On optimal interconnections

Many applications require algorithms for determining optimal interconnections. This dissertation centers on new geometric formulations and approximation algorithms for optimizing interconnection objectives which are of particular interest in the design of high-performance VLSI systems. These formulations include Steiner trees, pathlength-balanced trees, bounded-radius trees, and prescribed-width paths; we also address the closely related question of efficiently testing physical interconnections. For most cases, we have new, best-known results, and in all cases we have empirically demonstrated significant improvements over the best previous methods. We give the best-performing rectilinear Steiner tree heuristic to date: the algorithm has worst-case performance ratio strictly less than 3/2 times optimal, settling a long-standing open problem. We also give a class of instances which are pathological for virtually all existing Steiner tree heuristics in the literature, thus disproving several conjectures and claimed performance bounds. We propose a matching-based method for pathlength-balanced trees: the construction yields near-zero average pathlength skew while maintaining small total tree cost. To address a separate objective, we also offer the first general formulation of performance-driven routing, allowing a smooth tradeoff of tree cost for tree radius. Our algorithm melds the two classical constructions of the minimum spanning tree and the shortest paths tree, and has worst-case performance bounded by a constant times optimal with respect to both tree cost and tree radius. Motivated by recent circuit testing applications, we formulate connectivity testing as a problem of tree verification via k-probes. We present linear-time algorithms which compute a minimum probe set achieving complete coverage of both edge and node fault classes. Actual testing demands the efficient scheduling of probe operations: we show that this entails a special type of metric traveling salesman optimization, and we give provably good heuristics. Finally, we address a fundamental problem in routing and path planning: determining a minimum-cost path of prescribed width which connects a given source-destination pair in an arbitrarily costed region. We give the first known polynomial-time algorithm for this problem, and extend our approach to solve a discrete version of the classical Plateau problem on minimal surfaces.