Ranking Cost: One-Stage Circuit Routing by Directly Optimizing Global Objective Function

1 INTRODUCTION . As described in Moore ’ s Law ( Schaller , 1997 ) , the number of transistors in a dense integrated circuit ( IC ) increases exponentially over time and the complexity of chips and printed circuit boards ( PCBs ) becomes higher and higher . Such high complexity makes the IC design a time-consuming and errorprone work . Thus more capable automatic design systems , such as electronic design automation ( EDA ) tools , are needed to improve the performance . In the flow of IC designs , we need to find proper paths to place wires which connect electronic components on ICs , and these wires need to achieve expected connectivity under certain constraints . One of the most important constraints is that wires on the same layout should not intersect . In addition , to reduce the signal propagation delay , the wire-length should be minimized . This is a critical and challenging stage in the IC design flow ( Hu & Sapatnekar , 2001 ) , known as circuit routing , which has been studied by lots of researchers ( Kramer , 1984 ; Zhang & Chu , 2012 ; He & Bao , 2020 ) . Circuit routing involves a large number of nets ( a net is a set of vertices with the same electrical property ) to be routed , which is computationally expensive and makes manual design extremely time-consuming ( Kong et al. , 2009 ; Coombs & Holden , 2001 ) . Even under the simplest setting , where only two pairs of pins need to be routed , it is an NP-complete problem ( Kramer , 1984 ) . Although lots of circuit routing algorithms have been proposed ( Zhang , 2016 ) , there still remain three major challenges : ( 1 ) Early solutions ( Hu & Sapatnekar , 2001 ) are typically designed with hard-coded heuristics , which suffer from problems of non-optimal solutions ( Zhang , 2016 ) and lack of flexibility over new design needs . Therefore , a more powerful routing method that does not depend on domain knowledge is highly desired . ( 2 ) To reduce the difficulty of complex routing problems , traditional routing algorithms often adopt a two-stage procedure — first global routing and then detailed routing ( Chen & Chang , 2009 ; Kahng et al. , 2011 ) . The problem is that these two stages do not always coordinate well ( Zhang & Chu , 2012 ; Shi & Davoodi , 2017 ) . Sometimes a low-congested global routing result may lead to downstream detailed router un-routable . Hence , an end-to-end algorithm is preferred which can optimize the final global objective ( e.g. , the total wire-length ) directly . ( 3 ) Although a few learning-based methods have been proposed ( Liao et al. , 2020 ; He & Bao , 2020 ) recently , their methods are hard to extend to large-scale applications . In real settings , there are lots of components and nets on a single chip , which shows greater demand for the scalability of routing algorithms . To relieve the problems mentioned above , we propose a new algorithm , denoted as Ranking Cost ( RC ) , for circuit routing . In this paper , we innovatively combine search-based methods ( i.e. , A * algorithm ) and learning-based methods ( i.e. , Evolution Strategies ( Salimans et al. , 2017 ) ) to form an trainable router with proper parametrization . Our method is flexible for integrating new constraints and rules so long as they can be merged into the global objective function . Moreover , our method is an one-stage algorithm , which optimizes the global objective function directly . In our method , we introduce a new set of variables called cost maps , which can help the A * routers to find out proper paths to achieve the global objective . We also train a ranking parameter , which can produce the ranking order and further improve the performance of our method . In the experiments , we compare our method with the commonly used A * method and a canonical reinforcement learning approach , and results show that our method outperforms these baselines with higher connectivity rates . Our ablation study also shows that trained cost maps can capture the global information and guide the routing solution to approach global optimal . Experiments also show that our method is scalable to larger applications . 2 RELATED WORK . In this section , we summarize the related work on circuit routing . 2.1 TWO-STAGE ROUTING ALGORITHMS . The routing problem can be heuristically separated into two stages , the first being the global routing step , followed by detailed routing . On the one hand , there are multiple heuristic-based approaches for global routing including regionwise routing ( Hu & Sapatnekar , 2001 ) , force-directed routing ( Mo et al. , 2001 ) , and rip-up and reroute ( Cho et al. , 2007 ) . On the other hand , the most commonly used detailed routing algorithms are channel routing and its variants ( Ho et al. , 1991 ; Mandal et al. , 2020 ) , which decompose the routing region into routing channels and generate wires in these channels ( Chen & Chang , 2009 ) . One main issue of the two-stage methods is that these two stages do not always coordinate well ( Shi & Davoodi , 2017 ) , which results in enormous difficulty in joint optimization . Instead , our method is an one-stage algorithm and new design constraints can be simply involved into the objective function without changing the algorithm itself . 2.2 SEQUENTIAL ROUTING ALGORITHMS . A more straightforward strategy for circuit routing is to select a specific order and then route nets sequentially , e.g. , sequential A * algorithm and Lees algorithm ( Huang et al. , 2014 ; Malavasi & Sangiovanni-Vincentelli , 1993 ) . The major advantage of this type of approaches is that the congestion information for previously routed nets can be taken into consideration while routing the current one . However , the drawback of these sequential approaches is that the quality of the solution is very sensitive to the orders ( Zhang , 2016 ) . Moreover , earlier routed paths only focus on finding their own best solutions and are impossible to take into account the situation of subsequent paths . Such greedy strategies may make a solvable circuit routing problem insolvable . Figure 1 shows an example that the sequential A * algorithm will fail to handle . In this example , there are two pairs of points to be routed , i.e. , we should connect start vertices Si and end vertices Ei , i ∈ { 1 , 2 } , respectively . If we connect the pair ( S1 , E1 ) first in its shortest path , it will make ( S2 , E2 ) disconnected as showed in Figure 1 ( d ) , and vice versa ( Figure 1 ( e ) ) . But this case can be solved easily by our algorithm as showed in Figure 1 ( f ) , which means that our algorithm can take into account the global information when generating each path . 2.3 LEARNING-BASED ALGORITHMS FOR CIRCUIT ROUTING . There are some learning-based algorithms for circuit routing ( Qi et al. , 2014 ; Liao et al. , 2020 ) . Early methods are more focusing on supervised learning , such as learning models to predict routing congestion ( Qi et al. , 2014 ) or predicting the routability ( Zhou et al. , 2015 ; Xie et al. , 2018 ) . But the performances of these supervised models are constrained by the limits of human knowledge . Recently , Liao et al . ( 2020 ) trained a deep Q-network ( DQN ) to solve the global routing problem and their method still suffers from the miscoupling problem of two-stage routing algorithms . In addition , He & Bao ( 2020 ) reform circuit routing to a tree search problem and utilize a deep neural network as the rollout policy . However , their rollout policy is trained in a supervised manner with artificial data . Instead , our method optimizes the global objective function directly and does not require any artificial data or human demonstration . 3 BACKGROUND AND NOTATIONS . In this section , we will formalize the circuit routing problem first and then give a brief introduction to the OpenAI-ES algorithm ( Salimans et al. , 2017 ) . 3.1 CIRCUIT ROUTING . Circuit routing is a path search problem , where the goal is to find non-intersecting paths that connect an arbitrary number of pairs of vertices . It can be formalized as a grid graph G = ( V , E ) , where each vertex vi ∈ V represents an intersection on the grid , and each edge eij ∈ E represents the path between vi and its 1-hop neighbors vj ∈ V . And the non-routable obstructive vertices form a obstacle set , denoted as O ⊂ V . In circuit routing , a net N = { vn1 , vn2 , ... } ⊂ V is a set of vertices that need to be connected and a multi-vertex net can be decomposed into multiple two-vertex nets via a minimum spanning tree ( MST ) or a rectilinear Steiner tree ( RST ) ( de Vincente et al. , 1998 ; Hu & Sapatnekar , 2001 ) . Following He & Bao ( 2020 ) , we simplify this problem by letting each net only contain two vertices , i.e , a two-vertex net is defined as N = { vs , ve } with a start vertex vs and an end vertex ve . The net should be connected by a path P = [ v1 , v2 , ... , vn ] with vs = v1 , ve = vn and P ∩ O = ∅ . We use |P | to represent the length of the path P . Given a set N of nets { Ni } , we need to find a set P of paths { Pi } that connect these nets . A reasonable routing plan for a given set of nets is that all the paths do not share any vertex . In some cases , the routing problem do not have a solution because such a non-intersecting set P does not exist . In circuit routing , the routed paths should satisfy some constraints or achieve some goals . In this paper , we choose the most commonly used goal , i.e. , minimize the total length of paths . Given a solution P , the total length of paths is defined as L = ∑ P∈P |P | . 3.2 EVOLUTION STRATEGIES . Evolution Strategies ( ES ) is a class of black box optimization methods ( Rechenberg , 1994 ; Hansen & Ostermeier , 2001 ) . Recently , an evolution strategy variant , referred as the OpenAI-ES ( Salimans et al. , 2017 ) , has attracted attention because it could rival the performance of modern deep reinforcement learning methods on multiple control tasks . In the OpenAI-ES , there is a reward function , denoted as F ( ψ ) , where ψ is a solution vector , which is sampled from a probability distribution function ψ ∼ pθ ( ψ ) . The goal is to maximize the expected value J ( θ ) , which is defined as : max θ J ( θ ) = max θ Eψ∼pθ [ F ( ψ ) ] = max θ ∫ F ( ψ ) pθ ( ψ ) dψ . ( 1 ) A straightforward way to achieve this is to use the gradient ascent : θ ← θ + α∇θJ ( θ ) , ( 2 ) where α is the learning rate . In the OpenAI-ES , a score function estimator is used to calculate the gradient , which is similar to REINFORCE ( Williams , 1992 ) : ∇θJ ( θ ) = ∇θEψ∼pθ [ F ( ψ ) ] = Eψ∼pθ [ F ( ψ ) ∇θ log pθ ( ψ ) ] . ( 3 ) Usually , pθ is an isotropic multivariate Gaussian with mean θ and fixed covariance σ2I . And the expected value can be rewritten as Eψ∼pθ [ F ( ψ ) ] = E ∼N ( 0 , I ) [ F ( θ+σ ) ] . Thus the gradient estimator changes to : ∇θJ ( θ ) = ∇θE ∼N ( 0 , I ) [ F ( θ + σ ) ] = 1 σ E ∼N ( 0 , I ) [ F ( θ + σ ) ] , ( 4 ) where is sampled from a standard normal distribution . Once we form an objective function J ( θ ) , the gradient ∇θJ ( θ ) can be approximated via Equation 4 and parameters θ can be updated via Equation 2 .

Summary: The paper considers the problem of determining efficient routes for connecting pairs of source, destination points over a circuit board. Due to physical constraints no two routes may intersect, and the routes may not pass through obstacles if there are any. For efficient performance it is desirable that the paths are as short as possible; as such the authors consider minimizing the sum length of all the paths as the objective. The paper proposes a ranking cost algorithm that combines A* search with an evolution optimization technique for learning efficient routes. Experimental evaluations show the proposed approach outperforming an RL based approach.