| Summary: | 碩士 === 淡江大學 === 數學學系數學與數據科學碩士班 === 106 === Let G=(V,E) be a connected graph with vertices {v_1,v_2,…,v_n } and π be a permutation on [n]. Initially, each vertex v_i of G is occupied by a “pebble.” Pebbles can be moved around by the following rule: At each step a disjoint collection of edges of G is selected, and the pebbles at each edge’s two endpoints are interchanged. The goal is to move/route each pebble p_i to its destination v_i.Define rt(G,π) to be the minimum number of steps to route the permutation π.Finally, define rt(G) the routine number of G by rt(G)=(max)┬π〖rt(G,π)〗.
Given a sequence, π=p_1,p_2,…,p_n, we choose the number p_i, if the number of the original sequence is greater or equal to p_i then change to “1”, otherwise change to “0”, therefore we obtain a (0,1)-sequences, λ=c_1,c_2,…,c_n. It is known that using the (0,1)-sequence can prove the upper bound of the routine number of path,then given an arbitrary (0,1)-sequence λ=c_1,c_2,…,c_n, the number when it line up by odd-even transposition sort is less than or equal to n-1.
In this paper, in connection with arbitrary (0,1)-sequences,λ=c_1,c_2,…,c_N ,N=∑_(k=0)^n▒a_k ∑_(k=0)^n▒b_k , define S=a_0,b_n,a_1,b_(n-1),a_2,b_(n-2),…a_(n-2),b_2,a_(n-1),b_1,a_n,b_0, a_i,b_i≥1,i=0,1…n are quantity of 0,1 respectively and a_0,b_0≥0, after odd-even transposition sort, the number when it line up will be
N-(min)┬(i∈{0,1,…,n-1} )〖{∑_(k=0)^i▒a_k +∑_(k=0)^(n-i-1)▒b_k }〗-1
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