Closing the gap: near-optimal Steiner trees in polynomial time

Abstract

The minimum rectilinear Steiner tree (MRST) problem arises in global routing and wiring estimation, as well as in many other areas. The MRST problem is known to be NPhard, and the best performing MRST heuristic to date is the Iterated 1-Steiner (IIS) method recently proposed by Kahng and Robins. In this paper, we develop a straightforward, efficient implementation of IIS, achieving a speedup factor of three orders of magnitude over previous implementations. We also give a parallel implementation that achieves near-linear speedup on multiple processors. Several performance-improving enhancements enable us to obtain Steiner trees with average cost within 0.257 of optimal, and our methods produce optimal solutions in up to 90t/' of the cases for typical nets. We generalize 11S and its variants to three dimensions, as well as to the case where all the pins lie on k parallel planes, which arises in, e.g., multilayer routing. Motivated by the goal of reducing the running times of our algorithms, we prove that any pointset in the Manhattan plane has a minimum spanning tree (MST) with maximum degree 4, and that in three-dimensional Manhattan space every pointset has an MST with maximum degree of 14 (the best previous upper bounds on the maximum MST degree in two and three dimensions are 6 and 26, respectively); these results are of independent theoretical interest and also settle an open problem in complexity theory.

Minimum Steiner Tree Construction*

“…A more efficient algorithm based on [33] can find a new 1-Steiner point within O(n 2 ) time [57]. A linear number of Steiner points can therefore be found in O(n 3 ) time, and trees with a candidate Steiner point and the next change very little (by only a constant number of tree edges), incremental/dynamic MST updating schemes can be employed, resulting in further asymptotic time complexity improvements [36,57].…”

“…Iterated 1-Steiner (I1S) Heuristic [36,55,57] Input: set P of n points Output: rectilinear Steiner tree spanning P S = ∅ While Candidate Set = {x ∈ H(P ∪ S)|∆M ST (P ∪ S, {x}) > 0} = ∅ Do Find x ∈ Candidate Set which maximizes ∆M ST (P ∪ S, {x}) S = S ∪ {x} Remove points in S which have degree ≤ 2 in M ST (P ∪ S) Output MST(P ∪ S) To find a 1-Steiner point in the Manhattan plane, it suffices to construct an MST over |P ∪S|+1 points for each of the O(n 2 ) members of the Steiner candidate set (i.e., Hanan grid points), and then pick a candidate which minimizes the overall MST cost. Each MST computation can be performed in O(n log n) time [72], yielding an O(n 3 log n) time method to find a single 1-Steiner point.…”

MARS-a multilevel full-chip gridless routing system

“…Since the problem of minimum RST has been proved to be NP-hard, many heuristic algorithms such as in [32] and [36] were proposed to get approximate solutions. Most of the Steiner tree approximation algorithms are geometric distance based.…”

Timing-Driven Interconnect Synthesis

“…In order to achieve an improved runtime for the IDOM approach, Alexander and Robins [1] [66]; (b) the optimal Steiner tree, which is also the solution produced by the Graph Iterated 1-Steiner algorithm of [41,62]; (c) Dijkstra's shortest paths tree [29]); (d) the optimal Steiner arborescence, which is also the solution produced by the IDOM algorithm of [1]. Note that the IDOM solution in (d) is optimal in terms of both total wirelength as well as maximum pathlength (although this double-optimal outcome is unusual).…”