Adaptive heap sort
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The adaptive heap sort is a variant of heap sort that uses a randomized binary search tree, or RBST, to structure the input according to any preexisting order. The RBST is used to select candidates that are put into the heap, so the heap doesn't need to keep track of all elements.
See also
This article incorporates public domain material from Paul E. Black. "Adaptive heap sort". Dictionary of Algorithms and Data Structures. NIST.