Combinatorial Algorithms
FUNCTIONAL PEARL: On merging and selection
Journal of Functional Programming
MPC '02 Proceedings of the 6th International Conference on Mathematics of Program Construction
Algebraic methods for optimization problems
Algebraic and coalgebraic methods in the mathematics of program construction
Perfect trees and bit-reversal permutations
Journal of Functional Programming
Theory and applications of inverting functions as folds
Science of Computer Programming - Special issue on mathematics of program construction (MPC 2002)
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A common solution to the problem of handling list indexing efficiently in a functional program is to build a binary tree. The tree has the given list as frontier and is of minimum height. Each internal node of the tree stores size information (actually, the size of its left subtree) to direct the search for an element at a given position in the frontier. One application was considered in my previous pearl (Bird, 1997). There are two complementary methods for building such a tree, both of which can be implemented in linear time. One method is ‘recursive’, or top down, and works by splitting the list into two equal halves, recursively building a tree for each half, and then combining the two results. The other method is ‘iterative’, or bottom up, and works by first creating a list of singleton trees, and then repeatedly combining the trees in pairs until just one tree remains. The two methods lead to different trees, but in each case the result is a tree with smallest possible height.