SI545 VLSI CAD: Logic to Layout. Algorithms

Transcription

1 SI545 VLSI CAD: Logic to Layout Algorithms

2 Announcements TA for this course LeileiWang, Office hour: 1:30-:30pm, Thurs., Course page on Piazza

3 References and Copyright Slides used (Modified when necessary) Majid Sarrafzadeh, 001 Department of Computer Science, UCLA Kurt Keutzer, Dept. of EECS, UC-Berekeley Rajesh Gupta, UC-Irvine Kia Bazargan, University of Minnesota Naveed A. Sherwani, 1999

4 Computational Complexity

5 Combinatorial Optimization Problems Problems with discrete variables Examples: SORTING Given N integer numbers, write them in increasing order. KNAPSACK Given a bag of size S, and items {(s 1, v 1 ), (s, v ),..., (s n, v n )}, where s i and v i are the size and value of item i respectively, find a subset of items with maximum overall value that fit in the bag. 5

6 Graph Definition Graph: set of objects and their connections Formal definition: G = (V, E), V={v 1, v,..., v n }, E={e 1, e,..., 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 a e d b e d b c c 6

7 Decision Problem vs. Optimization Problem Decision problem ask for a yes or no answer Example: Hamiltonian cycle - simple cycle (no repeated vertices) that contains all nodes in the graph; TRAVELING SALESMAN. Optimization problem find optimal value of a target variable Example: Traveling salesman problem (TSP) - min-cost Hamiltonian cycle 11 Optimization problem is harder than its decision problem version

8 Algorithm An algorithm defines a procedure for solving a computational problem Examples: Quick sort, bubble sort, insert sort, heap sort Dynamic programming method for the knapsack problem 8

9 Bubble Sort for (j=1; j < N; j++) { for (i=1; i < N-j+1; i++) { if (a[i] > a[i+1]) { hold = a[i]; a[i] = a[i+1]; a[i+1] = hold; } } } e.g., initial ordering: 10, 5, 3, 8, 6, 1 9

10 [ Bubble Sort Animation Value 10 Index

11 Complexity Measure the efficiency of a algorithm Time and memory Computational complexity Time complexity, space complexity Average case, worst case Scalability (with respect to input size n) is important How does the running time of an algorithm change when the input size doubles? Function of input size (n). Examples: n +3n, n, nlog n,... Generally, large input sizes are of interest (n > 1,000 or even n > 1,000,000) 11

12 Function Growth Examples Function Growth f(n) 5n 0.1n^+n+40 n log n ^n n f(n) Function Growth 5n 0.1n^+n+40 n log n ^n n

13 Idea: Asymptotic Notions 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 = O (n ) 0.0 n + 17 n = O (n ) 13 Tractability: solvable in polynomial time Polynomial time = O(n k ), otherwise exponential

14 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 Question: If f(n)=o(g(n)), then is g(n)= (f(n))? 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 ) 4 n = (n ) 14

15 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 4 n n + log n n + n log(n) n! > K n > n K > log n > log log n > K What do asymptotic notations mean in practice? If algorithm A has time complexity O(n ) and algorithm B has time complexity O(n log n), then which algorithm 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. 15

16 The Complexity Classes P vs. NP A problem s complexity = the time complexity of the most efficient possible algorithm. 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) Examples: search problem, minimum spanning tree, etc. Class NP (non-deterministic polynomial time) A decision problem Can be solved in polynomial time on a nondeterministic computer Nondeterministic: multiple possibilities available for next step in computation A solution is verifiable in polynomial time on a deterministic computer Example: TRAVELING SALESMAN (=Hamilton cycle with length k). 16 P NP Is P = NP? (Find out and become famous!)

17 NP-Complete Practically, for a problem in NP but not in P: polynomial solution not found yet (probably does not exist) exact (optimal) solution can be found using an algorithm with exponential time complexity Class NP-complete (NPC) The most difficult problems in the NP class Till now, the best algorithm for worst-case NPC problems has exponential time complexity 17

18 Examples of NPC Problems Does a graph have a Hamiltonian cycle? Hamiltonian cycle/traveling SALESMAN 3SAT: Given a Boolean expression expressed as POS with 3 literals in each sum, is it satisfiable? Example: (x + y + z)( x + y)( y + z)( x + y + z) SAT can be solved in polynomial time! Find a maximal clique in a graph Clique = set of vertices so that every pair of vertices in the set is connected by an edge (complete subgraph) Find a maximal independent set in a graph A max set of vertices so that no pair of vertices is connected VLSI fixed-outline floorplanning Bible of NP-completeness M. R. Garey and D. S. Johnson, Computers and Intractability, W. H. Freeman and Company, New York, NY,

19 NP-Hard Class NP-hard A special definition - solution not verifiable in polynomial time on a deterministic computer At least as difficult to solve as NPC Example: TSP problem (solution check O(n n!)) Most CAD problems are NPC, NP-hard, or worse Be happy with a reasonably good solution Reading material on time complexity Textbook [Ch. 4., Wang], plus Princeton slides. 19 Stephen Cook, 198 s Turing award

20 0 Algorithm Types Deterministic vs. probabilistic (or randomized) Graph algorithms Heuristics: apply heuristics to find a good, but not necessarily optimal solution Greedy algorithm Dynamic programming Branch-and-bound Simulated annealing Genetic Mathematical programming Read [Ch. 4.4, Wang] for the detailed discussion on these algorithms.

21 1 Graph Algorithms

22 Graph Representation: Adjacency List a a b d c e d b b c a d a c d e e b d a a a b d e d b b c d a c d e a e

23 Graph Representation: Adjacency Matrix a a a 0 b c d e e d b b c c d e a a a 0 b c d e e d b b c c d e

24 4 Edge/Vertex Weights in Graphs Edge weights Usually represent the cost of an edge Examples: Distance between two cities e Width of a data bus 1 Representation Adjacency matrix: instead of 0/1, keep weight Adjacency list: keep the weight in the linked list item Node weight Usually used to enforce some capacity constraint Examples: The size of gates in a circuit The delay of operations in a data dependency graph d a - 3 c a b b 5 1 *

25 Hypergraphs Hypergraph definition Similar to graphs, but edges not between pairs of vertices, but between a set of vertices Directed/undirected versions possible Just like graphs, a node can be the source (or be connected to) of multiple hyperedges Data structure?

26 6 Graph Search Algorithms Purpose: to visit all the nodes Algorithms Depth-first search Breadth-first search Topological (on directed acyclic graph) Examples B D A F E G C B E G F A D C A B G F D E C

27 Depth-First Search Algorithm struct vertex {... int marked; }; dfs ( v ) v.marked 1 print v for each (v, u) E if (u.marked!= 1) // not visited yet? dfs (u) 7 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)

28 Depth-First Search Algorithm dfs ( a ) a.marked 1 print a for each (a, u) E if (u.marked!= 1) dfs (u) dfs( a )... a a a b d c b a d e d b c d a e b a c e d 8

29 Depth-First Search Algorithm u=b dfs ( a ) a.marked 1 print a for each (a, u) E if (b.marked!= 1) dfs (b) dfs( a )... dfs( b )... a a a b d c b a d e d b c d a e b a c e d 9

30 Depth-First Search Algorithm dfs ( b ) b.marked 1 print b for each (b, u) E if (u.marked!= 1) dfs (u) dfs( a )... dfs( b )... a b a a b d c b a d e d b c d a e b a c e d 30

31 Depth-First Search Algorithm u=a dfs ( b ) b.marked 1 print b for each (b, u) E if (a.marked!= 1) dfs (a) dfs( a )... dfs( b )... a b a a b d c b a d e d b c d a e b a c e d 31

32 Depth-First Search Algorithm u=d dfs ( b ) b.marked 1 print b for each (b, u) E if (d.marked!= 1) dfs (d) dfs( a )... dfs( b )... dfs( d )... a b a a b d c b a d e d b c d a e b a c e d 3

33 Depth-First Search Algorithm dfs ( d ) d.marked 1 print d for each (d, u) E if (u.marked!= 1) dfs (u) dfs( a )... dfs( b )... dfs( d )... a b d a a b d c b a d e d b c d a e b a c e d 33

34 Depth-First Search Algorithm u=e dfs ( d ) d.marked 1 print d for each (d, u) E if (e.marked!= 1) dfs (e) dfs( a )... dfs( b )... dfs( d )... dfs( e )... a b d a a b d c b a d e d b c d a e b a c e d 34

35 Depth-First Search Algorithm dfs ( e ) e.marked 1 print e for each (e, u) E if (u.marked!= 1) dfs (u) dfs( a )... dfs( b )... dfs( d )... dfs( e )... a b d e a a b d c b a d e d b c d a e b a c e d 35

36 Depth-First Search Algorithm u=d dfs ( e ) e.marked 1 print e for each (e, u) E if (d.marked!= 1) dfs (u) dfs( a )... dfs( b )... dfs( d )... dfs( e )... a b d e a a b d c b a d e d b c d a e b a c e d 36

37 Depth-First Search Algorithm dfs ( e ) e.marked 1 print e for each (e, u) E if (u.marked!= 1) dfs (u) dfs( a )... dfs( b )... dfs( d )... dfs( e )... a b d e a a b d c b a d e d b c d a e b a c e d 37

38 Depth-First Search Algorithm dfs ( d ) d.marked 1 print d for each (d, u) E if (b.marked!= 1) dfs (b) u=b,a dfs( a )... dfs( b )... dfs( d )... a b d e a a b d c b a d e d b c d a e b a c e d 38

39 Depth-First Search Algorithm dfs ( d ) d.marked 1 print d for each (d, u) E if (u.marked!= 1) dfs (u) dfs( a )... dfs( b )... dfs( d )... a b d e a a b d c b a d e d b c d a e b a c e d 39

40 Depth-First Search Algorithm dfs ( b ) b.marked 1 print b for each (b, u) E if (u.marked!= 1) dfs (u) dfs( a )... dfs( b )... a b d e a a b d c b a d e d b c d a e b a c e d 40

41 Depth-First Search Algorithm dfs ( a ) a.marked 1 print a for each (a, u) E if (d.marked!= 1) dfs (d) u=d dfs( a )... a b d e a a b d c b a d e d b c d a e b a c e d 41

42 Depth-First Search Algorithm dfs ( a ) a.marked 1 print a for each (a, u) E if (c.marked!= 1) dfs (c) u=c dfs( a )... dfs( c )... a b d e a a b d c b a d e d b c d a e b a c e d 4

43 Depth-First Search Algorithm dfs ( c ) c.marked 1 print c for each (c, u) E if (u.marked!= 1) dfs (u) dfs( a )... dfs( c ) a b d e c a e d c b a b d c b c d e a d a e b a d

44 Depth-First Search Algorithm dfs ( c ) c.marked 1 print c for each (c, u) E if (a.marked!= 1) dfs (a) u=a dfs( a )... dfs( c ) a b d e c a e d c b a b d c b c d e a d a e b a d

45 Depth-First Search Algorithm dfs ( c ) c.marked 1 print c for each (c, u) E if (u.marked!= 1) dfs (u) dfs( a )... dfs( c ) a b d e c a e d c b a b d c b c d e a d a e b a d

46 Depth-First Search Algorithm dfs ( a ) a.marked 1 print a for each (a, u) E if (u.marked!= 1) dfs (u) dfs( a ) a b d e c a e d c b a b d c b c d e a d a e b a d

47 Breadth-First Search Algorithm bfs ( v, Q) v.marked 1 for each (v, u) E if (u.marked!= 1) // not visited yet? Q Q + u Algorithm BREADTH_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? 47

48 Distance in (non-weighted) Graphs Distance d G (u,v): Length of a shortest u--v path in G. u d(u,v)= v x y d(x,y) = 48

49 Moore s Breadth-First Search Algorithm Objective: Find d(u,v) for a given pair (u,v) and find 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 u v 49

50 Moore s Breadth-First Search Algorithm Algorithm: 1. Initialize (w) for w u (u) 0 Q Q+u. If Q, x pop(q) else stop: no path u--v 3. (x,y) E, if (y)= (y) x (y) (x)+1 Q Q+y 4. If (v)= return to step 5. Follow (w) pointers from v to u. u v 50 E. F. Moore (1959), The shortest path through a maze. In Proceedings of the International Symposium on the Theory of Switching, Harvard University Press, pp

51 Moore s Breadth-First Search Algorithm u v v 3 v 1 v 4 v v 6 5 v 7 v 8 v 9 Q = u v 10 Algorithm: 1. Initialize (w) for w u (u) 0 Q Q+u. If Q, x pop(q) else stop: no path u--v 3. (x,y) E, if (y)= (y) x (y) (x)+1 Q Q+y 4. If (v)= return to step 5. Follow (w) pointers from v to u. Node u v 1 v v 3 v 4 v 5 v 6 v 7 v 8 v 9 v

52 Moore s Breadth-First Search Algorithm u v v 3 v 1 v 4 v v 6 5 v 7 v 8 v 9 Q = v 1, v, v 5 v 10 Algorithm: 1. Initialize (w) for w u (u) 0 Q Q+u. If Q, x pop(q) else stop: no path u--v 3. (x,y) E, if (y)= (y) x (y) (x)+1 Q Q+y 4. If (v)= return to step 5. Follow (w) pointers from v to u. Node u v 1 v v 3 v 4 v 5 v 6 v 7 v 8 v 9 v u u - - u

53 53 Moore s Breadth-First Search Algorithm u v 1 v v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 Q = v 4, v 3, v 6, v 10 Algorithm: 1. Initialize (w) for w u (u) 0 Q Q+u. If Q, x pop(q) else stop: no path u--v 3. (x,y) E, if (y)= (y) x (y) (x)+1 Q Q+y 4. If (v)= return to step 5. Follow (w) pointers from v to u. Node u v 1 v v 3 v 4 v 5 v 6 v 7 v 8 v 9 v u u v v 1 u v v 5

54 Moore s Breadth-First Search Algorithm u v v 3 v 1 v 4 v v 6 5 v 7 v 8 v 9 Q = v 7 v 10 Algorithm: 1. Initialize (w) for w u (u) 0 Q Q+u. If Q, x pop(q) else stop: no path u--v 3. (x,y) E, if (y)= (y) x (y) (x)+1 Q Q+y 4. If (v)= return to step 5. Follow (w) pointers from v to u. Node u v 1 v v 3 v 4 v 5 v 6 v 7 v 8 v 9 v u u v v 1 u v 5 v v 5 54

55 Notes on Moore s Algorithm Time complexity? Space complexity? 55

56 Maze Routing Problem: find a connection from A to B A Obstacle B Quite useful for two pin nets, rip-up-and-reroute, etc. 56

57 Lee s Algorithm C. Y. Lee (1961), An algorithm for path connection and its applications. IRE Transactions on Electronic Computers, EC-10(3), pp

58 A more space-efficient version

59 59 Other improvements

60 Line search algorithms Reduced memory requirements Sends out a line and uses it to guide the search No guarantees on being able to find a route 60

61 Distance in Weighted Graphs Why does a simple BFS not work any more? Your locally best solution is not your globally best one First, v 1 and v will be visited =3 =u v 1 3 v u 11 =11 =u u 61 v should be revisited =3 =u v 1 3 v 11 =5 =v 1

62 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,w) as you find shorter distances to w. (w) changed accordingly. Data structure: S (w) (w) the set of nodes whose d(u,v) is found current length of the shortest path u--w current parent node of w on u w 6

63 Dijkstra s Algorithm Algorithm: 1. Initialize (v) for v u (u) 0 S {u}, v j = u. For each v S s.t. (v j,v) E If (v) > (v j ) + w(v j,v) (v) (u i ) + w(v j,v) (v) v j O(e) 3. Find m=min{ (v) v S } and (v k )==m S S {v k } O(v ) If S < V, goto step

64 Dijkstra s Algorithm - an example u v 1 10 v 5 v v 7 64 v Edges examined 14 v v 4

65 Dijkstra s Algorithm - an example u v v 5 v v 7 v v 4 updated, min picked 65 v 3

66 Dijkstra s Algorithm - an example u v v 5 v v v S Augmented 14 v v 4

67 Dijkstra s Algorithm - an example u v v 5 v v v 14 Edges examined v v 4

68 Dijkstra s Algorithm - an example u v v 5 v v 7 v v 4 updated, min picked 68 v 3

69 Dijkstra s Algorithm - an example u v v 5 v v 7 69 v S Augmented v v 4

70 Dijkstra s Algorithm - an example u v v 5 v v v Edges examined 14 5 v v 4

71 Dijkstra s Algorithm - an example u v v 5 v v 7 v v 4 updated, min picked 71 v 3

72 Dijkstra s Algorithm - an example u v v 5 v v 7 7 v S augmented v v 4

73 Dijkstra s Algorithm - an example u v v 5 v v v Edges examined 14 5 v v 4

74 Dijkstra s Algorithm - an example u v v 5 v v 7 v v 4 updated, min picked 74 v 3

75 Dijkstra s Algorithm - an example u v v 5 v v 7 75 v S augmented v v 4

76 Dijkstra s Algorithm - an example u v v 5 v v v Edges examined 14 0 v v 4

77 Dijkstra s Algorithm - an example u v v 5 v v 7 v v 4 updated, min picked 77 v 3

78 Dijkstra s Algorithm - an example u v v 5 v v 7 78 v S augmented v v 4

79 Dijkstra s Algorithm - why does it work? Proof by contradiction Suppose v is the first node being added to S such that (v) > d(u 0,v) (d(u 0,v) is the real shortest u 0 --v path) The assumption that (v) and d(u 0,v) are different, means there are different paths with lengths (v) and d(u 0,v) Consider the path that has length d(u 0,v). Let x be the first node in S on this path (v) > d(u 0,v) by assumption and d(u 0,v) (x) => (v) > (x) => contradiction (since the min value is the one added to S in each iteration) u 0 S v x S 79

80 Static Timing Analysis (STA) Finding the longest path in a general graph is NP-complete, even when edges are not weighted Polynomial for DAG (directed acyclic graphs) Topological search on a DAG In circuit graphs, static timing analysis (STA) refers to the problem of finding the max delay from the input pins of the circuit (esp nodes) to each gate Max delay of the output pins determines clock period In sequential circuits, FF input acts as output pin, FF output acts as input pin Critical path is a path with max delay among all paths In addition to the arrival time of each node, we are interested in knowing the slack of each node/edge 80

81 STA Example: Arrival Times Assumptions: All inputs arrive at time 0 All gate delays = 1 All wire delays = 0 Question: Arrival time of each gate? Circuit delay? 0 a 1 c f t i = max {t j } + d i e 1 g d 1 b 4 4 h

82 8 STA Example: Required Times Assumptions: All inputs arrive at time 0 All gate delays = 1, wire delay = 0 Clock period = 7 Question: maximum required time (RT) of each gate? (i.e., if the gate output is generated later than RT, clk period is violated) a 3 e 3 c 4 g 4 d f 6 5 b r i = min {r j -d j } 6 h

83 83 STA Example: Slack Assumptions: All inputs arrive at time 0 All gate delays = 1, wire delay = 0 Clock period = 7 Question: What is the maximum amount of delay each gate can be slower not to violate timing? -0= -0= -0= 3-0=3 5-0=5 5-0=5 a 3-1= e c 3-1= 4-= g d f 4-= 5-3= s i = r i -t i 6-1=5 b 6-4= h 7-5= 7-3=4 7-4=3 7-5=

84 STA: Issues If the delay of one gate changes, what is the time complexity of updating the slack of all nodes? How can slack be used? Maintain a list of K-most critical paths? Refer to Chapter 5 of Sachin S. Sapatnekar, Timing, Kluwer Academic Publishers, Boston, MA,

85 Another Application of Longest Paths Layout compaction Given a set of blocks, place them in the most compact way possible The D compaction problem is often solved as a sequence of 1D problems, in the x and y directions w a w b x b x a W a x a a x b b A set of such constraints represents a longest path problem 85

86 Example Horizontal constraint graph G h s Vertical constraint graph G v s 86

87 Example: negative-weight edges Suppose you need a and b to be stuck together Then, x b x a = w a In other words, x b x a w a a b x b x a w a To represent these in a longest path problem (use ), rewrite as x a x b -w a x b x a w a w a 87 a -w a b

88 Bellman-Ford Algorithm Dijkstra s doesn t work when a graph has negative edges Intuition we cannot be greedy any more on the assumption that the lengths of paths will only increase in the future Bellman-Ford algorithm detects negative cycles (returns false) or returns the shortest path-tree 88

89 Bellman-Ford Algorithm (shortest path) Bellman-Ford(G,w,s) for each vertex u V d(u) = parent(u) = NIL d(s) = 0 for i 1 to V -1 for each edge (u,v) E d(v) = min[d(v),d(u)+w(u,v)] for each edge (u,v) E do if d(u) + w(u,v) < d(v) then // negative cycle return false return true 89

90 Bellman-Ford Example s z y x t s z y x t s z y x t s z y x t -4 5

91 Bellman-Ford Example s t y x z 7 Bellman-Ford running time: ( V -1) E + E = (VE) 91

92 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 9

93 Difference Between MST and Shortest Path Why can t Dijkstra solve the MST problem? Shortest path: min sum of edge weight to individual nodes Each node appear at most once in the shortest path MST: min sum of TOTAL edge weights that connect all vertices 93

94 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) 94

95 Prim s Algorithm for MST Data structure: S set of nodes added to the tree so far S set of nodes not added to the tree yet T the 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) 95

96 Prim s Algorithm Initialize S, S and T S {u 0 }, S V \ {u 0 } // u 0 is any vertex T { } v S, (v) Initialize and for the vertices adjacent to u 0 For each v S s.t. (u 0,v) E, (v) w((u 0,v)) (v) u 0 While (S!= ) Find u S, s.t. v S, (u) (v) S S {u}, S S \ {u}, T T { ( (u), u) } For each v s.t. (u, v) E, If (v) > w((u,v)) then (v) w((u,v)) (v) u 96

97 Prim s Algorithm Example v 1 5 v 1 4 v 3 v v 4 S = {v 1 } Node v 1 v v 3 v 4 v v 1 v 1 v 1-97

98 Prim s Algorithm Example v 1 5 v 1 4 v 3 v v 4 S = {v 1 } Node v 1 v v 3 v 4 v v 1 v 1 v 1-98

99 Prim s Algorithm Example v 1 5 v 1 4 v 3 v v 4 S = {v 1, v } Node v 1 v v 3 v 4 v v 1 v 1 v 1-99

100 Prim s Algorithm Example v 1 5 v 1 4 v 3 v v 4 S = {v 1, v } Node v 1 v v 3 v 4 v v 1 v v v 100

101 Prim s Algorithm Example v 1 5 v 1 4 v 3 v v 4 S = {v 1, v } Node v 1 v v 3 v 4 v v 1 v v v 101

102 Prim s Algorithm Example v 1 5 v 1 4 v 3 v v 4 S = {v 1, v, v 3 } Node v 1 v v 3 v 4 v v 1 v v v 10

103 Prim s Algorithm Example v 1 5 v 1 4 v 3 v v 4 S = {v 1, v, v 3 } Node v 1 v v 3 v 4 v v 1 v v 3 v 3 103

104 Prim s Algorithm Example v 1 5 v 1 4 v 3 v v 4 S = {v 1, v, v 3 } Node v 1 v v 3 v 4 v v 1 v v 3 v 3 104

105 Prim s Algorithm Example v 1 5 v 1 4 v 3 v v 4 S = {v 1, v, v 3, v 4 } Node v 1 v v 3 v 4 v v 1 v v 3 v 3 105

106 Prim s Algorithm Example v 1 5 v 1 4 v 3 v v 4 S = {v 1, v, v 3, v 4 } Node v 1 v v 3 v 4 v v 1 v v 3 v 3 106

107 Prim s Algorithm Example v 1 5 v 1 4 v 3 v v 4 S = {v 1, v, v 3, v 4, v 5 } Node v 1 v v 3 v 4 v v 1 v v 3 v 3 107

108 Kruskal s algorithm S = {}, S = E Sort S in nondecreasing order of weights While S {} For each edge (u,v) S in sorted order S = S \ (u,v) If (u,v) does not create a cycle with the edges in S S = S (u,v) 108

109 Kruskal s Algorithm Example v 1 5 v 1 4 v 3 v v 4 S = {} 109

110 Kruskal s Algorithm Example v 1 5 v 1 4 v 3 v v 4 S = {(v,v 3 )} 110

111 Kruskal s Algorithm Example v 1 5 v 1 4 v 3 v v 4 S = {(v,v 3 ), (v 3,v 4 )} 111

112 Kruskal s Algorithm Example v 1 5 v 1 4 v 3 v v 4 S = {(v,v 3 ), (v 3,v 4 ), (v 3,v 5 )} 11

113 Kruskal s Algorithm Example v 1 5 v 1 4 v 3 v v 4 S = {(v,v 3 ), (v 3,v 4 ), (v 3,v 5 ),(v,v 1 )} 113

114 Other Graph Algorithms of Interest... Min-cut partitioning Graph coloring Maximum clique, independent set Steiner tree [ 114

115 115 Dynamic Programming

116 Dynamic Programming Read the first two examples in the document written by Michael A. Trick see the Useful Links slide Knapsack problem Given N discrete items of size s i and value v i, how to fill a knapsack of size M to get the maximum value? There is only one of each item that can be either taken in whole or left out. 116

117 Dynamic Programming: Knapsack Partial solution constructed for items 1..(i-1) for knapsack sizes from 0..M (call this array A) i M-1 M c 0 c 1 c c M-1 c M For each knapsack size, how to extend the solution from 1..(i-1) to include i? Option 1: do not take item i w Option : take item i w-s i w i-1 c w i-1 i c w i c w-si +v i 117

118 Example of Knapsack i s i v i M=10 1

119 119 Mathematical Programming

120 10 General Form

121 11 ILP Example

122 1 Branch and Bound

123 13 Convex Optimization

124 Useful Links Books on STL and templates (thanks to Arvind U of M for suggesting thebooks) : Nicolai M. Josuttis, The C++ Standard Library: A Tutorial and Reference, Addison-Wesley, 1999, ISBN: David Vanevoorde and Nicolai M. Josuttis, C++ Templates, the Complete Guide, Addison- Wesley, 003, ISBN: Time Complexity and Asymptotic Notations Asymptotic bounds: definitions and theorems /asymptotic.html CMSC 341 (at CSEE/UMBC) Lecture oticanalysis.ppt Michael A. Trick, A Tutorial on Dynamic Programming, 14

125 To Read Further (Chapter 4) Heuristic algorithms Greedy Branch-and-Bound Simulated Annealing Genetic Algorithms Mathematical programming Linear programming (LP) Integer LP Convex optimization 15

126 Good References on Algorithms