Sheqin Dong, Song Chen, Xianlong Hong EDA Lab., Tsinghua Univ. Beijing Solution Space Smoothing Method and Its Application in VLSI Floorplanning Sheqin Dong, Song Chen, Xianlong Hong EDA Lab., Tsinghua Univ. Beijing 2018年11月28日

Content Principle of Solution Space Smoothing (SSS) Applying SSS Strategy to VLSI Floorplanning Problem Smoothing the solution space of Floorplan Experimental Results Conclusion 2018年11月28日

Principle of Solution Space Smoothing Local Minimum Points make a search problem hard The less the number of local minimum points, the more effective a local search algorithm is Search Space Smoothing technique limits the number of local minimum points in the search space In 1994, Gu Jun et al. proposed the method of Search Space Smoothing technique, which is special technique of multispace search developed in recent years. 2018年11月28日

Principle of Solution Space Smoothing (Con’t) Informally the basic idea of the method can explained as follows. Assume there is a solution space with many local minimum points, where a solution point could be easily trapped. We use a smoothed search space to approximate the original search space. After Search Space smoothing, some local minimum points are temporarily ‘filled’ and they will no longer cause any ‘trapping’ problems. So as illustrated in the figure, the number of local minimum points is ‘reduced. 2018年11月28日

Principle of Solution Space Smoothing (con’t) To apply search space smoothing techniques to local search, one is often faced with a contradictory situation. A weaker smoothing operation, however, results in less reduction in the number of local minimum points in the original search space. In order to increase the chance of finding the global minimum points in the smoothed search space, we expect a strong smoothing operation that produces a flatter search space, but we may lose some heuristic guidance information. To apply search space smoothing techniques to local search, one is often faced with a contradictory situation. That is, if one applies a weaker smoothing operation, the topological structure of the smoothed search space is similar to the original one. The heuristic guidance information of the original search space is thus strong. A weaker smoothing operation, however, results in less reduction in the number of local minimum points in the original search space. In order to increase the chance of finding the global minimum points in the smoothed search space, we expect a strong smoothing operation that produces a flatter search space, but we may lose some heuristic guidance information. This contradictory situation can be resolved using a series of search space smoothing operations. By altering the shape of the objective function in a parameter space, i.e., the alfa space, a series of smoothed search spaces with their structures varying from a flatter search space to the originial search space, as illustrated in Fig.6, is generated. Each (upper) search space is a further smoothing of the lower search space. The solution of a smoothed, flatter search space are used to guide the search of those in the more rugged search spaces. 2018年11月28日

Principle of Solution Space Smoothing (Con’t) Advantages. Smoothing makes the search of the global minimum solution point in a smoothed search space easier. After a smoothing operation, the number of local minimum points in a smoothed search space is “reduced”. Thus , for a search process, the probability of being trapped in a local minimum point is minimized and the chance of finding a global optimum point is increased. Secondly, since a smoothed search space has qualitatively accumulated the topological structure information of the original search space, a smoothing processing facilitates the search of the global minimum point in a original search space. Using an appropriate smoothing scheme, e.g., a gradually approximated smoothing scheme, the global minimum point in the smoothed search space could be set very close to the global minimum point in the original search space. If we use the global minimum point in the smoothed search space as the initial starting point in the original search space, then the probability of finding the global minimum point in the original search space could be increased considerably. 2018年11月28日

Applying the SSS strategy to VLSI floorplanning Problem Previous Work Smoothing the search space of Floorplanning Search Strategy Experimental results 2018年11月28日

Problem A set of rectangular blocks M={M1, M2, …, Mn} of A set of nets specifying the interconnections between pins of blocks and a set of pads (external pins) are also given. A placement P={Mi (xi, yi) , 1 < = i < = n} is an assignment of coordinates to the lower left corners of n rectangular blocks such that there is no two rectangular blocks overlapping. Given a set of circuit building blocks, the placement problem can be defined as follows. Each Mi is defined by a tuple (hi , wi) , where hi and wi are the height and the width of block Mi, respectively . The objective of the placement is to find an assignment so that the chip area and interconnection wire-length between blocks are minimized while satisfying the given constraints, if any. 2018年11月28日

Previous Work Application of random optimization algorithm Simple Rectangle-Packing problem is NP-hard Simulated Annealing – Not Stable - , and etc. Floorplan/Placement Representation Sequence Pair, Bounded Slice-line Grid, O-tree, B*-tree, Corner Block List, and etc. As you know, though the simplest rectangle-packing problem is NP-hard. Consequently, the approximate algorithms of random optimization are introduced to solve VLSI floorplan/placement problem. Simulated Annealing gets wide applications. And also many coding scheme for the floorplan solution spaces have been proposed, such as Sequence Pair, Bounded Slice-line Grid, Corner Block List, and etc. However, it is unstable and far away from the practical use. 2018年11月28日

Smoothing the search space of Floorplanning Intuitively, without consideration of connections among blocks, a placement of blocks with the same dimensions will have the most flat solution space. Therefore, we think that the search space will be more flatter if the blocks have more similar dimensions. The smoothing of the solution space of Floorplan/Placement is achieved by change the dimensions of blocks. A Serial of flattened solution space will be searched before the search of original solution space. 2018年11月28日

Smoothing Functions For calculation of blocks size 2018年11月28日

Variations of block dimensions Initial Dimensions Original Dimensions 2018年11月28日

Search Strategy applied on VLSI Floorplanning SSS() Begin /* Initializtion*/ P = get_a_placement_instance(); α = α0； /*Search*/ While(α>=1) do /*Generate a Simplified Placement Instance*/ For i:=1 to n do w(i) = compute_wi(α); h(i) = compute_hi(α); P = Placement_local_search(w, P); α = f(α ) End; 2018年11月28日

Comparison between the SSS and Simulated Annealing (1) 2018年11月28日

Comparison between the SSS and Simulated Annealing (2) 2018年11月28日

Comparison among different smoothing functions 2018年11月28日

Conclusion We apply the optimization algorithm of SSS to the problem of VLSI Floorplan/placement by means of changing the dimensions of blocks size to smoothen the solution space of floorplanning/placement Experimental shows that Solution Space Smoothing is a promising optimization strategy because of its stableness. The SSS method is very heuristic. The further works should be contributed to the theory aspect. 2018年11月28日

Thanks for your attenstion! 2018年11月28日