A Simple Construction of Broadcast Graphs

Abstract

Broadcasting is one of the basic primitives of communication in usual networks. It is a process of information dissemination in which one informed node of the network, called the originator, distributes the message to all other nodes of the network by placing a series of calls along the communication lines. The network is modeled as a graph. The broadcast time of a given vertex is the minimum time required to broadcast a message from the originator to all other vertices of the graph. The broadcast time of a graph is the maximum time required to broadcast from any vertex in the graph. Many papers have investigated the construction of minimum broadcast graphs, the cheapest possible broadcast network architecture (having the fewest communication lines) in which broadcasting can be accomplished as fast as theoretically possible from any vertex. Since this problem is very difficult, numerous papers give sparse networks in which broadcasting can be completed in minimum time from any originator. In this paper, we improve the existing upper bounds on the number of edges by constructing sparser graphs and by presenting a minimum time broadcast algorithm from any originator.

Appendix: Comparison

By our assumption $k_p<\cdots <k_1\le m-3$, $k_i$ is strictly larger than $k_{i+1}$. Then in general $k_i\le m-i-2$ for $1\le i\le p$. Similarly, since $1\le k_p<\cdots <k_1$, $p-i+1\le k_i$ for $1\le i\le p$. Thus, $p-i+1\le k_i\le m-i-2$, where $1\le i\le p$.

$$\begin{aligned}&OB-NB\\ =&\, \frac{1}{2}n(m-1)-\frac{1}{2}(\frac{1}{2}n(m-1)+\frac{1}{2}\sum _{i=1}^{p-1}(2^{m-i}-2^{k_i})(m-k_i)\\&\quad +\frac{1}{2}(2^{m-p+1}-2^{k_p})(m-k_p)+\sum _{i=1}^{p-1}(2^{k_i}-\sum _{j=i+1}^{p}2^{k_j}))\\ =&\, \frac{1}{4}n(m-1)-\frac{1}{4}\sum _{i=1}^{p-1}(2^{m-i}-2^{k_i})(m-k_i)\\&\quad -\frac{1}{4}(2^{m-p+1}-2^{k_p})(m-k_p)-\frac{1}{2}\sum _{i=1}^{p-1}(2^{k_i}-\sum _{j=i+1}^{p}2^{k_j}))\\ =&\, \frac{1}{4}n(m-1)-\frac{1}{4}\sum _{i=1}^{p-1}(2^{m-i}-2^{k_i})(m-k_i)\\&\quad -\frac{1}{4}(2^{m-p+1}-2^{k_p})(m-k_p)-\frac{1}{2}(2^{k_1}-\sum _{i=2}^{p-1}(i-2)2^{k_i}-(p-1)2^{k_p}) \end{aligned}$$

By substituting $n=\sum _{i=1}^{p-1}(2^{m-i}-2^{k_i})+2^{m-p+1}-2^{k_p}$,

$$\begin{aligned} =&\, \frac{1}{4}\sum _{i=1}^{p-1}(2^{m-i}-2^{k_i})(m-1)+\frac{1}{4}(2^{m-p+1}-2^{k_p})(m-1)-\frac{1}{4}\sum _{i=1}^{p-1}(2^{m-i}-2^{k_i})(m-k_i)\\&\quad -\frac{1}{4}(2^{m-p+1}-2^{k_p})(m-k_p)-\frac{1}{2}(2^{k_1}-\sum _{i=2}^{p-1}(i-2)2^{k_i}-(p-1)2^{k_p})\\ =&\, \frac{1}{4}\sum _{i=1}^{p-1}(2^{m-i}-2^{k_i})(k_i+1)+\frac{1}{4}(2^{m-p+1}-2^{k_p})(k_p+1)\\&\quad -\frac{1}{2}2^{k_1}+\frac{1}{2}\sum _{i=2}^{p-1}(i-2)2^{k_i}+\frac{1}{2}(p-1)2^{k_p} \end{aligned}$$

Since $k_i\ge 1$,

$$\begin{aligned} \ge&\, \frac{1}{2}\sum _{i=1}^{p-1}(2^{m-i}-2^{k_i})+\frac{1}{2}(2^{m-p+1}-2^{k_p})\\&\quad -\frac{1}{2}2^{k_1}+\frac{1}{2}\sum _{i=2}^{p-1}(i-2)2^{k_i}+\frac{1}{2}(p-1)2^{k_p}\\ =&\, \frac{1}{2}(2^{m-1}-2^{k_1}-2^{k_1})+\frac{1}{2}\sum _{i=2}^{p-1}(2^{m-i}-2^{k_i})+\frac{1}{2}(2^{m-p+1}-2^{k_p})\\&\quad +\frac{1}{2}\sum _{i=2}^{p-1}(i-2)2^{k_i}+\frac{1}{2}(p-1)2^{k_p}\\ =&\, \frac{1}{2}(2^{m-1}-2^{k_1}-2^{k_1})+\frac{1}{2}(2^{m-2}+2^{m-3}+2^{m-p-2})\\&\quad +\frac{1}{2}\sum _{i=2}^{p-1}(i-2)2^{k_i}+\frac{1}{2}(p-1)2^{k_p}\\ \ge&\, \frac{1}{2}(2^{m-1}-2^{k_1}-2^{k_1})+\frac{1}{2}(2^{m-2}+2^{m-3}) \end{aligned}$$

As $k_1\le m-3$,

$$\begin{aligned} OB-NB&\ge \frac{1}{2}(2^{m-1}-2^{m-3}-2^{m-3})+\frac{1}{2}(2^{m-2}+2^{m-3})\\&= 2^{m-2}+2^{m-4} \end{aligned}$$

Thus, our new upper bound is a better upper bound on B(n) when $2^m-2^{\frac{m+3}{2}}<n \le 2^m-8$.