Faster min–max resource sharing in theory and practice

Abstract

We consider the (block-angular) min-max resource sharing problem, which is defined as follows. Given finite sets R of resources and C of customers, a convex set B c , called block, and a convex function g c :As usual we assume that g c can be computed efficiently and we have a constant σ ≥ 1 and oracle functionsWe describe a simple algorithm which solves this problem with an approximation guarantee σ (1 + ω) for any ω > 0, and whose running time is O(θ (|C|+|R|) log |R|(log log |R|+ω −2 )) for any fixed σ ≥ 1, where θ is the time for an oracle call. This generalizes and improves various previous results. We also prove other bounds and describe several speed-up techniques. In particular, we show how to parallelize the algorithm efficiently. In addition we review another algorithm, variants of which were studied before. We show that this algorithm is almost as fast in theory, but it was not competitive in our experiments. Our work was motivated mainly by global routing in chip design. Here the blocks are mixedinteger sets (whose elements are associated with Steiner trees), and we combine our algorithm with randomized rounding. We present experimental results on instances 123 2 D. Müller et al.resulting from recent industrial chips, with millions of customers and resources. Our algorithm solves these instances nearly optimally in less than two hours.Keywords Min-max resource sharing · Fractional packing · Fully polynomial approximation scheme · Parallelization · Global routing · Chip design Mathematics Subject Classification (2010) 90C27 · 90C90 · 90C59 · 90C48 · 90C06

Placement: From Wirelength to Detailed Routability

“…In BonnPlace [10], BonnRouteGlobal [42] is used for congestion estimation. CGRIP [43] uses integer programming (IP) in global routing, which gives high quality global routing solution in a reasonable amount of time.…”

Detailed Routing Algorithms for Advanced Technology Nodes

“…We assume that we are given one source and one target component of a net and a routing corridor A ⊆ V (Ḡ) computed in the global routing step of BonnRoute (see [23], [2]). Our goal is to connect the two components by on-track wires withinḠ[A] (this restriction is relaxed gradually if no connection is found).…”

BonnRoute

“…This is indeed true in our case. In Müller et al [2011], we proved that inf b∈B int n r∈R y r g r n (b) = inf b∈B n r∈R y r g r n (b) holds, that is, optimization over the set B int n gives no worse results than optimizing over its convex hull B n . Therefore, it suffices to find a b * ∈ B int n with r∈R y r g r n (b * ) ≤ σ inf b∈B int n r∈R y r g r n (b).…”