Linear decision trees, subspace arrangements and Möbius functions

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) ${\text{max}}\{ n - 1,\quad n\;{\text{lo}}{{\text{g}}_3}(n/3k)\}$, and (2) ${\text{max}}\{ n - 1,\quad n\;{\text{lo}}{{\text{g}}_3}(k - 1) - k + 1\}$.

Our main lower bound for the size of linear decision trees for polyhedra $P$ in ${{\mathbf{R}}^n}$ is given by the sum of Betti numbers for the complement ${{\mathbf{R}}^n}\backslash P$. 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 ${{\mathbf{R}}^n}$ of points with no $k$ coordinates equal in terms of the roots of the truncated exponential $\sum\nolimits_{i < k} {{x^i}} /i!$.