Prim’s Minimum Spanning Tree Algorithm Neil Tang 4/1/2008

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Presentation transcript:

Prim’s Minimum Spanning Tree Algorithm Neil Tang 4/1/2008 CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Class Overview The minimum spanning tree problem An application Prim’s algorithm Implementation and time complexity CS223 Advanced Data Structures and Algorithms

Minimum Spanning Tree Problem The cost of a tree: The sum of the weights of all links on the tree. The Minimum Spanning Tree (MST) problem: Given a weighted undirected graph G, find a minimum cost tree connecting all the vertices on the graph CS223 Advanced Data Structures and Algorithms

Minimum Spanning Tree Problem CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms An Application Broadcasting problem in Computer Networks: Find the minimum cost route to send a package from a source node to all the other nodes in the network. CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Prim’s Algorithm CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Prim’s Algorithm CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Prim’s Algorithm CS223 Advanced Data Structures and Algorithms

Implementation and Time Complexities Difference between Prim and Dijkstra: 1) Arbitrarily pick a node to start with; 2) Relaxation dw=min(dw, cw,v) Trivial: O(|V|2 + |E|) = O(|V|2) Heap: deleteMin |V| times + decreaseKey |E| times O(|V|log|V| + |E|log|V|) = O (|E|log|V|) CS223 Advanced Data Structures and Algorithms