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Linear-Time Pointer-Machine Algorithms for Path-Evaluation Problems on Trees and Graphs

Adam L. Buchsbaum·Loukas Georgiadis·Haim Kaplan·Anne Rogers·Robert E. Tarjan·Jeffery R. Westbrook
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Abstract

We present algorithms that run in linear time on pointer machines for a collection of problems, each of which either directly or indirectly requires the evaluation of a function defined on paths in a tree. These problems previously had linear-time algorithms but only for random-access machines (RAMs); the best pointer-machine algorithms were super-linear by an inverse-Ackermann-function factor. Our algorithms are also simpler, in some cases substantially, than the previous linear-time RAM algorithms. Our improvements come primarily from three new ideas: a refined analysis of path compression that gives a linear bound if the compressions favor certain nodes, a pointer-based radix sort as a replacement for table-based methods, and a more careful partitioning of a tree into easily managed parts. Our algorithms compute nearest common ancestors off-line, verify and construct minimum spanning trees, do interval analysis on a flowgraph, find the dominators of a flowgraph, and build the component tree of a weighted tree.

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Adam L. Buchsbaum, Loukas Georgiadis, Haim Kaplan, Anne Rogers, Robert E. Tarjan, Jeffery R. Westbrook. 2006-11-14. Linear-Time Pointer-Machine Algorithms for Path-Evaluation Problems on Trees and Graphs. https://arxiv.org/abs/cs/0207061

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