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28 March 2024
 
  » arxiv » cs.DS/0207061

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Linear-Time Pointer-Machine Algorithms for Least Common Ancestors, MST Verification, and Dominators
Adam L. Buchsbaum ; Haim Kaplan ; Anne Rogers ; Jeffery R. Westbrook ;
Date 15 Jul 2002
Subject Data Structures and Algorithms ACM-class: D.3.4; E.1; F.1.1; F.2.2; G.2.2 | cs.DS
AbstractWe present two new data structure tools--disjoint set union with bottom-up linking, and pointer-based radix sort--and combine them with bottom-level microtrees to devise the first linear-time, pointer-machine algorithms for off-line least common ancestors, minimum spanning tree verification, and computing dominators in a flowgraph. Our results also imply the first randomized, linear-time algorithm for computing minimum spanning trees on a pointer machine. Specifically, we show that a sequence of $n-1$ unions and $m$ finds where the unions always include one of $ell$ designated special elements takes $O(malpha(m,ell)+n)$ time on a pointer machine. This result applies in particular to union trees on $ell$ leaves with a bottom-up union pattern. When $ell$ is slightly sublinear in $m$, the time bound becomes linear. We also show that for any tractable graph problem (generally restricted to consider only the topology of the input graph), a set of instances of total size $N$ can be solved on a pointer machine in O(N) time, as long as each instance is small with respect to $N$ (e.g., $O(log^{1/3} N)$ vertices each). We state these results generally and use them in turn-key fashion.
Source arXiv, cs.DS/0207061
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