P. Eades, M. Hickey, and R. C. Read, "Some Hamilton paths and a minimal change algorithm," J. Assoc. Comput. Mach., 31, No. 1 (1984), pp. 19-29.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Gray codes include (1) listing all permutations of 1 : n so that consecutive permutations differ only by the swap of one pair of adjacent elements [Joh63, Tro62] 2) listing all k element subsets of an n element set in such a way that consecutive sets differ by exactly one element [BER76, BW84, EHR84, EM84, NW78, Rus88a], 3) listing all binary trees so that consecutive trees differ only by a rotation at a single node [Luc87, LRR93] 4) listing all spanning trees of a graph so that successive trees differ only by a single edge [HH72, Cum66] 5) listing all partitions of an integer n so that in successive ....

....adjacent positions (see Figure 6(c) However, this is not always possible: it was shown that k subsets of an n set can be generated by adjacent interchanges if (i) k=0, 1, n, or n Gamma 1 or (ii) n is even and k is odd. In all other cases, parity problems prevent adjacent interchange generation [BW84, EHR84, HR88, Rus88a]. It was shown by Chase [Cha89] and by a simpler construction in [Rus93] that combinations can be generated so that successive elements differ either by an adjacent transposition or by the transposition of two bits that have a single 0 bit between them. There are several open problems about ....

P. Eades, M. Hickey, and R. C. Read. Some Hamilton paths and a minimal change algorithm. Journal of the ACM, 31(1):19--29, 1984.

....to produce a Hamilton path in I(T(n) when n is even. The similar problem of a strong Gray code for combinations of k objects out of n objects as represented by bitstrings of length n with exactly k ones was recently resolved in three independent and different ways by Eades, Hickey, and Read [3], Buck and Wiedemann [2] and Ruskey [16] We will imitate the approach of [16] In order to do this we need to add an additional parameter k to our problem. The addition of this parameter helps immensely; we can now follow the lead of [1] for the Gray code question, and of [16] for the strong ....

....in this paper. This is the same route followed in [1] see also [13] for bitstrings representing combinations: A careful analysis of the recursive generation algorithm produced a constant average time algorithm. It is remarkable that Theorem 2 is almost exactly the same as theorems in [2] [3], and [16] except that the set T(n; k) is replaced by C(n; k) the set of bitstrings representing combinations of k items out of n items. One is g vertex connection g vertex A Gamma1 (s; t) l A(s; t) A(s; t) A(s; t 2) A(s; t) Y (s; t 2) X(s; t) l X(s; t) X(s; t) X(s; t 1) ....

P. Eades, M. Hickey, and R.C. Read. Some Hamilton paths and a minimal change algorithm. JACM, 31:19--29, 1984.

.... For example, if the poset consists of two chains of lengths n and m, then the linear extensions can be generated by transpositions if and only if n and m are both odd (except for the trivial cases where n = 1 or m = 1) as shown independently by Buck and Wiedemann [2] Eades, Hickey, and Read [4], and Ruskey [17] Here we generalize this result to permutations of a multiset. There is a graph that naturally arises when generating extensions by transpositions. We call it the transposition graph of the poset. The vertices of this graph are the permutations that correspond to extensions of ....

....the edge [a; a 0 ] to get a STOC, S, for G 0 (3; n) Furthermore, G 0 (2; n)0 has the degree two vertex 01 n 00 on a cycle, and thus S does as well. If m; n 4 then we proceed by induction on m and n, using the decomposition of G 0 (m; n) implicit in the paper of Eades, Hickey and Read [4] (see also Hough and Ruskey [6] and the m = 3 and n 3 construction of the previous paragraph as the base case. In [4] the vertex set of G 0 (m; n) is decomposed into 4 parts, depending on the final two bits of each vertex. That is (again, abusing the notation) G 0 (m; n) G 0 (m Gamma ....

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P. Eades, M. Hickey, and R.C. Read. Some Hamilton paths and a minimal change algorithm. JACM, 31:19--29, 1984.

....extensions of the poset that is an n element antichain. In general, it is not always possible to generate the linear extensions of a poset by transpositions, adjacent or not; for example, the linear extensions of the poset consisting of two non trivial chains and only if n and m are both odd ( 5] [6], and [16] Thus, the linear extensions of the poset in Figure 1 (two 2 element chains) cannot be generated by transpositions. The linear extensions of some classes of posets have been shown to be generable by transpositions (see [18] 15] 20] It is an open problem to characterize the posets ....

P. Eades, M. Hickey, and R. Read, Some Hamilton paths and a minimal change algorithm, JACM, 31 (1984), pp. 19--29.

....;k) i=max(0;k Gamman n t ) d(k Gamma i; n 0 ; n t Gamma1 ) if t 0 odd P min(n t ;k) i=max(0;k Gamman n t ) Gamma1) i d(k Gamma i; n 0 ; n t Gamma1 ) if t 0 even This recurrence can be solved if all n i = 1. See, for example, Buck and Wiedemann [2] Eades, Hickey and Read [3], Ko and Ruskey [10] Then d(k; 1) 0 if t and k are odd i b(t 1) 2c bk=2c j otherwise: By similar reasoning, it can also be solved when all n i = 1. d(k; 1) 0 if t and k are odd i b(t k) 2c bk=2c j otherwise We close by mentioning three open problems: ffl Determine exact ....

P. Eades, M. Hickey and R. Read, Some Hamilton Paths and a Minimal Change Algorithm, Journal of the ACM, 31 (1984) 19-29.

....the two bits that are transposed could be quite far apart. The elements of B(n; k) cannot be listed so that successive strings differ by the transposition of two adjacent bits, unless n is even and k is odd, or k 2 f0; 1; n Gamma 1; ng (e.g. Buck and Wiedemann [BuWi] Eades, Hickey and Read [EaHiRe]) Let us say that two distinct bitstrings are two close if they differ by a transposition of two bits that are either adjacent or have a single 0 between them. Chase [Ch] showed that the elements of B(n; k) could always be listed so that successive strings are two close. He also presented a ....

P. Eades, M. Hickey and R. Read, Some Hamilton Paths and a Minimal Change Algorithm, Journal of the ACM, 31 (1984) 19-29.

....the list, first(E(n; k) is lexicographically largest, 1 k 0 n Gammak . The last element, last(E(n; k) is lexicographically smallest, 0 n Gammak 1 k . The recursive algorithm derived from this definition produces a homogeneous listing for combinations and this is proven by induction in [3]. Note that the Gray code sequence produced is not cyclic if 1 k n; that is, first(E(n; k) and last(E(n; k) do not differ by a homogeneous transposition. 2 Well Formed Parentheses The set of all well formed parentheses strings of length 2n is denoted by T(n) The subset of T(n) consisting ....

P. Eades, M. Hickey and R. Read, "Some Hamilton Paths and a Minimal Change Algorithm," Journal of the ACM, 31 (1984) 19-29.

.... For example, if the poset consists of two chains of lengths n and m, then the linear extensions can be generated by transpositions if and only if n and m are both odd (except for the trivial cases where n = 1 or m = 1) as shown independently by Buck and Wiedemann [1] Eades, Hickey, and Read [3], and Ruskey [15] Here we generalize this result to permutations of a multiset. There is a graph that naturally arises when generating extensions by transpositions. We call it the transposition graph of the poset. The vertices of this graph are the permutations that correspond to extensions of ....

P. Eades, M. Hickey, and R.C. Read. Some Hamilton paths and a minimal change algorithm. JACM, 31:19--29, 1984.

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P. Eades, M. Hickey, and R. C. Read, "Some Hamilton paths and a minimal change algorithm," J. Assoc. Comput. Mach., 31, No. 1 (1984), pp. 19-29.

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