Applications of forbidden 0-1 matrices to search tree and path compression-based data structures

Abstract

In this paper we improve, reprove, and simplify several theorems on the performance of data structures based on path compression and search trees. We apply a technique very familiar to computational geometers but still foreign to many researchers in (non-geometric) algorithms and data structures, namely, to bound the complexity of an object via its forbidden substructures.To analyze an algorithm or data structure in the forbidden substructure framework one proceeds in three discrete steps. First, one transcribes the behavior of the algorithm as some combinatorial object M; for example, M may be a graph, sequence, permutation, matrix, set system, or tree. (The size of M should ideally be linear in the running time.) Second, one shows that M excludes some forbidden substructure P, and third, one bounds the size of any object avoiding this substructure. The power of this framework derives from the fact that M lies in a more pristine environment and that upper bounds on the size of a P-free object M may be reused in different contexts.Among our results, we present the first asymptotically sharp bound on the length of arbitrary path compressions on arbitrary trees, improving analyses of Tarjan [35] and Seidel and Sharir [31]. We reprove the linear bound on postordered path compressions, due to Lucas [23] and Loebel and Nesetril [22], the linear bound on deque-ordered path compressions, due to Buchsbaum, Sundar, and Tarjan [5], and the sequential access theorem for splay trees, originally due to Tarjan [38]. We disprove a conjecture of Aronov et al. [3] related to the efficiency of their data structure for half-plane proximity queries and provide a significantly cleaner analysis of their structure. With the exception of the sequential access theorem, all our proofs are exceptionally simple. Notably absent are calculations of any kind.