc(T)=α′(T) for every tree T
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