Short Note: A Note on the Expected Behaviour of Binary Tree Traversals

Anderson, A.

doi:10.1093/comjnl/33.5.471

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2002

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“…; and the nodes can be traversed in different orders: inorder, preorder, postorder, breath-first, depth-first, etc. ; see [2,10,11,12,22,23,31,50,64] and [30]. The analysis of the cost of these algorithms then reduces to the calculation of certain parameters on trees such as the number of nodes with null (or non-null) left (or right) branches, the number of nodes with both non-null left and right branches, and the number of nodes that are a left child and whose right branch is not empty.…”

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confidence: 99%

Phase Change of Limit Laws in the Quicksort Recurrence under Varying Toll Functions

Hwang¹,

Neininger²

2002

SIAM J. Comput.

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We characterize all limit laws of the quicksort type random variables defined recursively by X n d = X In + X * n−1−In + T n when the "toll function" T n varies and satisfies general conditions, where (X n), (X * n), (I n , T n) are independent, X n d = X * n , and I n is uniformly distributed over {0,. .. , n − 1}. When the "toll function" T n (cost needed to partition the original problem into smaller subproblems) is small (roughly lim sup n→∞ log E(T n)/ log n ≤ 1/2), X n is asymptotically normally distributed; non-normal limit laws emerge when T n becomes larger. We give many new examples ranging from the number of exchanges in quicksort to sorting on broadcast communication model, from an in-situ permutation algorithm to tree traversal algorithms, etc.

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