Generalized Binary Search Trees and Clock Trees Revisited

In this paper, we study two diverse problems from a random matrix perspective. The first one is the problem of binary testing (or object/entity identification) that arises in applications such as active learning, fault diagnosis and computer vision, and the second is the problem of zero or bounded skew clock tree construction which arises in applications such as VLSI circuit design and network multicasting. Though both these problems involve construction of binary trees, the objectives and the greedy algorithms used for binary tree construction are very different. In the problem of binary testing, the goal is to identify an unknown object while minimizing the number of binary questions posed about that object. A binary decision tree is a solution to this problem, where often the goal is to minimize the average depth of the binary tree. Generalized binary search (GBS) is a greedy algorithm that is popularly used in the literature to construct near optimal binary decision trees. Here, we study the depth distribution of trees constructed using GBS and show that it converges to a strange distribution known as the Airy distribution under certain random matrix models. Refer Section 3 for more details. Next, we study the zero or bounded skew clock tree problem. The skew of an edge-weighted rooted tree is defined to be the maximum difference between any two root-to-leaf path weights. Zero or boundedskew trees are needed for achieving synchronization in applications such as network multicasting and VLSI clock routing, where the edge weights correspond to propagation delays. In these applications, the signal generated at the root should be received by multiple recipients located at the leaves (almost) simultaneously. The goal in these problems is to find a zero or bounded-skew tree of minimum total weight, since the weight of the tree corresponds to the amount of resources that must be allocated. Here, we study the skew distribution in clock trees and show that once again, this distribution converges to an Airy distribution as the size of the clock tree increases (refer Section 4). These observations are both surprising and unexpected. Further, they raise several interesting questions regarding the connection of these problems to that of Catalan trees studied in the literature. Also, Airy distribution has been observed to arise as a limit in several other problems involving binary trees in the literature. Hence, these findings pave way for future investigations into these problems by exploiting their relation to previously studied problems.