Minimum spanning tree Prof Amir Geva Eitan Netzer

Definition A sub group of edges from weighted graph G Spanning – reach all vertex Minimal – the sum of its edges is the lowest of all spanning trees Uses – connect a network with while spending minimum money Graph need to be connective

Prim algorithm (1957) Greedy algorithm Start with an empty list of vertex. Choose starting vertex from G. Randomly or a given choice. Add edge with minimal weight that not used yet to an un explored vertex. Continue until list of vertex contain all vertex in G. Minimum edge weight data structureTime complexity (total) adjacency matrixadjacency matrix, searchingO(|V| 2 ) binary heapbinary heap and adjacency listadjacency listO((|V| + |E|) log |V|) = O(|E| log |V|) Fibonacci heapFibonacci heap and adjacency listadjacency listO(|E| + |V| log |V|)

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Kruskal's algorithm (1956) Greedy algorithm Create a “forest” F a set of trees Create a set S containing all edges of G While S is not empty and F is not a spanning tree yet Remove minimum edge from S If edge connects to trees in F combine them Else discard edge

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