Lower Bound Techniques in the Comparison-Query Model and Inversion Minimization on Trees

Abstract

Given a rooted tree and a ranking of its leaves, what is the minimum number of inversions of the leaves that can be attained by ordering the tree? This variation of the problem of counting inversions in arrays originated in mathematical psychology, with the evaluation of the Mann--Whitney statistic for detecting differences between distributions as a special case. We study the complexity of the problem in the comparison-query model, used for problems like sorting and selection. For many types of trees with $n$ leaves, we establish lower bounds close to the strongest known in the model, namely the lower bound of $\log_2(n!)$ for sorting $n$ items. We show: (a) $\log_2((\alpha(1-\alpha)n)!) - O(\log n)$ queries are needed whenever the tree has a subtree that contains a fraction $\alpha$ of the leaves. This implies a lower bound of $\log_2((\frac{k}{(k+1)^2}n)!) - O(\log n)$ for trees of degree $k$. (b) $\log_2(n!) - O(\log n)$ queries are needed in case the tree is binary. (c) $\log_2(n!) - O(k \log k)$ queries are needed for certain classes of trees of degree $k$, including perfect trees with even $k$. The lower bounds are obtained by developing two novel techniques for a generic problem $\Pi$ in the comparison-query model and applying them to inversion minimization on trees. Both techniques can be described in terms of the Cayley graph of the symmetric group with adjacent-rank transpositions as the generating set. Consider the subgraph consisting of the edges between vertices with the same value under $\Pi$. We show that the size of any decision tree for $\Pi$ must be at least: (i) the number of connected components of this subgraph, and (ii) the factorial of the average degree of the complementary subgraph, divided by $n$. Lower bounds on query complexity then follow by taking the base-2 logarithm.