In this paper, we introduce and study the Rectangle Escape Problem (REP), which is motivated by PCB bus escape routing. Given a rectangular region R and a set S of rectangles within R, the REP is to choose a direction for each rectangle to escape to the boundary of R, such that the resultant maximum density over R is minimized. We prove that the REP is NP-Complete, and show that it can be formulated as an Integer Linear Program (ILP). A provably good approximation algorithm for the REP is developed by applying Linear Programming (LP) relaxation and a special rounding technique to the ILP. This approximation algorithm is also shown to work for a more general version of REP with weights (weighted REP). In addition, an iterative refinement procedure is proposed as a postprocessing step to further improve the results. Our approach is tested on a set of industrial PCB bus escape routing problems. Experimental results show that the optimal solution can be obtained within 3 seconds for each of the test cases.

In PCB designs, pin positions greatly affect routability of the design. State-of-the-art pin assignment algorithms are guided by simple (heuristic) metrics to estimate routability and thus have no guarantee to obtain a routable solution. In this paper, we present a novel approach to obtain a pin assignment solution that guarantees routability. We show that the problem of simultaneous pin assignment and escape routing can be solved optimally in polynomial time. We then focus on the pin assignment and escape routing for the terminals in a bus, and present algorithmic enhancements as well as discuss the tradeoffs between single-layer and multi-layer implementations. We tested our approach on a state-of-the-art industrial board with 80 buses (over 7000 nets). The pin assignment and escape routing solutions for all the 80 buses are successfully obtainted in less than 5 minutes of CPU time.

The maximum disjoint subset (MDS) of rectangles is a subset of non-overlapping rectangles with the maximum total weight. The problem of finding the MDS of general rectangles has been proven to be NP-complete in [6]. In this paper, we focus on the problem of finding the MDS of boundary rectangles, which is an open problem and is closely related to some difficult problems in PCB routing. We propose a polynomial time algorithm to optimally solve the MDS problem of boundary rectangles. Then we show that this algorithm can be applied to find the optimal solution of the bus escape routing problem.

Since no commercial PCB routing tools can solve the routing problem for today's complex PCBs, these circuit boards have to be routed manually, taking about 2 months of time per board. Bus planning is one of the most time-consuming steps of PCB routing. It consists of assigning buses to multiple layers of the PCB and routing them in a planar fashion on each layer. Routing congestion between on-board components and the min-max length bounds of the buses must also be considered during routing. In this paper, we present the first automatic bus planner. We tested our system on a state-of-the-art industrial circuit board with over 7000 nets and 12 signal layers. All the nets on this board were already manually routed. Our bus planner is able to achieve 100% routing completion using the layer assignment extracted from manual design. For simultaneous layer assignment and bus routing, we are able to successfully route 98.5% of the nets. The remaining 1.5% can be routed either manually or by using vias. The runtime of our bus planner is less than 3 hours on a 3 Ghz workstation.

Escape routing is a critical problem in PCB design. In IC-CAD'07, a layer assignment algorithm was proposed for escape routing of buses. The algorithm is optimal for single layer design in the sense that it determines if a set of buses can all be escaped on one layer. If they cannot, the algorithm is able to select a maximum subset of the buses that can be escaped on one layer. This, in turn, leads to a heuristic for the layer assignment problem with multiple layers, which is to repeatedly assign a maximum subset of the unassigned buses to a new layer. In this work, we present an algorithm that solves the multi-layer layer assignment problem optimally. Our algorithm guarantees to produce a layer assignment with minimum number of layers. We applied our algorithm on industrial data and obtained encouraging results.

The PCB routing problem has become so difficult that no commercial CAD software can provide an automatic solution for high-end boards. Existing algorithms for escape routing, an important step in PCB routing, are net-centric. Directly applying these algorithms will result in mixing nets of different buses together. But in practice, it is preferred to bundle together nets in a bus. Thus the bus-centric escape routing problem can be naturally divided into two subproblems: (1) finding a subset of buses that can be routed on the same layer without net mixings and crossings, which we refer to as the bus sequencing problem, and (2) finding the escape routing solutions for each chosen bus, which can be solved by a net-centric escape router. In this paper, we solve the bus sequencing problem. We introduce a new optimization problem called the longest common interval Sequence (LCIS) problem and model the bus sequencing problem as an LCIS problem. By using dynamic programming and balanced search tree data structure, we present an LCIS algorithm which can find an optimal solution in O(n log n) time. We also show that O(n log n) is a lower-bound for this problem and thus the time complexity of our algorithm is also the best possible.