New modeling and optimization techniques for the global routing problem
01 Jan 2003-
TL;DR: A global routing technique that combines wire length, congestion estimation in the path of the wires and the number of vias in the trees is proposed and investigated and results shown to reduce the congestion and wire length of the representative large-scale test circuits.
Abstract: Modern integrated circuits can contain millions of elements making their design an overwhelming task. The design procedure is therefore divided into a sequence of design steps. Circuit layout is the design step in which a physical realization of a circuit is obtained from its functional description. Global routing is one of the key subproblems of the circuit layout which involves finding an approximate path for the wires connecting the elements of the circuit while minimizing the total interconnecting wire length. The global routing problem is NP-hard, therefore, heuristics capable of producing high quality routes with little computational effort are required as Integrated Circuits (IC) increase in size.
In this thesis, a global routing technique that combines wire length, congestion estimation in the path of the wires and the number of vias in the trees is proposed and investigated. In the first stage of the global routing a good set of routes is constructed for each net. This set is produced based on minimizing the wire length and an approximate estimation of congestion. The global routing problem is then formulated as an Integer Linear Programming (ILP) problem with the objective of minimizing the wire length. Congestion estimation and number of vias are incorporated as penalty functions in the objective function. Two models are developed for the global routing problem. The first model focuses only on minimizing wire length while the second model considers both minimization of wire length and maximization of channel capacity.
To solve the ILP model, first a linear relaxation of the model is solved using Interior Point algorithms. Computational efficiency is achieved by applying an eigenvalue based matrix re-ordering technique to the constraint matrix of the global routing formulation. The re-ordering of the constraint matrix of the optimization problem, increases the speed of the Interior Point algorithm. Subsequently, a rounding algorithm is developed to turn the fractional solutions of the Linear Programming (LP) problem to integer solutions. Global routing results produced by our heuristic are shown to reduce the congestion and wire length of the representative large-scale test circuits.
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TL;DR: Experimental results obtained show that the proposed combined WVEM (wirelength, via, edge capacity) model can optimize several global routing objectives simultaneously and effectively and reduce the CPU time by about 66% on average for edge capacity model (ECM).
Abstract: The use of integrated circuits in high-performance computing, telecommunications, and consumer electronics has been growing at a very fast pace. The level of integration as measured by the number of logic gates in a chip has been steadily rising due to the rapid progress in processing and interconnect technology. The interconnect delay in VLSI circuits has become a critical determiner of circuit performance. As a result, circuit layout is starting to play a more important role in today’s chip designs. Global routing is one of the key sub-problems of circuit layout which involves finding an approximate path for the wires connecting the elements of the circuit without violating resource constraints. In this paper, several integer programming (ILP) based global routing models are fully investigated and explored. The resulting ILP problem is relaxed and solved as a linear programming (LP) problem followed by a rounding heuristic to obtain an integer solution. Experimental results obtained show that the proposed combined WVEM (wirelength, via, edge capacity) model can optimize several global routing objectives simultaneously and effectively. In addition, several hierarchical methods are combined with the proposed flat ILP based global router to reduce the CPU time by about 66% on average for edge capacity model (ECM).
20 citations
19 Dec 2005
TL;DR: This paper proposes new and realistic models for both problems and develops asymptotic approximation algorithms depending on the best known approximation ratio for the minimum Steiner tree problem, which are the first known theoretical approximation bound results for these problems.
Abstract: In this paper, we study the global routing problem in VLSI design and the multicast routing problem in communication networks. We first propose new and realistic models for both problems. Both problems are NP-hard. We present the integer programming formulation of both problems and solve the linear programming (LP) relaxations approximately by the fast approximation algorithms for min-max resource-sharing problems in [10]. For the global routing problem, we investigate particular properties of lattice graphs and propose a combinatorial technique to overcome the hardness due to the bend-dependent vertex cost. Finally we develop asymptotic approximation algorithms for both problems depending on the best known approximation ratio for the minimum Steiner tree problem. They are the first known theoretical approximation bound results for these problems.
14 citations
Cites background or methods from "New modeling and optimization techn..."
...For the global routing problem, our model generalizes the previous models developed in [3, 4, 42, 47]....
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...Based on the above traditional models, two new models are proposed in [3, 4]....
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...More details of the global routing problem in VLSI design can be found in [3]....
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...Three traditional models of the global routing by integer programming are listed in [3]....
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TL;DR: These are the first known theoretical approximation bound results for the problems of minimizing the total costs (including both the edge and the bend costs) while spanning all given subsets of vertices.
14 citations
Cites background or methods from "New modeling and optimization techn..."
...Three traditional models of the global routing by integer programming are listed in [3]....
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...For the global routing problem, we first present a new model Section 2 generalizing the previous models developed in [3,4,35,41]....
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01 Jan 2010
TL;DR: This paper presents a polynomial time algorithm for the global routing problem based on integer programming formulation with a theoretical approximation bound that ensures that all routing demands are satisfled concurrently, and the overall cost is approximately minimized.
Abstract: Global routing in VLSI (very large scale integration) design is one of the most challenging discrete optimization problems in computational theory and practice. In this paper, we present a polynomial time algorithm for the global routing problem based on integer programming formulation with a theoretical approximation bound. The algorithm ensures that all routing demands are satisfled concurrently, and the overall cost is approximately minimized. We provide both serial and parallel implementation as well as develop several heuristics used to improve the quality of the solution and reduce running time. We provide computational results on two sets of well-known benchmarks and show that, with a certain set of heuristics, our new algorithms perform extremely well compared with other integer-programming models.
7 citations
Cites methods from "New modeling and optimization techn..."
...Additionally, we will provide a set of heuristics which improve the quality of the approximate solutions, as well as reduce the time taken to obtain them....
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27 Jun 2008
TL;DR: In this paper, it is proved that GCTR is-approximable if i¾?i¼? i½?+ βi¾?
Abstract: In this paper, we introduce the generalized capacitated tree-routing problem(GCTR), which is described as follows Given a connected graph G= (V,E) with a sink si¾? Vand a set Mi¾? Vi¾? {s} of terminals with a nonnegative demand q(v), vi¾? M, we wish to find a collection ${\mathcal T}=\{T_{1},T_{2},\ldots,T_{\ell}\}$ of trees rooted at sto send all the demands to s, where the total demand collected by each tree T i is bounded from above by a demand capacity i¾?> 0 Let i¾?> 0 denote a bulk capacity of an edge, and each edge ei¾? Ehas an installation cost w(e) i¾? 0 per bulk capacity; each edge eis allowed to have capacity ki¾?for any integer k, which installation incurs cost kw(e) To establish a tree routing T i , each edge econtained in T i requires i¾?+ βqi¾? amount of capacity for the total demand qi¾? that passes through edge ealong T i and prescribed constants i¾?,βi¾? 0, where i¾?means a fixed amount used to separate the inside of the routing T i from the outside while term βqi¾? means the net capacity proportional to qi¾? The objective of GCTR is to find a collection ${\mathcal T}$ of trees that minimizes the total installation cost of edges Then GCTR is a new generalization of the several known multicast problems in networks with edge/demand capacities In this paper, we prove that GCTR is $(2[ \lambda/(\alpha+\beta \kappa)] /\lfloor \lambda/(\alpha+\beta \kappa)\rfloor +\rho_{\mbox{\tiny{\sc ST}}})$-approximable if i¾?i¾? i¾?+ βi¾?holds, where $\rho_{\mbox{\tiny{\sc ST}}}$ is any approximation ratio achievable for the Steiner tree problem
2 citations
Additional excerpts
...In this model, for each edge e of the underlying network, we assign capacity of λe = α|T ′|+ β ∑Ti∈T ′ ∑ v∈Zi∩DTi (vei ) q(v) on e, where T ′ is the set of trees containing e....
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