All-to-all broadcast problems on Cartesian product graphs Jen-Chun Lin 林仁俊 指導教授：郭大衛教授 國立東華大學 應用數學系碩士班

Outline Introduction Main result Reference

Introduction

Suppose each vertex has a private message needed to send to every other vertices(all-to-all broadcast). At each time unit, vertices exchange their messages under the following constraints: (1) only one message can travel a link at a time unit (2) a message requires one time unit to be transferred between two nodes (3) each vertex can use all of its links at the same time

Chang et al. They gave upper and lower bounds for all-to- all broadcast number (the shortest time needed to complete the all-to-all broadcast) of graphs and give formulas for the all-to-all broadcast number of trees, complete bipartite graphs and double loop networks under this model.

All-to-all broadcasting numbers of Cartesian product of cycles and complete graphs

Given a graph G and a positive integer t, we use G ◦ t to denote the multigraph obtained from G by replacing each edge in E(G) with k edges

① ② ③ ① ② ③ ① ①② ② ③ ③

For a graph G with a set of edge-disjoint subgraphs of G◦t is a t-broadcasting system of G.

Theorem

0,0 2,0 1,0 0,1 1,1 1,2 0,3 3,0 0,4 1,3 1,42,4 2,2 2,3 3,1 3,4 3,3 3,2 2,1 0,2 ① ① ② ② ③ ③ ④ ④ ⑤ ⑤ ⑥ ⑥⑦ ⑦ ⑧ ⑧ ⑨ ⑨ ⑩ The broadcasting tree

0,0 2,0 1,0 0,1 1,1 1,2 0,3 3,0 0,4 1,3 1,42,4 2,2 2,3 3,1 3,4 3,3 3,2 2,1 0,2 ① ① ② ② ⑨ ⑧ ④ ⑤⑥ ③ ⑨ ③ ④ ⑧ ⑩ ⑦ ⑤⑥ ⑦ The broadcasting tree

Lemma Given a graph G, if and only if there exists a t-broadcasting systm of G.

Lemma For any graph G with,

Theorem

0,0 2,0 1,0 0,1 1,1 1,2 0,3 3,0 0,4 1,3 1,42,4 2,2 2,3 3,1 3,4 3,3 3,2 2,1 0,2 4,0 4,1 4,4 4,3 4,2 The perfect broadcasting tree ① ② ③④ ⑤⑥ ⑦⑧ ① ① ② ② ③ ③ ④ ④ ⑤ ⑤⑥ ⑥ ⑦ ⑦ ⑧ ⑧

Theorem For positive integers m and n with We set

3 1 The subgraph 0,0 0,2 0,1 0,4 0,3 6,0 3,0 2,0 1,0 7,0 4,0 8,0 5,

0,0 0,1 6,0 3,0 2,0 1,0 7,0 4,0 8,0 5,0 0,2 0,4 0,

All-to-all broadcasting numbers of hypercubes

Theorem

For a vertex the weight of, denoted by, is defined by Definition n個n個

For the set the right-shift of elements in is a function from to defined By = Definition =ex:

the orbit of v, denoted by A vertex v in is said to be Definition

Theorem

Lemma (0,0,………,0) (1,0,………,0)(0,1,………,0) (0,0,………,1) ……… 方向 1 方向 2 方向 n (2,0,………,0)(0,2,………,0) (0,0,………,2) (1,0,………,1)(1,1,………,0)

Lemma (1,1,………,1) (0,1,………,1)

6 2 0,0 0,11,0 0,2 1,1 2,0 0,31,23,0 2,1 3,1 1,3 2,2 3,2 2,3 3,

Thanks for your listening!