CSC 8301 Design & Analysis of Algorithms: Warshall s, Floyd s, and Prim s algorithms

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1 CSC 8301 Design & Analysis of Algorithms: Warshall s, Floyd s, and Prim s algorithms Professor Henry Carter Fall 2016

2 Recap Space-time tradeoffs allow for faster algorithms at the cost of space complexity overhead Dynamic programming achieves this by saving the result of overlapping subproblems Can be executed bottom-up or top-down (using memory functions) 2

3 Digraphs Revisited Recall: directed graphs New feature: edge weights Applications: scheduling, process flow, revision history 3

4 Transitive Closure Is node b reachable from node a? Consider all pairs (a,b) Store results in an n x n matrix of {0,1} 4

5 Example Graph a b c d a b c d A B a a b b C D c d c d

6 BFS/DFS Approach The transitive closure from a to any other node can be found through graph traversal Repeating for all nodes yields the complete closure matrix How could we apply dynamic programming? 6

7 Warshall s Algorithm Construct transitive closure using a series of matrices Matrix k considers paths through G traversing nodes 1,,k If a path exists from i to k and from k to j, mark a path from i to j R 0 is the paths between each vertex with no intermediate vertices (i.e., the adjacency matrix) 7

8 Rule for Changing 0 to 1 8

9 Example Application 9

10 Algorithm Warshall(A[1,...,n,1,...,n]) input : The adjacency matrix A with n vertices. output: The transitive closure of the digraph. R (0) A for k 1 to n do for i 1 to n do for j 1 to n do R (k) [i, j] R (k 1) [i, j] or (R (k 1) [i, k] and R (k 1) [k, j]) end end end return R (n) 10

11 Speeding Things Up More efficient inner loop Treat rows as bit strings and apply boolean operations simultaneously Combine matrices into one 11

12 All-Pairs Shortest Paths Given a weighted digraph, find the shortest path from a to b Solve for all pairs (a,b) Stored in an n x n integer distance matrix 12

13 Example Graph A 2 B a b c d a b c d a 0 3 a C D b 2 0 c d 6 0 b c d

14 Floyd s Algorithm Construct the distance matrix using a series of matrices Matrix k considers paths through G that traverse any vertex numbered 1,,k If there is a path from i to k and from k to j, we compare it to the current shortest path and (possibly) update the distance R0 is the distance between each node with no intermediate vertices (i.e., the edge weight matrix) 14

15 Rule for Updating Distance 15

16 Example Application 16

17 Algorithm Floyd(W [1,...,n,1,...,n]) input : The weight matrix W. output: The distance matrix of the shortest paths lengths. D W for k 1 to n do for i 1 to n do for j 1 to n do D[i, j] min(d[i, j],d[i, k]+d[k, j]) end end end return D 17

18 Recap Transitive closure and all-pairs shortest paths problems Warshall s and Floyd s algorithms use dynamic programming to store intermediate results in a series of matrices We will revisit all-pairs shortest paths in the next chapter 18

19 Greedy Algorithms Global optimization that makes a series of locally optimal choices Three requirements at each iteration: Feasible Locally optimal Irrevocable Recall: change making Other examples? 19

20 Proofs of Optimality Induction Other iterative approaches cannot do better Show the output is always optimal 20

vertices with the cheapest edge weights Applications:

21 Minimum Spanning Tree Given: a weighted, undirected, connected graph Find: a subgraph that connects all vertices with the cheapest edge weights Applications: network infrastructure, data set clustering, approximation algorithms 21

22 Greedy Approaches How could we greedily add vertices to the MST? How could we greedily add edges to the MST? 22

the MST, vertices outside the MST Iterate: add the

23 Prim s Algorithm Greedy approach to adding vertices Maintains three sets: vertices in the MST, edges in the MST, vertices outside the MST Iterate: add the vertex outside the MST with the cheapest connection 23

24 Example b 1 d a 5 c 5 e f 24

25 Proof of Optimality 25

26 Prim s Algorithm Prim(G) input : A weighted connected graph G = hv,ei. output: E T, the set of edges composing a minimum spanning tree of G. V T {v 0 } E T ; for i 1 to V 1 do Find a minimum-weight edge e =(v,u ) among all the edges (v, u) such that v is in V T and u is in V V T V T V T [ {u } E T E T [ {e } end return E T 26

27 Analysis Main loop: V Unordered list priority queue: V 2 Min-heap: log V Total: O( E log V ) 27

28 Next Time... Levitin Chapter Remember, you need to read it BEFORE you come to class! Homework: 8.4: 1, 6, 7 9.1: 3, 9a 28

CSC 1700 Analysis of Algorithms: Minimum Spanning Tree

CSC 1700 Analysis of Algorithms: Minimum Spanning Tree Professor Henry Carter Fall 2016 Recap Space-time tradeoffs allow for faster algorithms at the cost of space complexity overhead Dynamic programming