A Search Problem in the Complete Graph

• Main Authors: YU-CIAN WANG, 王郁茜
• Other Authors: SHOOU-REN HSIAU
• Format: Others
• Language: zh-TW
• Published: 2011
• Online Access: http://ndltd.ncl.edu.tw/handle/98045519271517623991

碩士 === 國立彰化師範大學 === 數學系所 === 99 === Abstract Two people walk randomly in a complete graph with n equal-distanced vertices. We want to find the distribution of the first meeting time under two different definitions of meeting . The first is that meeting occurs only at vertices, and the second is that meeting can occur anywhere. Under each definition of meeting , we consider the search problem with given ratio of two people’s speeds. We denote the first meeting time by T. In the first search problem, if the ratio of speeds is b/a , where a,b are coprime, then we can prove that T/b is a geometric distribution. In the second search problem, we can establish a recurrence relation among the probabilities of meeting times. Theoretically, the distribution of meeting time can be represented in terms of the roots of the characteristic polynomial of the recurrence relation. Finally, we discuss some specialcases and compute the distribution and expectation of the first meeting time.