1 / 11

Dr. O.Bushehrian

Algorithm design minimum Spanning Trees. Dr. O.Bushehrian. A spanning tree for G is a connected subgraph that contains all the vertices in G and is a tree. 2. Y. 1. 6. 2. 1. Y. 1. 6. 3. 3. 3. 4. 4. 4. 2. 3. 1. 1. 5. 4. 5. 1.

zamir
Download Presentation

Dr. O.Bushehrian

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Algorithm designminimum Spanning Trees Dr. O.Bushehrian

  2. A spanning treefor G is a connected subgraph that contains all the vertices in G and is a tree.

  3. 2 Y 1 6 2 1 Y 1 6 3 3 3 4 4 4 2 3 1 1 5 4 5 1 فاصله گره 5 از مجموعه Y: min(4,3)=3 Distance[5]=3, nearest[5]=2 Distance[4]=1, nearest[4]=3 فاصله گره 4 از مجموعه Y: min(1,2)=1

  4. A weighted graph and the steps in Prim's algorithm for that graph

  5. The array representation W of the graph W[i][J] = weight on edge if there is an edge between vi and vJ ∞ if there is no edge between vi and vJ 0 if i=J

  6. Prim's Algorithm: nearest [i] =index of the vertex in Y nearest to vi distance [i] = weight on edge between vi and the vertex indexed by nearest [i] void prim (intn, number W[] [] ) { index i, vnear; number min; edge e; index nearest [2 . . n]; number distance [2 . . n]; F = Ø; for (i = 2; i <= n; i++){ nearest [i] = 1; distance [i] = W[1] [i] ; }

  7. repeat (n - 1 times){ min = ∞ for (i = 2; i <= n; i++) { if (0 ≤ distance [i] < min) { min = distance [i]; vnear = i; } } e = edge connecting vertices indexed by vnear and nearest [vnear]; add e to F; distance [vnear] = - 1; for (i = 2; i <= n; i++) { if (W[i] [vnear] < distance [i]){ distance = W[i] [vnear]; nearest [i] = vnear; } } } } T(n) = 2 (n - 1) (n - 1) ∊ Θ (n2).

  8. Kruskal'sAlgorithm weighted graph and the steps in Kruskal's algorithm for that graph.

  9. void kruskal (int n, intm, set_of_edges E, set_of_edgesF) { index i, j; set_pointer p, q; edge e; Sort the m edges in E by weight in nondecreasing order; F = Ø; while (number of edges in F is less than n - 1){ e = edge with least weight not yet considered; i, j = indices of vertices connected by e; p = find(i); q = find(j); if (! equal(p, q)){ merge(p, q); add e to F; } } }

  10. Analysis of Krukal's Algorithm Worst-case Time-Complexity W ( m, n ) €Ѳ ( m log m) m = n (n-1) / 2 €Ѳ ( n2 ) W ( m, n ) €Ѳ(n2log n2) = Ѳ (n22 log n) = Ѳ (n2 log n)

  11. Comparing Prim's Algorithm with Kruskal'sAlgorithm Prim's Algorithm: T(n) ∊ θ(n2) Kruskal's Algorithm: W (m, n) ∊ θ(mlgm) and W (m, n) ∊ θ (n2lgn) in a connected graph: n-1 <= m <= n (n-1) /2 For a graph whose number of edges m is near the low end of these limits (the graph is very sparse), Kruskal's algorithm is θ(n lg n), which means that Kruskal's algorithm should be faster. However, for a graph whose number of edges is near the high end (the graph is highly connected), Kruskal's algorithm is θ (n2 lg n), which means that Prim's algorithm should be faster.

More Related