### 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 log_{3}(n/3k)), and (2) max(n − 1, nlog_{3}(k − 1) − k + 1). Our main lower bound for the size of linear decision trees for polyhedra P in R^{n} is given by the sum of Betti numbers for the complement R^{n} \ 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 R^{n} of points with no k coordinates equal in terms of the roots of the truncated exponential (Equation peresented).

Original language | English |
---|---|

Pages (from-to) | 677-706 |

Number of pages | 30 |

Journal | Journal of the American Mathematical Society |

Volume | 7 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1994 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

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

Research output: Contribution to journal › Article

*Journal of the American Mathematical Society*, vol. 7, no. 3, pp. 677-706. https://doi.org/10.1090/S0894-0347-1994-1243770-0

}

TY - JOUR

T1 - Linear decision trees, subspace arrangements, and Möbius functions

AU - Björner, Anders

AU - Lovász, L.

PY - 1994

Y1 - 1994

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=84968491648&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84968491648&partnerID=8YFLogxK

U2 - 10.1090/S0894-0347-1994-1243770-0

DO - 10.1090/S0894-0347-1994-1243770-0

M3 - Article

AN - SCOPUS:84968491648

VL - 7

SP - 677

EP - 706

JO - Journal of the American Mathematical Society

JF - Journal of the American Mathematical Society

SN - 0894-0347

IS - 3

ER -