Linear decision trees, subspace arrangements, and Möbius functions

Anders Björner, L. Lovász

Research output: Contribution to journalArticle

36 Citations (Scopus)

Abstract

Topological methods are described for estimating the size and depth of decision trees where a linear test is performed at each node. The methods are applied, among others, to the questions of deciding by a linear decision tree whether given n real numbers (1) some k of them are equal, or (2) some k of them are unequal. We show that the minimum depth of a linear decision tree for these problems is at least (1) max(n − 1, n log3(n/3k)), and (2) max(n − 1, nlog3(k − 1) − k + 1). Our main lower bound for the size of linear decision trees for polyhedra P in Rn is given by the sum of Betti numbers for the complement Rn \ P. The applications of this general topological bound involve the computation of the Möbius function of intersection lattices of certain subspace arrangements. In particular, this leads to computing various expressions for the Möbius function of posets of partitions with restricted block sizes. Some of these formulas have topological meaning. For instance, we derive a formula for the Euler characteristic of the subset of Rn of points with no k coordinates equal in terms of the roots of the truncated exponential (Equation peresented).

Original languageEnglish
Pages (from-to)677-706
Number of pages30
JournalJournal of the American Mathematical Society
Volume7
Issue number3
DOIs
Publication statusPublished - 1994

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Decision trees
Decision tree
Arrangement
Subspace
Exponential equation
Topological Methods
Betti numbers
Euler Characteristic
Unequal
Poset
Polyhedron
Complement
Intersection
Partition
Roots
Lower bound
Subset
Computing
Vertex of a graph

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Linear decision trees, subspace arrangements, and Möbius functions. / Björner, Anders; Lovász, L.

In: Journal of the American Mathematical Society, Vol. 7, No. 3, 1994, p. 677-706.

Research output: Contribution to journalArticle

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