Minimum spanning tree

Transcription

1 Minimum spanning tree Jean Cousty MorphoGraph and Imagery 2011 J. Cousty : MorphoGraph and Imagery 1/17

2 Outline of the lecture 1 Minimum spanning tree 2 Cut theorem for MST 3 Kruskal algorithm J. Cousty : MorphoGraph and Imagery 2/17

3 Minimum spanning tree Edge weighted graph G denotes a connected undirected graph (E, Γ) J. Cousty : MorphoGraph and Imagery 3/17

4 Minimum spanning tree Edge weighted graph G denotes a connected undirected graph (E, Γ) W denotes a map from Γ into R W (u) is called the weight (or value) of u ( u Γ) J. Cousty : MorphoGraph and Imagery 3/17

5 Minimum spanning tree Edge weighted graph G denotes a connected undirected graph (E, Γ) W denotes a map from Γ into R W (u) is called the weight (or value) of u ( u Γ) The pair (G, W ) is called an (edge-) weighted graph J. Cousty : MorphoGraph and Imagery 3/17

6 Minimum spanning tree Spanning subgraph Definition A spanning subgraph of G is a graph (E, Γ ) such that Γ Γ J. Cousty : MorphoGraph and Imagery 4/17

7 Minimum spanning tree Weight of a subgraph Definition Let G = (E, Γ ) be a spanning subgraph of G The weight of G, denoted by W (G ), is the sum of the weights of the edges that belong to Γ W (G) = {W (u) u Γ } J. Cousty : MorphoGraph and Imagery 5/17

8 Minimum spanning tree Minimum spanning tree (MST) Definition A minimum spanning tree (for (G, W )) is a connected spanning subgraph G = (E, Γ ) of G whose weight is minimum: for any connected spanning subgraph G of G, if W (G ) W (G ) then W (G ) = W (G ) J. Cousty : MorphoGraph and Imagery 6/17

9 Minimum spanning tree Minimum spanning tree (MST) Definition A minimum spanning tree (for (G, W )) is a connected spanning subgraph G = (E, Γ ) of G whose weight is minimum: for any connected spanning subgraph G of G, if W (G ) W (G ) then W (G ) = W (G ) Property If G is an MST, then G is a tree J. Cousty : MorphoGraph and Imagery 6/17

10 Minimum spanning tree Illustration J. Cousty : MorphoGraph and Imagery 7/17

11 Minimum spanning tree Maximum spanning tree (MaxST) Definition A maximum spanning tree (for (G, W )) is a spanning tree (E, Γ ) of maximum weight: for any spanning tree G, if W (G ) W (G ) then W (G ) = W (G ) J. Cousty : MorphoGraph and Imagery 8/17

12 Minimum spanning tree Maximum spanning tree (MaxST) Definition A maximum spanning tree (for (G, W )) is a spanning tree (E, Γ ) of maximum weight: for any spanning tree G, if W (G ) W (G ) then W (G ) = W (G ) Property G is a MaxST of (G, W ) if and only if G is a MST of (G, W ) J. Cousty : MorphoGraph and Imagery 8/17

13 Minimum spanning tree MST algorithm and safe edges Generic MST Algorithm ( Data: (E, Γ, W ) ; Result: Γ such that (E, Γ ) is an MST) n := E ; Γ := ; k := 0 ; While k < n 1 do Find an edge u that is safe for Γ Γ := Γ {u} J. Cousty : MorphoGraph and Imagery 9/17

14 Minimum spanning tree MST algorithm and safe edges Generic MST Algorithm ( Data: (E, Γ, W ) ; Result: Γ such that (E, Γ ) is an MST) n := E ; Γ := ; k := 0 ; While k < n 1 do Find an edge u that is safe for Γ Γ := Γ {u} Definition Let (E, Γ ) be a subgraph of a MST Let u Γ \ Γ u is safe for Γ if (E, Γ {u}) is also a subgraph of an MST J. Cousty : MorphoGraph and Imagery 9/17

15 Cut theorem for MST Cut and MST Question How to efficiency detect an edge that is safe? J. Cousty : MorphoGraph and Imagery 10/17

16 Cut theorem for MST Cut Definition Let Γ Γ The cut induced by Γ is the subset of Γ that contains the edges that link two distinct connected components of the graph (E, Γ ) J. Cousty : MorphoGraph and Imagery 11/17

17 Cut theorem for MST Cut Definition Let Γ Γ The cut induced by Γ is the subset of Γ that contains the edges that link two distinct connected components of the graph (E, Γ ) A subset C of Γ is a cut if there exists Γ Γ such that C is the cut induced by Γ J. Cousty : MorphoGraph and Imagery 11/17

18 Cut theorem for MST Cut theorem for MST Theorem Let (E, Γ ) be a subgraph of an MST J. Cousty : MorphoGraph and Imagery 12/17

19 Cut theorem for MST Cut theorem for MST Theorem Let (E, Γ ) be a subgraph of an MST Let C be a cut that is a subset of the cut induced by Γ J. Cousty : MorphoGraph and Imagery 12/17

20 Cut theorem for MST Cut theorem for MST Theorem Let (E, Γ ) be a subgraph of an MST Let C be a cut that is a subset of the cut induced by Γ Let u C be an edge of minimum weight in C (i.e., v C, W (u) W (v)) J. Cousty : MorphoGraph and Imagery 12/17

21 Cut theorem for MST Cut theorem for MST Theorem Let (E, Γ ) be a subgraph of an MST Let C be a cut that is a subset of the cut induced by Γ Let u C be an edge of minimum weight in C (i.e., v C, W (u) W (v)) Then, u is safe for Γ J. Cousty : MorphoGraph and Imagery 12/17

22 Kruskal algorithm MST computation KRUSKAL Algorithm ( Data: a connected graph (E, Γ, W ) ; Result: Γ such that (E, Γ ) is an MST) n := E ; Γ := ; k := 0 ; i := 1; Sort the edges in Γ in increasing order of weight: Find the sequence (u 1,..., u m ) such that Γ = {u 1,..., u m } and W (u i 1 ) W (u i ), i {2,..., m} While k < n 1 Do {x i, y i } := u i ; If x i / the connected component containing y i in (E, Γ ) Then Γ := Γ {u i }; k := k + 1; i := i + 1; J. Cousty : MorphoGraph and Imagery 13/17

23 Kruskal algorithm Execution example J. Cousty : MorphoGraph and Imagery 14/17

24 Kruskal algorithm Proof of the algorithm When the test If is positive, the edge u is safe for Γ This can be proved by the cut theorem applied to the cut induced by Γ J. Cousty : MorphoGraph and Imagery 15/17

25 Kruskal algorithm Complexity of Kruskal Algorithm Complexity Complexity of a sort (O(m log(m)) for quick sort) J. Cousty : MorphoGraph and Imagery 16/17

26 Kruskal algorithm Complexity of Kruskal Algorithm Complexity Complexity of a sort (O(m log(m)) for quick sort) Complexity of the While loop: O(m) J. Cousty : MorphoGraph and Imagery 16/17

27 Kruskal algorithm Complexity of Kruskal Algorithm Complexity Complexity of a sort (O(m log(m)) for quick sort) Complexity of the While loop: O(m) Complexity of the If test: O(m n) (connected component algorithm seen in the course) J. Cousty : MorphoGraph and Imagery 16/17

28 Kruskal algorithm Complexity of Kruskal Algorithm Complexity Complexity of a sort (O(m log(m)) for quick sort) Complexity of the While loop: O(m) Complexity of the If test: O(m n) (connected component algorithm seen in the course) Overall complexity: O(m n + m log(m)) J. Cousty : MorphoGraph and Imagery 16/17

29 Kruskal algorithm Complexity of Kruskal Algorithm Complexity Complexity of a sort (O(m log(m)) for quick sort) Complexity of the While loop: O(m) Complexity of the If test: O(m n) (connected component algorithm seen in the course) Overall complexity: O(m n + m log(m)) The complexity of the If test can be reduced to quasi linear time using Tarjan s union of disjoint set algorithm J. Cousty : MorphoGraph and Imagery 16/17

30 Kruskal algorithm Prim and Boruvka Algorithm These are other algorithms for computing an MST Also a version of the generic algorithm Their invariants also rely on the cut theorem Read eppstein/161/ html J. Cousty : MorphoGraph and Imagery 17/17