CAD of VLSI Systems. d Virendra Singh Indian Institute of Science Bangalore E0-285: CAD of VLSI Systems

Transcription

1 CAD of VLSI Systems Introduction(Contd..) d Virendra Singh Indian Institute of Science Bangalore virendra@computer.org E0-8: CAD of VLSI Systems

2 CAD of VLSI Systems Design of VLSI Systems Complex system Need systematic methodology Algorithms Can be automated Should handle very large designs Aug 7, 00

3 Design Domains (Y - Chart) Behavaioral Domain Structural t Domain System Algorithms Reg. Transfers Logic Transfer functions Processors ALU, RAM,.. Gates, FFs,.. Transistors Transistor Layout Cell Layout Module Layout Floorplan Physical Domain Physical Partitions Gajski, 98 Aug 7, 00 E0-8@SERC

4 Algorithms Mostly intractable problems Exact Algorithms Approximate algorithms Branch and Bound Dynamic Programming Greedy Algorithms Soft computing techniques Genetic Algorithms Ant colony optimization Tabu search etc.. Aug 7, 00

5 Algorithms A Graph G(V, E) is a pair (V, E), where V is a set and E is a relation on V Directed Graph - the edges are ordered pairs of vertices Undirected Graph edges are unordered pairs Aug 7, 00 E0-8@SERC

6 Architectural t Synthesis Computation: Differential Equation Solver xl = x + dx ul = u (*x*u*dx) (*y*dx) c = xl < a Data Flow Graph (DFG): represent operation and data dependencies Aug 7, 00 E0-8@SERC 6

7 Data Flow Graph x u dx y * * * * + u dx * * + < x dx dx y a xl 7 9 u - yl c - ul Aug 7, 00 E0-8@SERC 7

8 Graph Representation a 0 d e b c Aug 7, 00 E0-8@SERC 8

9 Graph Representation d a e c b Aug 7, 00 E0-8@SERC 9

10 Graph Algorithms Shortest Path Algorithms Longest Path Algorithms Traveling Salesman Problem Maximal Cliques Graph Colouring Vertex Covering Minimum Spanning Tree Aug 7, 00 0

11 Graph Algorithms Mostly intractable problems Approximate algorithms Branch and Bound Dynamic Programming Greedy Algorithms Soft computing techniques Genetic Algorithms Ant colony optimization Tabu search etc.. Aug 7, 00

12 Graph Algorithms Shortest Path algorithms Dijkstra s algorithm Greedy algorithm Make local decision greedily 0 00 Gives shortest path from a source node Aug 7, 00 E0-8@SERC

13 Dijkstra s Algorithms Iter. S V-S w D[] D[] D[] D[] Init {} {} {,,,} {,} {,,} {,,} {,} {,,,} {} {,,,,} Φ Aug 7, 00 E0-8@SERC

14 Dijkstra s Algorithms Begin S = {} For I = to n do D[i] = C [,i] initialize For I = to n- do begin Choose a vertex w in V-S s.t. D[w] is minimum Add w to S For each vertex v in V-S do D[v] = Min{D[v], D[w]+C[w,v]} end Aug 7, 00 E0-8@SERC

15 Floyd s Algorithms All Pair Shortest Path algorithms Floyd s algorithm Make local decision and refine it later Gives shortest path for all pairs 8 Aug 7, 00 E0-8@SERC

16 Floyd s Algorithms A 0 [i,j] A [i,j] Aug 7, 00 E0-8@SERC 6

17 Floyd s Algorithms A [i,j] A [i,j] Aug 7, 00 E0-8@SERC 7

18 Floyd s Algorithms Begin S = {} For I = to n do Forj=tondo A[I,j] = C [i,j] initialize For I = to n do A[Ii]=0 A[I,i] For k = to n- do begin For i = to n do For j = to n do If A[I,k]+A[k,j] < A[I,j] then end A[I,j] = A[I,k]+A[k,j] P[I,j] = k 8 Aug 7, 00 E0-8@SERC 8

19 Spanning Tree Free three that t connects all the vertices Cost of a spanning tree is sum of edges Minimum Spanning Tree (MST) Prim s Algorithm Greedy algoritthm Start from an intial node U = {} Grows ST, one edge at a time At each step, it finds a shortest edge (u,v) that connects U and V-U and adds v to V-U from U Aug 7, 00 E0-8@SERC 9

20 Prim s Algorithm Aug 7, 00 E0-8@SERC 0

21 Prim s Algorithm Aug 7, 00 E0-8@SERC 6 6 6

22 Kruskal s Algorithm Start with a graph t = (V,Φ) only vertices Each vertex is connected component in itself As algorithm progresses, Have collection of connected components For each component select an edge for ST To build progressively larger connected component Examine edges for E in order of increasing cost If the edge connects two vertices in two different connected component, tthen add edge to T Aug 7, 00 E0-8@SERC

23 Kruskal s Algorithm Aug 7, 00 E0-8@SERC

24 Vertex Covering Problem Vertex Covering of an undirected graph G is a subset of the vertices s.t. each edge in Eh has at tleast one edge in that tsubset Heuristic Select vertex with largest degree Deletion of a vertex corresponds to the removal of vertex itself and all edges incident to it Aug 7, 00 E0-8@SERC

25 Vertex Covering Problem Aug 7, 00

26 Graph Colouring Problem Graph colouring problem of an undirected d graph G is a labeling of vertices such that no edge in E has two end-points with the same label Search for a vertex colouring with minimum number of colours Most algorithms are based on sequential scan of vertex set where vertices are coloured one at a time Aug 7, 00 E0-8@SERC 6

27 THANK YOU Aug 7, 00 7