Explore Minimum Spanning Trees Algorithms

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Discover the concept of Minimum Spanning Trees (MST) in graph theory, focusing on finding the lowest-cost way to connect all points while ensuring the solution forms a tree structure. Learn about greedy algorithms like Kruskal's Algorithm and high-level approaches for constructing MST. Dive into the principles of selecting safe edges and avoiding cycles in MST construction.

  • Spanning Trees
  • Graph Theory
  • Minimum Spanning Trees
  • Greedy Algorithm
  • Kruskal
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  1. Minimum Spanning Trees

  2. b 3 4 f a 5 2 G=(V,E) 1 1 e 5 7 2 c 9 d 3 g h 4 . , , . Minimum Spanning Trees 2

  3. Minimum Spanning Trees (MST) Find the lowest-cost way to connect all of the points (the cost is the sum of weights of the selected edges). The solution must be a tree. (Why ?) A spanning tree : a subgraph that is a tree and connect all of the points. A graph may contains exponential number of spanning trees. (e.g. # spanning trees of a complete graph = nn-2.) Minimum Spanning Trees 3

  4. A High-Level Greedy Algorithm for MST A = ; while(T=(V,A) is not a spanning tree of G) { select a safe edge for A ; } The algorithm grows a MST one edge at a time and maintains that A is always a subset of some MST. An edge is safe if it can be safely added to without destroying the invariant. How to check that T is a spanning tree of G ? How to select a safe edge edge ? Minimum Spanning Trees 4

  5. MST Let V = V1 + V2, (V1,V2) = {uv | u V1& v V2}. if xy (V1,V2) and w(xy) = min {w(uv)| uv (V1,V2)}, then xy is contained in a MST. b 3 4 f a 5 2 1 1 e 5 7 2 c 9 d 3 g h 4 V2 V1 Minimum Spanning Trees 5

  6. Kruskals Algorithm (pseudo code 1) A= ; for( each edge in order by nondecreasing weight ) if( adding the edge to A doesn't create a cycle ){ add it to A; if( |A| == n 1 ) break; } How to check that adding an edge does not create a cycle? Minimum Spanning Trees 6

  7. Kruskals Algorithm ( 1/3) b 3 4 f a 5 2 1 1 e 4 7 2 9 c d 3 4 g h Minimum Spanning Trees 7

  8. Kruskals Algorithm ( 2/3) b 3 4 f a 5 2 1 1 e 4 7 2 9 c d 3 4 g h Minimum Spanning Trees 8

  9. Kruskals Algorithm ( 3/3) b 3 4 f a 5 2 1 1 e 4 7 2 9 c d 3 4 g h MST cost = 17 Minimum Spanning Trees 9

  10. Kruskals Algorithm (pseudo code 2) A = ; initial(n); // for each node x construct a set {x} for( each edge xy in order by nondecreasing weight) if ( ! find(x, y) ) { union(x, y); add xy to A; if( |A| == n 1 ) break; } find(x, y) = true iff. x and y are in the same set union(x, y): unite the two sets that contain x and y, respectively. Minimum Spanning Trees 10

  11. Prims Algorithm (pseudo code 1) ALGORITHM Prim(G) // Input: A weighted connected graph G=(V,E) // Output: A MST T=(V, A) VT { v0} // Any vertex will do; A ; for i 1 to |V| 1 do find an edge xy (VT, V VT) s.t. its weight is minimized among all edges in (VT, V VT); VT VT { y} ; A A { xy} ; Minimum Spanning Trees 11

  12. Prims Algorithm ( 1/8) b 3 4 f a 5 2 1 1 e 4 7 2 9 c d 3 4 g h Minimum Spanning Trees 12

  13. Prims Algorithm ( 2/8) b 3 4 f a 5 2 1 1 e 4 7 2 9 c d 3 4 g h Minimum Spanning Trees 13

  14. Prims Algorithm ( 3/8) b 3 4 f a 5 2 1 1 e 4 7 2 9 c d 3 4 g h Minimum Spanning Trees 14

  15. Prims Algorithm ( 4/8) b 3 4 f a 5 2 1 1 e 4 7 2 9 c d 3 4 g h Minimum Spanning Trees 15

  16. Prims Algorithm ( 5/8) b 3 4 f a 5 2 1 1 e 4 7 2 9 c d 3 4 g h Minimum Spanning Trees 16

  17. Prims Algorithm ( 6/8) b 3 4 f a 5 2 1 1 e 4 7 2 9 c d 3 4 g h Minimum Spanning Trees 17

  18. Prims Algorithm ( 7/8) b 3 4 f a 5 2 1 1 e 4 7 2 9 c d 3 4 g h Minimum Spanning Trees 18

  19. Prims Algorithm ( 8/8) b 3 4 f a 5 2 1 1 e 4 7 2 9 c d 3 4 g h MST cost = 17 Minimum Spanning Trees 19

  20. Prims Algorithm (pseudo code 2) Built a priority queue Q for V with key[u] = u V; key[v0] = 0; [v0] = Nil; // Any vertex will do While (Q ) { u = Extract-Min(Q); for( each v Adj(u) ) if (v Q && w(u, v) < key[v] ) { [v] = u; key[v] = w(u, v); Change-Priority(Q, v, key[v]); } } Minimum Spanning Trees 20

  21. Minimum Spanning Tree ( ) Let n = |V(G)|, m =|E(G)|. Execution time of Kruskal s algorithm: (use union-find operations, or disjoint-set unions) O(m logm ) = O(m logn ) Running time of Prim s algorithm: adjacency lists + (binary or ordinary) heap: O((m+n) logn ) = O(m logn ) adjacency matrix + unsorted list: O(n2) adjacency lists + Fibonacci heap: O(n logn + m) Minimum Spanning Trees 21

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