Two-Level Rectilinear Steiner Trees

Given a set $P$ of terminals in the plane and a partition of $P$ into $k$ subsets $P_1, ..., P_k$, a two-level rectilinear Steiner tree consists of a rectilinear Steiner tree $T_i$ connecting the terminals in each set $P_i$ ($i=1,...,k$) and a top-level tree $T_{top}$ connecting the trees $T_1, ..., T_k$. The goal is to minimize the total length of all trees. This problem arises naturally in the design of low-power physical implementations of parity functions on a computer chip. For bounded $k$ we present a polynomial time approximation scheme (PTAS) that is based on Arora's PTAS for rectilinear Steiner trees after lifting each partition into an extra dimension. For the general case we propose an algorithm that predetermines a connection point for each $T_i$ and $T_{top}$ ($i=1,...,k$). Then, we apply any approximation algorithm for minimum rectilinear Steiner trees in the plane to compute each $T_i$ and $T_{top}$ independently. This gives us a $2.37$-factor approximation with a running time of $\mathcal{O}(|P|\log|P|)$ suitable for fast practical computations. The approximation factor reduces to $1.63$ by applying Arora's approximation scheme in the plane

Algorithms for the power-p Steiner tree problem in the Euclidean plane

We study the problem of constructing minimum power-$p$ Euclidean $k$-Steiner trees in the plane. The problem is to find a tree of minimum cost spanning a set of given terminals where, as opposed to the minimum spanning tree problem, at most $k$ additional nodes (Steiner points) may be introduced anywhere in the plane. The cost of an edge is its length to the power of $p$ (where $p\geq 1$), and the cost of a network is the sum of all edge costs. We propose two heuristics: a ``beaded" minimum spanning tree heuristic; and a heuristic which alternates between minimum spanning tree construction and a local fixed topology minimisation procedure for locating the Steiner points. We show that the performance ratio $\kappa$ of the beaded-MST heuristic satisfies $\sqrt{3}^{p-1}(1+2^{1-p})\leq \kappa\leq 3(2^{p-1})$. We then provide two mixed-integer nonlinear programming formulations for the problem, and extend several important geometric properties into valid inequalities. Finally, we combine the valid inequalities with warm-starting and preprocessing to obtain computational improvements for the $p=2$ case

Timing-Constrained Global Routing with Buffered Steiner Trees

This dissertation deals with the combination of two key problems that arise in the physical design of computer chips: global routing and buffering. The task of buffering is the insertion of buffers and inverters into the chip's netlist to speed-up signal delays and to improve electrical properties of the chip. Insertion of buffers and inverters goes alongside with construction of Steiner trees that connect logical sources with possibly many logical sinks and have buffers and inverters as parts of these connections. Classical global routing focuses on packing Steiner trees within the limited routing space. Buffering and global routing have been solved separately in the past. In this thesis we overcome the limitations of the classical approaches by considering the buffering problem as a global, multi-objective problem. We study its theoretical aspects and propose algorithms which we implement in the tool BonnRouteBuffer for timing-constrained global routing with buffered Steiner trees. At its core, we propose a new theoretically founded framework to model timing constraints inherently within global routing. As most important sub-task we have to compute a buffered Steiner tree for a single net minimizing the sum of prices for delays, routing congestion, placement congestion, power consumption, and net length. For this sub-task we present a fully polynomial time approximation scheme to compute an almost-cheapest Steiner tree with a given routing topology and prove that an exact algorithm cannot exist unless P=NP. For topology computation we present a bicriteria approximation algorithm that bounds both the geometric length and the worst slack of the topology. To improve the practical results we present many heuristic modifications, speed-up- and post-optimization techniques for buffered Steiner trees. We conduct experiments on challenging real-world test cases provided by our cooperation partner IBM to demonstrate the quality of our tool. Our new algorithm could produce better solutions with respect to both timing and routability. After post-processing with gate sizing and Vt-assignment, we can even reduce the power consumption on most instances. Overall, our results show that our tool BonnRouteBuffer for timing-constrained global routing is superior to industrial state-of-the-art tools

Facility Location and Clock Tree Synthesis

The construction of clock trees and repeater trees are major challenges in chip design. Such trees distribute an electrical clock signal from a source to a set of sinks on a chip. On recent designs there can be millions of repeater trees with only a few up to some hundred sinks and several clock trees with up to some hundred thousand of sinks. In repeater trees the signal has to arrive at each sink not later than an individual required arrival time, while in clock trees it has to arrive at each sink within an individual required arrival time window. In this thesis, we present new theory and algorithms for the construction of clock trees and repeater trees and an essential sub-problem, the Sink Clustering Problem. We also describe our clock tree construction tool BonnClock, which has been used by IBM Microelectronics for the design of hundreds of most complex chips. First, we introduce the Sink Clustering Problem, the main sub-problem of clock tree design. Given a metric space (V,c), a finite set D of terminals with positions p(v) ∈ V and demands d(v) ∈ R ≥ 0 for all v ∈ D, a facility opening cost f ∈ R>0 and a load limit u ∈ R>0 , the task is to find a partition D=D1 ∪ ... ∪ Dk of D and, for all 1 ≤ i ≤ k, a Steiner tree Si for {p(v)| v ∈ Di }. Each cluster (Di ,Si ), 1 ≤ i ≤ k, has to keep the load limit, that means ∑e ∈ E(Si) c(e) +∑s ∈ Di d(s) ≤ u. The goal is to minimize the weighted sum of the length of all Steiner trees plus the number of clusters, i.e. minimize ∑i=1,...,k (∑e ∈ E(Si ) c(e)) +kf. We present the first constant-factor approximation algorithm for the Sink Clustering Problem. It is based on decomposing a minimum spanning tree on the sinks and has an approximation guarantee of 1+2α, where α is the Steiner ratio of the underlying metric. Moreover, we introduce two variants of the algorithm that rely on decomposing an approximate minimum Steiner tree and an approximate minimum traveling salesman tour. These algorithms have approximation guarantees of 3β and 3γ, respectively, where β and γ are the approximation guarantees of the Steiner tree and TSP approximation algorithms, respectively. We also propose two post-optimization algorithms that can further improve an existing clustering. We analyze the structure of the Sink Clustering Problem and exhibit its connections to matroid theory. In particular, we use the property of matroids that for any two bases B1 , B2 there is a bijection p : B1 → B2 so that (B1 \ {b}) ∪ {p(b)} is again a basis for each b ∈ B1. We replace each Steiner tree of an optimum solution by a minimum spanning tree and connect all trees to a new artificial vertex s and get a tree S. In a modified metric the total length of S is a good lower bound for the cost of an optimum solution. Due to the matroid property we can compare a minimum spanning tree T on D ∪ {s} with S; the length of any edge of T is bounded by the length of an edge of S. We introduce the concept of K-dominated functions that helps us to increase the `cost' of certain edges of T while still having the property that the total length of all edges of T ending in a vertex of K ⊆ D is bounded by the total length of all edges of S ending in a vertex of K. Applying this procedure to the sets of a laminar family on D yields an improved lower bound. The bound can be further improved by combining it with a lower bound for the length of a minimum Steiner tree on D. For this bound we prove the following lemma: For any family of trees T = {T1 ,..., Tk } with V(Ti ) ⊂ D, 1 ≤ i ≤ k, with the property that for any subset T' ⊆ T the trees in T' cover at least | T' |+1 vertices, there exists an edge ei ∈ E(Ti ) for i=1,..., k such that these edges E={ei | 1 ≤ i ≤ k} form a forest, i.e. the set does not contain an edge twice and it does not contain a circuit. Our experimental results on real-world instances from clock tree design show that the cost of the solutions computed by our algorithms is in average only 10% over the best lower bound. Moreover, we compare our algorithm to another clustering algorithm used in industry. The results show that the total cost of our solutions is 10% less than the cost of the solutions computed by the competitive tool. Clock trees have to satisfy several timing constraints. More precisely, the signal has to reach each sink within an individual required arrival time window. Sinks can only be clustered together if their required arrival time windows have a point of time in common. Typically, all required arrival time windows are the same. In this case we have the Sink Clustering Problem defined above. However, there are clock trees where the sinks have different required arrival time windows. This motivates a generalization of the Sink Clustering Problem where each sink additionally has an individual time window. As further constraint the time windows of the sinks of a cluster must have at least one point of time in common. We study the Sink Clustering Problem with Time Windows and present a polynomial O(log s)-approximation algorithm for this problem, where s is the size of a minimum clique partition in the interval graph induced by the time windows. Our algorithm is based on a divide and conquer approach and uses the approximation algorithms for the Sink Clustering Problem on sub-sets of the instance. We show that the approximation guarantee of the algorithm is tight. For the practical construction of clock trees we present our algorithm BonnClock. BonnClock builds a clock tree combining a bottom-up clustering and a top-down partitioning strategy. In the bottom-up phase BonnClock is using the Sink Clustering Algorithm in order to determine the drivers of unconnected sinks or inverters. The `global' topology of the tree is determined by the top-down partitioning considering big blockages and timing restrictions. BonnClock uses a dynamic program in order to determine the sizes of the inverters that are inserted. All components of the algorithm are discussed in detail. As part of this thesis, we have also implemented this algorithm. BonnClock has become the standard tool to construct clock trees within IBM. We show experimental results with comparisons to another industrial clock tree construction tool and to lower bounds for the power consumption. It turns out that - mainly due to the Sink Clustering Algorithm - our power consumption is much smaller than with the other tool and only one third over the lower bound. Finally, we consider the repeater tree construction problem. In contrast to clock trees, each sink has a latest required arrival time instead of a time window. We describe a simple algorithm to build such trees where we insert the sinks one by one into an existing tree. Depending on the optimization goal we show a variant of the algorithm computing trees of almost optimal length or trees with guaranteed best possible performance. Moreover, we analyze the topology of trees with best or almost best performance more closely. Such trees are equivalent to minimax and almost minimax trees: Let a1 , ... , an ∈ N ≥ 0 be a set of numbers. The weight of a tree with n leaves is the maximum over all leaves i of the depth of leaf i plus ai. For a non-negative integral constant c the goal is to build a binary tree with weight at most the optimum weight plus c. This problem can be solved optimally by a greedy algorithm. However, we are interested in the online version of this problem where we have to insert the leaf i with weight ai into the tree without knowing n and the following weights aj, j> i. We give necessary and sufficient conditions for an online algorithm to compute trees of weight at most the optimum weight plus c. Moreover, we show how these conditions can be verified efficiently. We obtain an online algorithm that computes an optimum tree in O(nlog n) time. Finally, we study a further mathematical model of repeater trees that considers that additional delay caused by a bifurcation of a tree can be distributed partially to the two branches. For c∈ R>0 and a set L ⊆ {(l1 ,l2 ) ∈ R2 ≥ 0 | l1 +l2 = c} of two-element sets of non-negative real numbers we consider rooted binary trees with the property that the two edges emanating from every non-leaf are assigned lengths l1 and l2 with { l1 ,l2 } ? L. We study the asymptotic growth of the maximum number of leaves of bounded depths in such trees and the existence of such trees with leaves at individually specified maximum depths. Our results yield better lower bounds for repeater trees

A parallel genetic algorithm for the Steiner Problem in Networks

This paper presents a parallel genetic algorithm to the Steiner Problem in Networks. Several previous papers have proposed the adoption of GAs and others metaheuristics to solve the SPN demonstrating the validity of their approaches. This work differs from them for two main reasons: the dimension and the characteristics of the networks adopted in the experiments and the aim from which it has been originated. The reason that aimed this work was namely to build a comparison term for validating deterministic and computationally inexpensive algorithms which can be used in practical engineering applications, such as the multicast transmission in the Internet. On the other hand, the large dimensions of our sample networks require the adoption of a parallel implementation of the Steiner GA, which is able to deal with such large problem instances