Minimum Spanning Trees

Transcription

1 CSMPS 2200 Fall Minimum Spanning Trees Carola Wenk Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk 11/6/ CMPS 2200 Intro. to Algorithms 1

2 Minimum spanning trees Input: A connected, undirected graph G = (V, E) with weight function w : E R. For simplicity, assume that all edge weights are distinct. Output: A spanning tree T a tree that connects all vertices of minimum weight: w( T ) w( u, v). ( u, v) T 11/6/ CMPS 2200 Intro. to Algorithms 2

3 Example of MST /6/ CMPS 2200 Intro. to Algorithms

4 Hallmark for greedy algorithms Greedy-choice property A locally optimal choice is globally optimal. Theorem [Cut property]. Let G = (V, E) and let A V. Suppose that (u, v) E is the least-weight edge connecting A to V \ A. Then, (u, v) is contained in an MST T of G. 11/6/ CMPS 2200 Intro. to Algorithms 4

5 Proof of theorem Proof. Suppose (u, v) T. Cut and paste. T: A V \ A u v (u, v) = least-weight edge connecting A to V \ A 11/6/ CMPS 2200 Intro. to Algorithms 5

6 Proof of theorem Proof. Suppose (u, v) T. Cut and paste. T: v A V \ A u (u, v) = least-weight edge connecting A to V \ A Consider the unique simple path from u to v in T. 11/6/ CMPS 2200 Intro. to Algorithms 6

7 Proof of theorem Proof. Suppose (u, v) T. Cut and paste. T: v A V \ A u (u, v) = least-weight edge connecting A to V \ A Consider the unique simple path from u to v in T. Swap (u, v) with the first edge on this path that connects a vertex in A to a vertex in V \ A. 11/6/ CMPS 2200 Intro. to Algorithms

8 Proof of theorem Proof. Suppose (u, v) T. Cut and paste. T: v A V \ A u (u, v) = least-weight edge connecting A to V \ A Consider the unique simple path from u to v in T. Swap (u, v) with the first edge on this path that connects a vertex in A to a vertex in V \ A. A lighter-weight spanning tree than T results. 11/6/ CMPS 2200 Intro. to Algorithms

9 Prim s algorithm IDEA: Maintain V \ A as a priority queue Q. Key each vertex in Q with the weight of the leastweight edge connecting it to a vertex in A. Q V key[v] for all v V key[s] 0 for some arbitrary s V while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if v Q and w(u, v) < key[v] then key[v] w(u, v) Dijkstra: while Q do u EXTRACT-MIN(Q) S S {u} for each v Adj[u] do if d[v] > d[u] + w(u, v) then d[v] d[u] + w(u, v) DECREASE-KEY [v] u At the end, {(v, [v])} forms the MST edges. 11/6/ CMPS 2200 Intro. to Algorithms 9

10 Example of Prim s algorithm A V \ A /6/ CMPS 2200 Intro. to Algorithms u EXTRACT-MIN(Q) for each v Adj[u] do if v Q and w(u, v) < key[v] then key[v] w(u, v) DECREASE-KEY [v] u

11 Example of Prim s algorithm A V \ A /6/ CMPS 2200 Intro. to Algorithms 11 u EXTRACT-MIN(Q) for each v Adj[u] do if v Q and w(u, v) < key[v] then key[v] w(u, v) DECREASE-KEY [v] u

12 Example of Prim s algorithm A V \ A /6/ CMPS 2200 Intro. to Algorithms 12 u EXTRACT-MIN(Q) for each v Adj[u] do if v Q and w(u, v) < key[v] then key[v] w(u, v) DECREASE-KEY [v] u

13 Example of Prim s algorithm A V \ A /6/ CMPS 2200 Intro. to Algorithms 1 u EXTRACT-MIN(Q) for each v Adj[u] do if v Q and w(u, v) < key[v] then key[v] w(u, v) DECREASE-KEY [v] u

14 Example of Prim s algorithm A V \ A /6/ CMPS 2200 Intro. to Algorithms u EXTRACT-MIN(Q) for each v Adj[u] do if v Q and w(u, v) < key[v] then key[v] w(u, v) DECREASE-KEY [v] u

15 Example of Prim s algorithm A V \ A /6/ CMPS 2200 Intro. to Algorithms u EXTRACT-MIN(Q) for each v Adj[u] do if v Q and w(u, v) < key[v] then key[v] w(u, v) DECREASE-KEY [v] u

16 Example of Prim s algorithm A V \ A /6/ CMPS 2200 Intro. to Algorithms 16 u EXTRACT-MIN(Q) for each v Adj[u] do if v Q and w(u, v) < key[v] then key[v] w(u, v) DECREASE-KEY [v] u

17 Example of Prim s algorithm A V \ A /6/ CMPS 2200 Intro. to Algorithms 1 u EXTRACT-MIN(Q) for each v Adj[u] do if v Q and w(u, v) < key[v] then key[v] w(u, v) DECREASE-KEY [v] u

18 Example of Prim s algorithm A V \ A /6/ CMPS 2200 Intro. to Algorithms 1 u EXTRACT-MIN(Q) for each v Adj[u] do if v Q and w(u, v) < key[v] then key[v] w(u, v) DECREASE-KEY [v] u

19 Example of Prim s algorithm A V \ A /6/ CMPS 2200 Intro. to Algorithms 19 u EXTRACT-MIN(Q) for each v Adj[u] do if v Q and w(u, v) < key[v] then key[v] w(u, v) DECREASE-KEY [v] u

20 Example of Prim s algorithm A V \ A /6/ CMPS 2200 Intro. to Algorithms 20 u EXTRACT-MIN(Q) for each v Adj[u] do if v Q and w(u, v) < key[v] then key[v] w(u, v) DECREASE-KEY [v] u

21 Example of Prim s algorithm A V \ A /6/ CMPS 2200 Intro. to Algorithms 21 u EXTRACT-MIN(Q) for each v Adj[u] do if v Q and w(u, v) < key[v] then key[v] w(u, v) DECREASE-KEY [v] u

22 Example of Prim s algorithm A V \ A /6/ CMPS 2200 Intro. to Algorithms 22 u EXTRACT-MIN(Q) for each v Adj[u] do if v Q and w(u, v) < key[v] then key[v] w(u, v) DECREASE-KEY [v] u

23 V times Analysis of Prim ( V ) total degree(u) times Q V key[v] for all v V key[s] 0 for some arbitrary s V while Q do u EXTRACT-MIN(Q) for each v Adj[u] do if v Q and w(u, v) < key[v] then key[v] w(u, v) [v] u Handshaking Lemma ( E ) implicit DECREASE-KEY s. Time = ( V ) T EXTRACT -MIN + ( E ) T DECREASE-KEY 11/6/ CMPS 2200 Intro. to Algorithms 2

24 Analysis of Prim (continued) Time = ( V ) T EXTRACT -MIN + ( E ) T DECREASE-KEY Q T EXTRACT -MIN T DECREASE-KEY Total array O( V ) O(1) O( V 2 ) binary heap Fibonacci heap O(log V ) O(log V ) O( E log V ) O(log V ) amortized O(1) O( E + V log V ) amortized worst case 11/6/ CMPS 2200 Intro. to Algorithms 24

25 Kruskal s algorithm IDEA (again greedy): Repeatedly pick edge with smallest weight as long as it does not form a cycle. The algorithm creates a set of trees (a forest) During the algorithm the added edges merge the trees together, such that in the end only one tree remains Correctness: Next edge e connects two components T 1, T 2. It is the lightest edge which does not produce a cycle, hence it is also the lightest edge between T 1 and V\T 1 and therefore satisfies the cut property. 11/6/ CMPS 2200 Intro. to Algorithms 25

26 Example of Kruskal s algorithm MST edges a set repr. b e S={ {a},{b},{c},{d},{e} a {f},{g},{h} } 5 9 c d f g h Every node is a single tree. 11/6/ CMPS 2200 Intro. to Algorithms 26

27 Example of Kruskal s algorithm MST edges a set repr. b e S={ {a},{b},{c},{d},{f} a {g}, {e, h} } 5 9 c d f g h Edge merged two singleton trees. 11/6/ CMPS 2200 Intro. to Algorithms 2

28 Example of Kruskal s algorithm MST edges a set repr. b e S={ {a},{d},{f}, {g} a {e, h}, {b, c} } 5 9 c d f g h 11/6/ CMPS 2200 Intro. to Algorithms 2

29 Example of Kruskal s algorithm MST edges a set repr. b e S={ {d},{f}, {g} a {e, h}, {a, b, c} } 5 9 c d f g h 11/6/ CMPS 2200 Intro. to Algorithms 29

30 Example of Kruskal s algorithm MST edges a set repr. b e S={ {d}, {g} a {e, h}, {a, b, c, f} } 5 9 c d f g h 11/6/ CMPS 2200 Intro. to Algorithms 0

31 Example of Kruskal s algorithm MST edges a set repr. b e S={ {d}, {g} a {e, h, a, b, c, f} } 5 9 c d f g h Edge merged the two bigger trees. 11/6/ CMPS 2200 Intro. to Algorithms 1

32 Example of Kruskal s algorithm MST edges a set repr. b e S={ {g} a {e, h, a, b, c, f, d} } 5 9 c d f g h 11/6/ CMPS 2200 Intro. to Algorithms 2

33 Example of Kruskal s algorithm MST edges a set repr. b e S={ {g} a {e, h, a, b, c, f, d} } 5 9 c d f g h Skip edge as it would cause a cycle. 11/6/ CMPS 2200 Intro. to Algorithms

34 Example of Kruskal s algorithm MST edges a set repr. b e S={ {g} a {e, h, a, b, c, f, d} } 5 9 c d f g h Skip edge 12 as it would cause a cycle. 11/6/ CMPS 2200 Intro. to Algorithms 4

35 Example of Kruskal s algorithm MST edges a set repr. b e S={ {g} a {e, h, a, b, c, f, d} } 5 9 c d f g h Skip edge as it would cause a cycle. 11/6/ CMPS 2200 Intro. to Algorithms 5

36 Example of Kruskal s algorithm MST edges a set repr. b e S={{e, h, a, b, c, f, d, g} } 6 a 12 5 c 9 d f g h 11/6/ CMPS 2200 Intro. to Algorithms 6

37 Disjoint-set data structure (Union-Find) Maintains a dynamic collection of pairwise-disjoint sets S = {S 1, S 2,, S r }. Each set S i has one element distinguished as the representative element. Supports operations: O(1) MAKE-SET(x): adds new set {x} to S O((n)) UNION(x, y): replaces sets S x, S y with S x S y O((n)) FIND-SET(x): returns the representative of the set S x containing element x 1 < (n) < log*(n) < log(log(n)) < log(n) 11/6/ CMPS 2200 Intro. to Algorithms

38 Union-Find Example S = {} MAKE-SET(2) S = {{2}} MAKE-SET() S = {{2}, {}} MAKE-SET(4) S = {{2}, {}, {4}} FIND-SET(4) = 4 S = {{2, 4}, {}} UNION(2, 4) The representative is underlined FIND-SET(4) = 2 MAKE-SET(5) S = {{2, 4}, {}, {5}} UNION(4, 5) S = {{2, 4, 5}, {}} 11/6/ CMPS 2200 Intro. to Algorithms

39 Kruskal s algorithm IDEA: Repeatedly pick edge with smallest weight as long as it does not form a cycle. S S will contain all MST edges O( V ) for each v V do MAKE-SET(v) O( E log E ) Sort edges of E in non-decreasing order according to w O( E For each (u,v) E taken in this order do if FIND-SET(u) FIND-SET(v) u,v in different trees O(( V )) S S {(u,v)} UNION(u,v) Edge (u,v) connects the two trees Runtime: O( V + E log E + E ( V )) = O( E log E ) 11/6/ CMPS 2200 Intro. to Algorithms 9

40 MST algorithms Prim s algorithm: Maintains one tree Runs in time O( E log V ), with binary heaps. Kruskal s algorithm: Maintains a forest and uses the disjoint-set data structure Runs in time O( E log E ) Best to date: Randomized algorithm by Karger, Klein, Tarjan [199]. Runs in expected time O( V + E ) 11/6/ CMPS 2200 Intro. to Algorithms 40