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This lecture is part of lecture series for Design and Analysis of Algorithms course. This course was taught by Dr. Bhaskar Sanyal at Maulana Azad National Institute of Technology. It includes: Prim, Algorithm, Running, Time, Priority, Implementation, Performance, Tree, Queue, Kruskal, Greedy
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MST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[ u ] if (v Q and w( u,v ) < key[ v ]) p[v] = u; key[v] = w(u,v);
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MST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[ u ] if (v Q and w( u,v ) < key[ v ]) p[v] = u; key[v] = w(u,v);
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MST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[ u ] if (v Q and w( u,v ) < key[ v ]) p[v] = u; key[v] = w(u,v);
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MST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[ u ] if (v Q and w( u,v ) < key[ v ]) p[v] = u; key[v] = w(u,v);
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MST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[ u ] if (v Q and w( u,v ) < key[ v ]) p[v] = u; key[v] = w(u,v);
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Docsity.com
MST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[ u ] if (v Q and w( u,v ) < key[ v ]) p[v] = u; key[v] = w(u,v);
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Docsity.com
MST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[ u ] if (v Q and w( u,v ) < key[ v ]) p[v] = u; DecreaseKey(v, w(u,v));
delete the smallest element from the min-heap
decrease an element’s value in the min-heap (outline an efficient algorithm for it) Docsity.com
MST-Prim(G, w, r) Q = V[G]; for each u Q key[u] = ; key[r] = 0; p[r] = NULL; while (Q not empty) u = ExtractMin(Q); for each v Adj[ u ] if (v Q and w( u,v ) < key[ v ]) p[v] = u; DecreaseKey(v, w(u,v));
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» Grow a single tree by repeatedly adding the least cost edge that connects a vertex in the existing tree to a vertex not in the existing tree Intermediary solution is a subtree
» Grow a tree by repeatedly adding the least cost edge that does not introduce a cycle among the edges included so far Intermediary solution is a spanning forest
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Kruskal’s Algorithm
(High-Level Pseudocode)
Kruskal(G) //Input: A weighted connected graph G = <V, E> //Output: ET --- the set of edges composing MST of G Sort E in nondecreasing order of the edge weight ET = ; encounter = 0 //initialize the set of tree edges and its size k = 0 //initialize the number of processed edges while encounter < |V| - 1 k = k+ if ET {ek} is acyclic ET = ET {ek}; encounter = encounter + 1 return ET
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Kruskal() { T = ; for each v V MakeSet(v); sort E by increasing edge weight w for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); }
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Kruskal() { T = ; for each v V MakeSet(v); sort E by increasing edge weight w for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); }
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Kruskal() { T = ; for each v V MakeSet(v); sort E by increasing edge weight w for each (u,v) E (in sorted order) if FindSet(u) FindSet(v) T = T U {{u,v}}; Union(FindSet(u), FindSet(v)); }
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