3 research outputs found
Maximum Weight Matching via Max-Product Belief Propagation
Max-product "belief propagation" is an iterative, local, message-passing
algorithm for finding the maximum a posteriori (MAP) assignment of a discrete
probability distribution specified by a graphical model. Despite the
spectacular success of the algorithm in many application areas such as
iterative decoding, computer vision and combinatorial optimization which
involve graphs with many cycles, theoretical results about both correctness and
convergence of the algorithm are known in few cases (Weiss-Freeman Wainwright,
Yeddidia-Weiss-Freeman, Richardson-Urbanke}.
In this paper we consider the problem of finding the Maximum Weight Matching
(MWM) in a weighted complete bipartite graph. We define a probability
distribution on the bipartite graph whose MAP assignment corresponds to the
MWM. We use the max-product algorithm for finding the MAP of this distribution
or equivalently, the MWM on the bipartite graph. Even though the underlying
bipartite graph has many short cycles, we find that surprisingly, the
max-product algorithm always converges to the correct MAP assignment as long as
the MAP assignment is unique. We provide a bound on the number of iterations
required by the algorithm and evaluate the computational cost of the algorithm.
We find that for a graph of size , the computational cost of the algorithm
scales as , which is the same as the computational cost of the best
known algorithm. Finally, we establish the precise relation between the
max-product algorithm and the celebrated {\em auction} algorithm proposed by
Bertsekas. This suggests possible connections between dual algorithm and
max-product algorithm for discrete optimization problems.Comment: In the proceedings of the 2005 IEEE International Symposium on
Information Theor
3D Global Router: a Study to Optimize Congestion, Wirelength and Via for Circuit Layout
The increasing size of integrated circuits and aggressive shrinking process feature size for IC manufacturing process poses signicant challenges on traditional physical design problems. Various design rules signicantly complicate the physical design problems and large problem size abides nothing but extremely e cient techniques. Leading physical design tools have to be powerful enough to handle complex design demands and be nimble enough to waste no runtime. This thesis studies the challenges faced by global routing problem, one of the traditional physical design problems that needs to be pushed to its new limit. This work proposes three e ective tools to tackle congestion, wire and via optimization in global routing process, from three di erent aspects.
The number of vias generated during the global routing stage is a critical factor for the yield of integrated circuits. However, most global routers only approach the problem by charging a cost for vias in the maze routing cost function. The first work of this thesis, FastRoute 4.0 presents a global router that addresses the via number optimization problem throughout the entire global routing ow. It introduces the via aware Steiner tree generation, 3-bend routing and layer assignment with careful ordering to reduce via count. The integration of these three techniques with existing academic global routers achieves signicant reduction in via count without any sacrice in runtime.
Despite of the recent development for popular rip-up and reroute framework, the congestion elimination process remains arbitrary and requires signicant tuning. Global routing has congestion elimination as the first and foremost priority and congestion issue becomes increasingly severe due to timing requirements, design for manufacturability. The second work of this thesis, an auction algorithm based pre-processing framework (APF) for global routing focuses on how to eliminate congestion e ectively. In order to achieve more consistent congestion elimination, the framework uses auction based detour techniques to alleviate the impacts of greedy sequential manner of maze routing, which remains as a major drawback in the most popular global routing framework. In the framework, APF first identies the most congested global routing locations by an interval over ow lower bound technique. Then APF uses auction based detour algorithm to compute which nets to detour and where to detour. The framework can be applied to any global routers and would help them to achieve signicant improvement in both solution quality and runtime.
The third work in this thesis combines the advantage of the two framework used to minimize via usage in global routing: 3D routers with good solution quality and e cient 2D routers with layer assignment process. It results in a new multi-level 3D global router called MGR (multi-level global router) that combines the advantage of both kinds. MGR resorts to an e cient multi-level framework to reroute nets in the congested region on the 3D grid graph. Routing on the coarsened grid graph speeds up the global router while 3D routing introduces less vias. The powerful multi-level rerouting framework wraps three innovative routing techniques together: an adaptive resource reservation technique in coarsening process, a new 3-terminal maze routing algorithm and a network flow based solution propagation method in uncoarsening process. As a result, MGR can achieve the solution quality close to 3D routers with comparable runtime of 2D routers
Statistical Physics of Hard Optimization Problems
Optimization is fundamental in many areas of science, from computer science
and information theory to engineering and statistical physics, as well as to
biology or social sciences. It typically involves a large number of variables
and a cost function depending on these variables. Optimization problems in the
NP-complete class are particularly difficult, it is believed that the number of
operations required to minimize the cost function is in the most difficult
cases exponential in the system size. However, even in an NP-complete problem
the practically arising instances might, in fact, be easy to solve. The
principal question we address in this thesis is: How to recognize if an
NP-complete constraint satisfaction problem is typically hard and what are the
main reasons for this? We adopt approaches from the statistical physics of
disordered systems, in particular the cavity method developed originally to
describe glassy systems. We describe new properties of the space of solutions
in two of the most studied constraint satisfaction problems - random
satisfiability and random graph coloring. We suggest a relation between the
existence of the so-called frozen variables and the algorithmic hardness of a
problem. Based on these insights, we introduce a new class of problems which we
named "locked" constraint satisfaction, where the statistical description is
easily solvable, but from the algorithmic point of view they are even more
challenging than the canonical satisfiability.Comment: PhD thesi