F d a b c e g 2 7 5 7 1 4 1 3 4 4 5 2 Prim’s Algorithm – an Example edge candidates choosen.
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f d a b c e g 2 7 5 7 1 4 1 3 4 4 5 2 Prim’s Algorithm – an Example edge candidates choosen
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Transcript of F d a b c e g 2 7 5 7 1 4 1 3 4 4 5 2 Prim’s Algorithm – an Example edge candidates choosen.
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Prim’s Algorithm – an Example
edge candidates
choosen
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Total weight of the MST: 14
Prim’s Algorithm - Description
MST-Prim(G, w) choose an arbitrary r V(G) Initialize T as a tree consisting of vertex r only. while T has < n vertices do let (u, v) be the lightest edge with u V(T) and
v V(G) – V(T) T T { (u, v) } return T
Correctness follows a previous corollary:
Let A be a subset of E that is included in some MST of G, and C be a connected component in the forest G = <V, A>. If (u, v) is a light edge connecting C to some other component in G , then (u, v) is safe for A.
Implementation
At every iteration, maintain the following information for each vertex v in G:
cost(v) is the weight of the lightest edge connectingv to T.
closest(v) is the vertex y in T such that cost(v) = w(v, y).
The next edge to add is (u, closest(u)). Here u is the vertex with minimum cost in V(G) – V(T).
u
closest(u)
T
cost(u)
Updating Closest Vertex
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v
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After adding (u, closest(u)), scan every neighbor v of u:
set closest(v) = u if cost(v) decreases,
update cost(v) if necessary.
MST
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Priority Queue Implementation
MST-Prim(G, w) choose r V(G) for each u V(G) do cost(u) cost(r) 0 Q V(G) // Build a priority queue keyed by cost closest(r) NIL
while Q { } do do u Extract-Min(Q)
for each v Adj(u) do if v Q and w(u, v) < cost(v) then closest(v) u
cost(v) w(u, v) // Decrease-Key
Running Time AnalysisQueue opertions:
Build-Queue creates the initial queueExtract-Min extracts the vertex of minimum costDecrease-Key decreases cost(v) for v in Q, if necessary
Heap (V) O(lg V) O(lg V) O(E lg V)
#operations 1 V 2E
Array (V) O(V) O(1) O(V ) 2
Q Build-Queue Extract-Min Decrease-Key
Time
Prim’s
