Approximation Algorithms in Graphs with Known Broadcast Time of the Base Graph

Abstract

AbstractBroadcasting is an information dissemination problem in a connected network in which one node, called the originator, must distribute a message to all other nodes of the network by placing a series of calls along the communication lines of the network. The broadcast time of a vertex is defined to be the minimum number of time units required to broadcast the message to all vertices of the graph (network) from that vertex. Finding the broadcast time of any vertex in an arbitrary graph is NP-complete. The polynomial time solvability is shown only for certain tree-like graphs. In this paper we study the broadcast problem in graph of trees where broadcast algorithms for the base graph is known. In such graphs we design a linear time constant approximation algorithm to determine the broadcast time of any originator in general case. In a particular case when the base graph is the hypercube or another minimum broadcast graph (graph with minimum possible broadcast time having the smallest number of edges) containing one tree we present a linear time exact algorithm to find the broadcast time of any originator vertex. When the base graph is the hypercube graph we improve the known result by presenting a 1.5-approximation algorithm to find the broadcast time of the whole graph which runs in linear time instead of known quadratic algorithm.