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1 MA 515: Introduction to Algorithms & MA5 : Design and Analysis of Algorithms [-0-0-6] Lecture 20 & 21 Partha Sarathi Manal psm@iitg.ernet.in Dept. of Mathematics, IIT Guwahati Mon 10:00-10:55 Tue 11:00-11:55 Fri 9:00-9:55 Class Room : 2101
2 Kruskal s Algorithm Run the algorithm: 2 19 Kruskal() { T = ; 8 25 for each v V MakeSet(v); 21 1 sort E by increasing edge weight w for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); return T; } 5 9 1
3 Kruskal s Algorithm Kruskal() { T = ; for each v V MakeSet(v); What will affect the running time? 1 Sort O(V) MakeSet() calls O(E) FindSet() calls O(V) Union() calls (Exactly how many Union()s?) sort E by increasing edge weight w for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); return T; }
4 Kruskal s Algorithm: Running Time To summarize: Sort edges: O(E lg E) V; MakeSet() s implies O(V) E; FindSet() s implies O(E) V-1; Union() s implies O(V) Upshot: Best disjoint-set union algorithm makes above operations take O(E (E,V)), (E,V) = lg E Overall thus O(E lg E), almost linear w/o sorting
5 Generic MST Algorithm input: weighted undirected graph G = (V,E,w) T := empty set while T is not yet a spanning tree of G find an edge e in E s.t. T {e} is a subgraph of some MST of G add e to T return T (as MST of G)
6 Kruskal's Algorithm as a Special Case of Generic Alg. Consider edges in increasing order of weight Add the next edge iff it doesn't cause a cycle At any point, T is a forest (set of trees); eventually T is a single tree
7 Idea of Prim's Algorithm Instead of growing the MST as possibly multiple trees that eventually all merge, grow the MST from a single node, so that there is only one tree at any point. Also a special case of the generic algorithm: at each step, add the minimum weight edge that goes out from the tree constructed so far.
8 Prim's Algorithm input: weighted undirected graph G = (V,e,w) T := empty set S := {any node in V} while T < V - 1 do let (u,v) be a min wt. outgoing edge (u in S, v not in S) add (u,v) to T add v to S return (S,T) (as MST of G)
9 Prim s Algorithm
10 Prim s Algorithm Run on example graph
11 Prim s Algorithm Run on example graph
12 Prim s Algorithm r Pick a start vertex r
13 Prim s Algorithm u Red vertices have been removed from Q
14 Prim s Algorithm u Red arrows indicate parent pointers
15 Prim s Algorithm u
16 Prim s Algorithm u 8
17 Prim s Algorithm u 8
18 Prim s Algorithm u 8
19 Prim s Algorithm u
20 Prim s Algorithm u
21 Prim s Algorithm u
22 Prim s Algorithm u
23 Prim s Algorithm u
24 Prim s Algorithm 4 u
25 Prim s Algorithm 4 u
26 Prim s Algorithm u
27 Prim s Algorithm u
28 Prim s Algorithm u
29 Prim s Algorithm u
30 Correctness of Prim's Algorithm Let T i be the tree represented by (S,T) at the end of iteration i. Show by induction on i that T i is a subtree of some MST of G. Basis: i = 0 (before first iteration). T 0 contains just a single node, and thus is a subtree of every MST of G.
31 Correctness of Prim's Algorithm Induction: Assume T i is a subtree of some MST M. We must show T i+1 is a subtree of some MST. Let (u,v) be the edge added in iteration i+1. T i u T i+1 v Case 1: (u,v) is in M. Then T i+1 is also a subtree of M.
32 Correctness of Prim's Algorithm Case 2: (u,v) is not in M. There is a path P in M from u to v, since M spans G. Let (x,y) be the first edge in P with one endpoint in T i and the other not in T i. x y P T i u v
33 Correctness of Prim's Algorithm Let M' = M - {(x,y)} U {(u,v)} M' is also a spanning tree of G. w(m') = w(m) - w(x,y) + w(u,v) w(m) since (u,v) is min wt outgoing edge So M' is also an MST and T i+1 is subtree M' T i x u y T i+1 v
34 Running Time of Prim s Algorithm What is the hidden cost in this code?
35 Running Time of Prim s Algorithm What is the hidden cost in this code?
36 Running Time of Prim s Algorithm What is the hidden cost in this code?
37 Running Time of Prim s Algorithm DecreaseKey(v, w(u,v));
38 Running Time of Prim s Algorithm Insert cost? How often is ExtractMin() called? How often is DecreaseKey() called?
39 Running Time of Prim s Algorithm Insert cost? How often is ExtractMin() called? How often is DecreaseKey() called? DecreaseKey(v, w(u,v));
40 Running Time of Prim s Algorithm Insert cost? How often is ExtractMin() called? How often is DecreaseKey() called? DecreaseKey(v, w(u,v));
41 Running Time of Prim's Algorithm Depends on priority queue implementation. Let T ins be time for insert T dec be time for decrease-key T ex be time for extract-min Then we have V inserts and one decrease-key in the initialization: O(V (T ins +T dec )) V iterations of while one extract-min per iteration: O(V T ex ) total
42 Running Time of Prim's Algorithm Each iteration of while includes a for loop. Number of iterations of for loop varies, depending on how many neighbors the current node has Total number of iterations of for loop is O(E). Each iteration of for loop: one decrease key, so O(E T dec ) total
43 Running Time of Prim's Algorithm O(V(T ins + T ex ) + E T dec ) If priority queue is implemented with a binary heap, then T ins = T ex = T dec = O(log V) total time is O(E log V) If priority queue is implemented with a Fibonacci heap, then Fibonacci heap: O(V lg V + E) Exercise: Fibonacci heap.
Partha Sarathi Mandal
MA 5: Data Strctres and Algorithms Lectre http://www.iitg.ernet.in/psm/indexing_ma5/y1/index.html Partha Sarathi Mandal Dept. of Mathematics, IIT Gwahati Idea of Prim s Algorithm Instead of growing the
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