An Exact Algorithm for the Statistical Shortest Path Problem

Transcription

1 An Exact Algorithm for the Statistical Shortest Path Problem Liang Deng and Martin D. F. Wong Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign

2 Outline Motivation Statistical shortest path (SSP) problem Our exact algorithm for SSP problem Applications Maze Routing Timing Analysis Buffer Insertion

3 Why Statistical Methods? Intra-die variations become dominant Corner-based design flow leads to over design or yield loss Statistical methods are needed not only in simulation but also in design tools. Temperature Variation in Cell Processor Dac C. Pham, et al. ISSCC05

4 Variations, Performance and Yield Variation sources Process variations Gate length variation Geometric variation in interconnection wires Temperature variations Supply voltage variations Statistical models for circuits have been proposed New algorithm considering variations are needed for performance/yield optimization

5 Statistical Model for Variations Use mean µ and variance σ 2 to capture the random property of variations Exact for Gaussian, uniform, binominal, exponential distributions and etc. Good approximation for arbitrary random variables

6 Statistical Model for Variations Mean and variance are additive, but not the standard deviation σ Recall the Chebyshev s Inequality: P( X µ kσ ) > 1 The cost function µ+kσ is important to yield optimization σ not additive presents difficulties in solving statistical graph problems 1 k 2

7 Statistical Shortest Path Problem Edge weights are random variables a (10,4) t To find a path with minimum µ+φ(σ 2 ) value Existing methods cannot solve this problem s (14,5) (10,2) (14,24) b (8,14) (18,1) c (2,1) Edge weight: (mean, variance)

8 From Deterministic to Statistical Deterministic Edge weight w w is additive Path weight Σw Minimize Σw Statistical Edge weight ( µ,σ 2 ) µ,σ 2 are additive Path weight ( µ P, σ P2 ) Minimize µ P +Φ(σ P2 )

9 Statistical Shortest Path Problem Given a directed graph G Not necessarily a DAG Find a path p from source vertex s to sink vertex t such that µ P +Φ(σ P2 ) is minimized Path weight of p is a random variable with mean µ P and variance σ P 2

10 Practical Observations for EDA problems µ, σ 2 are additive For yield optimization problems σ 2 is bounded σ 2 can be discretized without introducing much error We may assume the variance σ 2 of path weight are integers upper bounded by B, i.e., σ 2 B

11 Algorithm for Solving SSP Problem Vertex splitting for µ, σ 2 Graph expansion to generate a new graph G G has real numbers as its edge weights Each vertex u in G is split into a set of vertices in G : {u 1,u 2,, u B }

12 Graph Expansion Source Node s (μ, σ 2 ) a From source to other vertices Only expand vertex a Each new vertex a i corresponds to a with variance i Edge weight is µ s μ a 1 a 2 a σ 2 a B

13 Graph Expansion Internal Nodes u (μ, σ 2 ) v Assuming vertex u is already split Its neighbor v will be also split Edges are connected according to σ 2 of path weight Edge weight are µ u 1 u 2 u 3 u B-σ 2 μ μ μ μ v 1 2 v σ +1 2 v σ +2 v 2 σ +3 u B v B

14 Graph Expansion Sink Node t Original sink node is already split according to previous steps Add a super sink node t Edge weight for edge ti to t is Φ(i) Note that any path from source to ti has variance equals to i t 1 t i t B Φ(i) t'

15 From Arbitrary Graph to DAG There will be no loop in expanded graph since σ 2 > 0 a 1 a i a σ 2 3 +i?? μ 1 b 1 b σ 2 +i 1 a (μ 3, σ 3 2 ) (μ 1, σ 1 2 ) b (μ 2, σ 2 2 ) a B μ 3 c 1 μ 2 b B c 2 c σ 2 +i c B

16 SSP Algorithm The expanded graph G is a DAG Shortest path in G can be found by existing deterministic shortest path algorithms for DAG This path corresponds to a path in G that minimizes µ P +Φ(σ P2 ) Time complexity is O(B(V+E))

17 Improvement Only split a vertex whenever it is necessary; don t split all vertices Remove redundant vertices during splitting If paths have same variance, then the one with larger mean is redundant If Φ(σ P2 ) is a monotonically increasing function, paths with larger mean and variance are redundant

18 Example (14,5) b (10,4) e 14 b e' a (10,2) (14,24) c (8,14) (18,1) d (2,1) a c d e Edge weight: (mean, variance) Φ ( x) = 3 x

19 SSP Algorithm Improved Much less vertices are generated 100 vertices needed for previous example with original approach 10 vertices used with improved algorithm Expand graph simultaneously with searching the shortest path Much faster with less memory requirement

20 EDA applications Maze Routing Timing Analysis Buffer Insertion

21 Maze Routing Timing-driven maze routing Process Variations Systematic variations Random variations Temperature variations Find the shortest path to improve the performance

22 Maze Routing Hot Cell Cold Cell No Variations considered Variations considered

23 Timing Analysis Find the longest delay path considering intra-die variations Large circuits with several logic levels Gaussian distribution for the path delay µ P +3σ P is used to measure the timing-yield Our algorithm can also find the (path) candidates with longest delay

24 Timing Analysis ISCAS benchmarks Cell delays are not necessarily Gaussian 40X 1000X runtime improvement over Monte Carlo simulation Very little error compare to Monte Carlo method

25 Buffer Insertion Buffer insertion in 2-pin net can be formulated into shortest path problem With variations from both devices and interconnections, it should be formulated into statistical shortest path problem Our algorithm can solve this buffer insertion with variations consideration

26 Buffer Insertion Graph based approach Formulated as a shortest path problem Drive Buffer Load Shortest Path

27 Conclusion Exact algorithm to solve the statistical shortest path problem Arbitrary graph, arbitrary cost function Φ Efficient implementation Can be used in varieties of applications in nanometer design