Linearity and unprovability of set union problem

Authors:
Martin Loebl;Jaroslav Nešetřil
Affiliations:
KAM MFF UK, Department of Applied Mathematics, Charles University, Malostranské nám. 25 11800 Prague;KAM MFF UK, Department of Applied Mathematics, Charles University, Malostranské nám. 25 11800 Prague
Venue:
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Year:
1988

Citing 7
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Abstract

We consider the set union problem (SUP) which consists in designing data for manipulation of a family of disjoint sets which partition a given universe of n elements. In response to a work of Tarjan we prove that the POSTORDER strategy for SUP has a linear length (thus solving a problem of Hart and Sharir). On the other side, we provide a data structure and axioms for an on line strategy-LOCAL POSTORDER-for SUP which fails to be linear but it has a very slow-indeed in the theory of finite sets unprovable-growth. This complements a result of Tarjan who showed an Ackermann type growth for a related problem. Our results may be summarized by saying that (in finite set theory) we may assume that our algorithms are linear (although we know that in fact they fail to be linear). Perhaps this is the first occurrence of unprovability in the complexity analysis of algorithms.