Dr. O.Bushehrian

ALGORITHM DESIGN

MINIMUM SPANNING TREES

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

گره مجموعه 5فاصله :Yازmin(4,3)=3

Distance[5]=3, nearest[5]=2

گره مجموعه 4فاصله :Yازmin(1,2)=1

Distance[4]=1, nearest[4]=3

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

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

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 (int n, 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] ;}

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).

Kruskal's Algorithm

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

void kruskal (int n, int m, set_of_edges E, set_of_edges F){ 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; } }}

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 (n2 log n2) = Ѳ (n2 2 log n) = Ѳ (n2 log n )

Comparing Prim's Algorithm with Kruskal's Algorithm

Prim's Algorithm: T(n) ∊ θ(n2)

Kruskal's Algorithm: W (m, n) ∊ θ(m lg m) and W (m, n) ∊ θ (n2 lg n)

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.