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- W2463278523 abstract "The rotation graph of binary trees, $Rsb{n}$, has a node for each binary tree on n nodes. Two nodes are adjacent when their corresponding trees differ by a single rotation, a standard operation for rebalancing binary search trees. Each node of $rm Rsb{n}$ can also be labelled with a triangulation of a vertex-labelled convex (n + 2)-gon, where adjacent triangulations differ by a single diagonal flip. It is known that R$rmsb{n}$ is Hamiltonian, and that its diameter is bounded above by 2n $-$ 6, for n $geq$ 11. Using properties of triangulations, we show R$rmsb{n}$ has radius n $-$ 1; characterize the n + 2 nodes in the center; show it is the union of n + 2 copies of R$sb{rm n-1}$; and prove it is (n - 1)-connected. We also define RS$rmsb{n}$, the skeleton graph of R$rmsb{n}$, and give additional properties of both graphs.A coding scheme for binary trees is a bijection mapping a set of trees to a set of integer tuples called codewords. For any coding scheme, let S$rmsb{n}$ be a sequence containing the codewords for every n-node tree. If successive codewords represent trees differing by a single rotation, S$rmsb{n}$ is Hamiltonian, i.e., corresponds to a Hamiltonian path in R$rmsb{n}$. We present an algorithm, OzLex, which, for each of many different coding schemes, generates 2$rmsp{n-1}$Hamiltonian sequences S$rmsb{n}$, of which 2$rmsp{n-2}$ represent distinct Hamiltonian paths. For n $geq$ 4 all such sequences combined identify 2$rmsp{n}$ Hamiltonian paths in R$rmsb{n}$. We also discuss a few coding schemes where codeword sequences, as generated by OzLex, are not Hamiltonian. Finally, we modify OzLex to create TransOx, which generates (n + 2)2$rmsp{n}$ Hamiltonian sequences of codewords from a single coding scheme, representing (n + 2)2$rmsp{n-1}$ distinct Hamiltonian paths, for n $geq$ 5.The distance between extreme nodes in R$rmsb{n}$ is the diameter of the graph. We give properties of extreme nodes in terms of their corresponding triangulations, and present two polynomial-in-n heuristics, based on flipping diagonals in triangulations, which find a path between nodes in R$rmsb{n}$. Each computes a path with less than twice the minimum length. We also identify a class of input triangulation pairs where one procedure significantly outperforms the other." @default.
- W2463278523 created "2016-07-22" @default.
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- W2463278523 date "1996-01-01" @default.
- W2463278523 modified "2023-09-23" @default.
- W2463278523 title "Finding paths in the rotation graph of binary trees" @default.
- W2463278523 hasPublicationYear "1996" @default.
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