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Constructing RedBlack Trees
, 1999
"... This paper explores the structure of redblack trees by solving an apparently simple problem: given an ascending sequence of elements, construct a redblack tree which contains the elements in symmetric order. Several extreme redblack tree shapes are characterized: trees of minimum and maximum heig ..."
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Cited by 10 (3 self)
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This paper explores the structure of redblack trees by solving an apparently simple problem: given an ascending sequence of elements, construct a redblack tree which contains the elements in symmetric order. Several extreme redblack tree shapes are characterized: trees of minimum and maximum height, trees with a minimal and with a maximal proportion of red nodes. These characterizations are obtained by relating tree shapes to various number systems. In addition, connections to leftcomplete trees, AVL trees, and halfbalanced trees are highlighted. 1 Introduction Redblack trees are an elegant searchtree scheme that guarantees O(log n) worstcase running time of basic dynamicset operations. Recently, C. Okasaki (1998; 1999) presented an impressively simple functional implementation of redblack trees. In this paper we plunge deeper into the structure of redblack trees by solving an apparently simple problem: given an ascending sequence of elements, construct a redblack tree whic...
Numerical Representations as HigherOrder Nested Datatypes
, 1998
"... Number systems serve admirably as templates for container types: a container object of size n is modelled after the representation of the number n and operations on container objects are modelled after their numbertheoretic counterparts. Binomial queues are probably the first data structure that wa ..."
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Cited by 7 (2 self)
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Number systems serve admirably as templates for container types: a container object of size n is modelled after the representation of the number n and operations on container objects are modelled after their numbertheoretic counterparts. Binomial queues are probably the first data structure that was designed with this analogy in mind. In this paper we show how to express these socalled numerical representations as higherorder nested datatypes. A nested datatype allows to capture the structural invariants of a numerical representation, so that the violation of an invariant can be detected at compiletime. We develop a programming method which allows to adapt algorithms to the new representation in a mostly straightforward manner. The framework is employed to implement three different container types: binary randomaccess lists, binomial queues, and 23 finger search trees. The latter data structure, which is treated in some depth, can be seen as the main innovation from a datastruct...