Fall 2003EE VLSI Design Automation I 37 EE 5301 – VLSI Design Automation I Kia Bazargan University of Minnesota Part II: Algorithms

Fall 2003EE VLSI Design Automation I 38 References and Copyright Textbooks referred (none required)  [Mic94] G. De Micheli “Synthesis and Optimization of Digital Circuits” McGraw-Hill,  [CLR90] T. H. Cormen, C. E. Leiserson, R. L. Rivest “Introduction to Algorithms” MIT Press,  [Sar96] M. Sarrafzadeh, C. K. Wong “An Introduction to VLSI Physical Design” McGraw-Hill,  [She99] N. Sherwani “Algorithms For VLSI Physical Design Automation” Kluwer Academic Publishers, 3 rd edition, 1999.

Fall 2003EE VLSI Design Automation I 39 References and Copyright (cont.) Slides used: (Modified by Kia when necessary)  [©Sarrafzadeh] © Majid Sarrafzadeh, 2001; Department of Computer Science, UCLA  [©Sherwani] © Naveed A. Sherwani, 1992 (companion slides to [She99])  [©Keutzer] © Kurt Keutzer, Dept. of EECS, UC-Berekeley  [©Gupta] © Rajesh Gupta UC-Irvine

Fall 2003EE VLSI Design Automation I 40 Combinatorial Optimization Problems with discrete variables  Examples: oSORTING: given N integer numbers, write them in increasing order oKNAPSACK: given a bag of size S, and items {(s 1, w 1 ), (s 2, w 2 ),..., (s n, w n )}, where s i and w i are the size and value of item i respectively, find a subset of items with maximum overall value that fit in the bag.  A problem vs. problem instance Problem complexity:  Measures of complexity  Complexity cases: average case, worst case  Tractability ­ solvable in polynomial time [©Gupta]

Fall 2003EE VLSI Design Automation I 41 Algorithm An algorithm defines a procedure for solving a computational problem  Examples: oQuick sort, bubble sort, insertion sort, heap sort oDynamic programming method for the knapsack problem Definition of complexity  Run time on deterministic, sequential machines  Based on resources needed to implement the algorithm oNeeds a cost model: memory, hardware/gates, communication bandwidth, etc. oExample: RAM model with single processor  running time  # operations [©Gupta]

Fall 2003EE VLSI Design Automation I 42 Algorithm (cont.) Definition of complexity (cont.)  Example: Bubble Sort  Scalability with respect to input size is important oHow does the running time of an algorithm change when the input size doubles? oFunction of input size (n). Examples: n 2 +3n, 2 n, n log n,... oGenerally, large input sizes are of interest (n > 1,000 or even n > 1,000,000) oWhat if I use a better compiler? What if I run the algorithm on a machine that is 10x faster? [©Gupta] for (j=1 ; j< N; j++) { for (i=j; i < N-1; i++) { if (a[i] > a[i+1]) { hold = a[i]; a[i] = a[i+1]; a[i+1] = hold; } for (j=1 ; j< N; j++) { for (i=j; i < N-1; i++) { if (a[i] > a[i+1]) { hold = a[i]; a[i] = a[i+1]; a[i+1] = hold; }

Fall 2003EE VLSI Design Automation I 43 Function Growth Examples

Fall 2003EE VLSI Design Automation I 44 Asymptotic Notions Idea:  A notion that ignores the “constants” and describes the “trend” of a function for large values of the input Definition  Big-Oh notation f(n) = O ( g(n) ) if constants K and n 0 can be found such that:  n  n 0, f(n)  K. g(n) g is called an “upper bound” for f (f is “of order” g: f will not grow larger than g by more than a constant factor) Examples: 1/3 n 2 = O (n 2 ) (also O(n 3 ) ) 0.02 n n = O (n 2 )

Fall 2003EE VLSI Design Automation I 45 Asymptotic Notions (cont.) Definition (cont.)  Big-Omega notation f(n) =  ( g(n) ) if constants K and n 0 can be found such that:  n  n 0, f(n)  K. g(n) g is called a “lower bound” for f  Big-Theta notation f(n) =  ( g(n) ) if g is both an upper and lower bound for f Describes the growth of a function more accurately than O or  Example: n n   (n 2 ) 4 n =  (n 2 )

Fall 2003EE VLSI Design Automation I 46 Asymptotic Notions (cont.) How to find the order of a function?  Not always easy, esp if you start from an algorithm  Focus on the “dominant” term o4 n n 2 + log n  O(n 3 ) on + n log(n)  n log (n)  n! = K n > n K > log n > log log n > K  n > log n, n log n > n, n! > n 10. What do asymptotic notations mean in practice?  If algorithm A has “time complexity” O(n 2 ) and algorithm B has time complexity O(n log n), then algorithm B is better  If problem P has a lower bound of  (n log n), then there is NO WAY you can find an algorithm that solves the problem in O(n) time.

Fall 2003EE VLSI Design Automation I 47 Problem Tractability Problems are classified into “easier” and “harder” categories  Class P: a polynomial time algorithm is known for the problem (hence, it is a tractable problem)  Class NP (non-deterministic polynomial time): ~ polynomial solution not found yet (probably does not exist)  exact (optimal) solution can be found using an algorithm with exponential time complexity Unfortunately, most CAD problems are NP  Be happy with a “reasonably good” solution [©Gupta]

Fall 2003EE VLSI Design Automation I 48 Algorithm Types Based on quality of solution and computational effort  Deterministic  Probabilistic or randomized  Approximation  Heuristics: local search Problem vs. algorithm complexity [©Gupta]

Fall 2003EE VLSI Design Automation I 49 Deterministic Algorithm Types Algorithms usually used for P problems  Exhaustive search! (aka exponential)  Dynamic programming  Divide & Conquer (aka hierarchical)  Greedy  Mathematical programming  Branch and bound Algorithms usually used for NP problems (not seeking “optimal solution”, but a “good” one)  Greedy (aka heuristic)  Genetic algorithms  Simulated annealing  Restrict the problem to a special case that is in P

Fall 2003EE VLSI Design Automation I 50 Graph Definition Graph: set of “objects” and their “connections” Formal definition:  G = (V, E), V={v 1, v 2,..., v n }, E={e 1, e 2,..., e m }  V: set of vertices (nodes), E: set of edges (links, arcs)  Directed graph: e k = (v i, v j )  Undirected graph: e k ={v i, v j }  Weighted graph: w: E  R, w(e k ) is the “weight” of e k. a c de b a c de b

Fall 2003EE VLSI Design Automation I 51 Graph Representation: Adjacency List acdebacdeb b · c · d · e · a · b·d·c·a·d·a·e·b·b·a·d· b · c · d · e · a · b·d·a·a·b·e·d·

Fall 2003EE VLSI Design Automation I 52 Graph Representation: Adjacency Matrix acdebacdeb b c d e a bcdea b c d e a bcdea

Fall 2003EE VLSI Design Automation I 53 Edge / Vertex Weights in Graphs Edge weights  Usually represent the “cost” of an edge  Examples: oDistance between two cities oWidth of a data bus  Representation oAdjacency matrix: instead of 0/1, keep weight oAdjacency list: keep the weight in the linked list item Node weight  Usually used to enforce some “capacity” constraint  Examples: oThe size of gates in a circuit oThe delay of operations in a “data dependency graph” acdeb a-+*b

Fall 2003EE VLSI Design Automation I 54 Graph Search Algorithms Purpose: to visit all the nodes Algorithms  Depth-first search  Breadth-first search  Topological Examples [©Sherwani] A B G F D E C A B DE F C G B A E GF C D

Fall 2003EE VLSI Design Automation I 55 Depth-First Search Algorithm struct vertex {... int mark; }; dfs ( v ) v.marked  1 print v for each (v, u)  E if (u.mark != 1) // not visited yet? dfs (u) Algorithm DEPTH_FIRST_SEARCH ( V, E ) for each v  V v.marked  0 // not visited yet for each v  V if (v.marked == 0) dfs (v)

Fall 2003EE VLSI Design Automation I 56 Breadth-First Search Algorithm bfs ( v, Q) v.marked  1 for each (v, u)  E if (u.mark != 1) // not visited yet? Q  Q + u Algorithm BREDTH_FIRST_SEARCH ( V, E ) Q  {v 0 } // an arbitrary node while Q != {} v  Q.pop() if (v.marked != 1) print v bfs (v) // explore successors There is something wrong with this code. What is it?

Fall 2003EE VLSI Design Automation I 57 Distance in (non-weighted) Graphs Distance d G (u,v):  Length of a shortest u--v path in G. uv d(u,v)=2 x y d(x,y) = 

Fall 2003EE VLSI Design Automation I 58 Moor’s Breadth-First Search Algorithm Objective:  Find d(u,v) for a given pair (u,v) and a shortest path u--v How does it work?  Do BFS, and assign (w) the first time you visit a node. (w)=depth in BFS. Data structure  Q a queue containing vertices to be visited  (w)length of the shortest path u--w (initially  )   (w)parent node of w on u--w

Fall 2003EE VLSI Design Automation I 59 Moor’s Breadth-First Search Algorithm Algorithm: 1.Initialize o (w)   for w  u o (u)  0 oQ  Q+u 2.If Q  , x  pop(Q) else stop. “no path u--v” 3.  (x,y)  E, oif (y)= ,  (y)  x o (y)  (x)+1 oQ  Q+y 4.If (v)=  return to step 2 5.Follow  (w) pointers from v to u.

Fall 2003EE VLSI Design Automation I 60 Moor’s Breadth-First Search Algorithm u v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v9v9 v 10 Q = u Nodeuv 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 0  

Fall 2003EE VLSI Design Automation I 61 Moor’s Breadth-First Search Algorithm u v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v9v9 v 10 Q = v 1, v 2, v 5 Nodeuv 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v  1   -uu--u-----

Fall 2003EE VLSI Design Automation I 62 Moor’s Breadth-First Search Algorithm u v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v9v9 v 10 Q = v 4, v 3, v 6, v 10 Nodeuv 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v  2  -uuv 2 v 1 uv 5 ---v 5

Fall 2003EE VLSI Design Automation I 63 Moor’s Breadth-First Search Algorithm u v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 v9v9 v 10 Q = v 7 Nodeuv 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v  2  -uuv 2 v 1 uv 5 v 4 --v 5

Fall 2003EE VLSI Design Automation I 64 Notes on Moor’s Algorithm Why the problem of BREADTH_FIRST_SEARCH algorithm does not exist here? Time complexity? Space complexity?

Fall 2003EE VLSI Design Automation I 65 Distance in Weighted Graphs Why a simple BFS doesn’t work any more?  Your locally best solution is not your globally best one  First, v 1 and v 2 will be visited u v1v1 v2v =3  =u =11  =u v 2 should be revisited u v1v1 v2v =3  =u =5  =v 1

Fall 2003EE VLSI Design Automation I 66 Dijkstra’s Algorithm Objective:  Find d(u,v) for all pairs (u,v) (fixed u) and the corresponding shortest paths u--v How does it work?  Start from the source, augment the set of nodes whose shortest path is found.  decrease (w) from  to d(u,v) as you find shorter distances to w.  (w) changed accordingly. Data structure:  S the set of nodes whose d(u,v) is found  (w)current length of the shortest path u--w   (w)current parent node of w on u—w

Fall 2003EE VLSI Design Automation I 67 Dijkstra’s Algorithm O(e) O(v 2 ) Algorithm: 1.Initialize o (v)   for v  u o (u)  0 oS  {u} 2.For each v  S’ s.t. u i v  E, oIf (v) > (u i ) + w(u i v), (v)  (u i ) + w(u i v)  (v)  u i 3.Find m=min{ (v)|v  S’ } and (v j )==m oS  S  {v j } 4.If |S|<|V|, goto step 2

Fall 2003EE VLSI Design Automation I 68 Dijkstra’s Algorithm - why does it work? u0u0 v x Proof by contradiction  Suppose v is the first one being added to S such that (v) > d(u 0,v)  Let x be the first node in S’ which makes the shortest u 0 --v path  (x) contradiction

Fall 2003EE VLSI Design Automation I 69 Minimum Spanning Tree (MST) Tree (usually undirected):  Connected graph with no cycles  |E| = |V| - 1 Spanning tree  Connected subgraph that covers all vertices  If the original graph not tree, graph has several spanning trees Minimum spanning tree  Spanning tree with minimum sum of edge weights (among all spanning trees)  Example: build a railway system to connect N cities, with the smallest total length of the railroad

Fall 2003EE VLSI Design Automation I 70 Minimum Spanning Tree Algorithms Basic idea:  Start from a vertex (or edge), and expand the tree, avoiding loops (i.e., add a “safe” edge)  Pick the minimum weight edge at each step Known algorithms  Prim: start from a vertex, expand the connected tree  Kruskal: start with the min weight edge, add min weight edges while avoiding cycles (build a forest of small trees, merge them)

Fall 2003EE VLSI Design Automation I 71 Prim’s Algorithm for MST Data structure:  Sset of nodes added to the tree so far  S’set of nodes not added to the tree yet  Tthe edges of the MST built so far  (w)current length of the shortest edge (v, w) that connects w to the current tree   (w)potential parent node of w in the final MST (current parent that connects w to the current tree)

Fall 2003EE VLSI Design Automation I 72 Prim’s Algorithm  Initialize S, S’ and T oS  {u 0 }, S’  V \ {u 0 } // u 0 is any vertex oT  { } o  v  S’, (v)    Initialize and  for the vertices adjacent to u 0 oFor each v  S’ s.t. (u 0,v)  E, (v)   ((u 0,v))  (v)  u 0  While (S’ !=  ) oFind u  S’, s.t.  v  S’, (u)  (v) oS  S  {u}, S’  S’ \ {u}, T  T  { (  (u), u) } oFor each v s.t. (u, v)  E, If (v) >  ((u,v)) then (v)   ((u,v))  (v)  u

Fall 2003EE VLSI Design Automation I 73 Prim’s Algorithm Example v1v1 v2v2 v3v3 v4v4 v5v5 S = {v 1 } Nodev 1 v 2 v 3 v 4 v   -v 1 v 1 v

Fall 2003EE VLSI Design Automation I 74 Prim’s Algorithm Example v1v1 v2v2 v3v3 v4v4 v5v5 S = {v 1 } Nodev 1 v 2 v 3 v 4 v   -v 1 v 1 v

Fall 2003EE VLSI Design Automation I 75 Prim’s Algorithm Example v1v1 v2v2 v3v3 v4v4 v5v5 S = {v 1, v 2 } Nodev 1 v 2 v 3 v 4 v   -v 1 v 1 v

Fall 2003EE VLSI Design Automation I 76 Prim’s Algorithm Example v1v1 v2v2 v3v3 v4v4 v5v5 S = {v 1, v 2 } Nodev 1 v 2 v 3 v 4 v  -v 1 v 2 v 2 v

Fall 2003EE VLSI Design Automation I 77 Prim’s Algorithm Example v1v1 v2v2 v3v3 v4v4 v5v5 S = {v 1, v 2 } Nodev 1 v 2 v 3 v 4 v  -v 1 v 2 v 2 v

Fall 2003EE VLSI Design Automation I 78 Prim’s Algorithm Example v1v1 v2v2 v3v3 v4v4 v5v5 S = {v 1, v 2, v 3 } Nodev 1 v 2 v 3 v 4 v  -v 1 v 2 v 2 v

Fall 2003EE VLSI Design Automation I 79 Prim’s Algorithm Example v1v1 v2v2 v3v3 v4v4 v5v5 S = {v 1, v 2, v 3 } Nodev 1 v 2 v 3 v 4 v  -v 1 v 2 v 3 v

Fall 2003EE VLSI Design Automation I 80 Prim’s Algorithm Example v1v1 v2v2 v3v3 v4v4 v5v5 S = {v 1, v 2, v 3 } Nodev 1 v 2 v 3 v 4 v  -v 1 v 2 v 3 v

Fall 2003EE VLSI Design Automation I 81 Prim’s Algorithm Example v1v1 v2v2 v3v3 v4v4 v5v5 S = {v 1, v 2, v 3, v 4 } Nodev 1 v 2 v 3 v 4 v  -v 1 v 2 v 3 v

Fall 2003EE VLSI Design Automation I 82 Prim’s Algorithm Example v1v1 v2v2 v3v3 v4v4 v5v5 S = {v 1, v 2, v 3, v 4 } Nodev 1 v 2 v 3 v 4 v  -v 1 v 2 v 3 v

Fall 2003EE VLSI Design Automation I 83 Prim’s Algorithm Example v1v1 v2v2 v3v3 v4v4 v5v5 S = {v 1, v 2, v 3, v 4, v 5 } Nodev 1 v 2 v 3 v 4 v  -v 1 v 2 v 3 v

Fall 2003EE VLSI Design Automation I 84 Other Graph Algorithms of Interest... Min-cut partitioning Graph coloring Maximum clique, independent set Min-cut algorithms Steiner tree Matching...