Average depth in a binary search tree with repeated keys

Abstract

International audience Random sequences from alphabet $\{1, \ldots,r\}$ are examined where repeated letters are allowed. Binary search trees are formed from these, and the average left-going depth of the first $1$ is found. Next, the right-going depth of the first $r$ is examined, and finally a merge (or 'shuffle') operator is used to obtain the average depth of an arbitrary node, which can be expressed in terms of the left-going and right-going depths. The variance of each of these parameters is also found.

Quicksort Is Optimal For Many Equal Keys

“…subtrees, we note that the relative order of elements is retained in both processes, so the elements inserted in the left/right child of the root are exactly U (1) resp. U (2) in both cases. The claim thus follows by induction.…”

“…Many parameters like path length, height and profiles of fringe-balanced trees have been studied when the trees are built from a random permutation of n distinct elements, see, e.g., Drmota [11]. The case of equal elements has not been considered except for the unbalanced case k = 1, i.e., ordinary BSTs; see Kemp [26], Archibald and Clément [2].…”

“…So assume the claim holds for inputs with less than k elements. If now n ≥ k, Quicksort chooses a pivot P from the first k elements, compares all elements to P , and divides the input into segments U (1) and U (2) containing the elements strictly smaller resp. strictly larger than P .…”

“…where (A (1) q ) and (A (2) q ) are independent copies of (A q ), which are also independent of (V 1 , V 2 , Z  , Z  ).…”

“…At the KnuthFest celebrating Donald Knuth's 1000000 2 th birthday, Robert Sedgewick gave a talk titled "Quicksort is optimal" [40], 2 presenting a result that is at least very nearly so: Quicksort with fat-pivot (a.k.a. three-way) partitioning uses 2 ln 2 ≈ 1.39 times the number of comparisons needed by any comparison-based sorting method for any randomly ordered input, with or without equal elements.…”