Inken Jana PRIM

Prawo Unii Europejskiej : zagadnienia systemowe / pod redkcja Jana Barcza südosteuropäischen Kandidatenländern / Arndt Sändig Inken Baumgartner.

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Prim’s Algorithm Prim’s algorithm (Jarník 1930 Dijkstra 1957 Prim 1959) Initialize F = ? S = {s} for some arbitrary vertex s Repeat until S has V vertices: – let f be smallest edge with exactly one endpoint in S – add other endpoint to S – add edge f to F 1 3 8 2 6 7 4 5 8 S 1 S 2 6 5 4-F 1-2 1-6 6-5 5-4 6 Prim’s Algorithm

14 1 2 Prim’s Algorithm What Key Property tells us in most general terms is that any algorithm conforming to the following greedy schema is guaranteed to work Algorithm: Prim(G = (V;E)) Initialize the tree F to be empty; while jFj< jVj 1 do Find a cut that does not have any edge in the current tree; Add the min cost edge of the cut to the

PRIM (Patient Rule Induction Method) is a data mining technique introduced by Friedman and Fisher (1999) Its objective is to nd subregions in the input space with relatively high (low) values for the target variable By construction PRIM directly targets these regions rather than indirectly through the estimation of a regression function

Prim’salgorithm:anexample a 0 b 4 c 8 d 7 e 9 f 4 g 2 h 1 i 2 4 8 8 11 7 4 2 9 14 10 1 2 6 7 Therearenomorenodesleftsowe’redone 14 Prim’salgorithm

Implementing Prim’s Algorithm Implementing Prim’s Algorithm Prim ComputeMST E is the set of all edges in G S = f1g T is empty (* T will store edges of a MST *) while S 6=V do pick e = (v;w) 2E such that v 2S and w 2V S e has minimum cost T = T [e S = S [w return the set T Analysis 1 Number of iterations = O(n) where n is number of vertices

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Developed in 1957 by computer scientist Robert C Prim Bell System Technical Journal 36 (1957) pp 1389–1401 Developed about 1956 by EdsgerDijkstraand published in in 1959 NumerischeMathematik1 269–271 (1959) MST using “Prim’s algorithm” (should be called “JPD algorithm”)

Prim’s Algorithm: Q: What must be true about the connectivity of a graph when running an MST algorithm? what does this imply about the relationship between n and m? Q: Suppose we built a new heap that optimized the decrease-key operation where decreasing the value of a key in a heap updates the Prim’s Algorithm runtime? Shortest Path Home:

Prim and Dijkstra: Ef?ciency and Correctness Luay Nakhleh 1 Prim’s Algorithm In class we saw Prim’s algorithm for computing a minimum spanning tree (MST) of a weighted undirected graph g The pseudo-code is given in Algorithm 1 Algorithm 1: Prim Input:Weighted undirected graph g = (V;E;w) Output: E T the edges of an MST of g 1 V

Prim's Algorithm Breadth First Search Depth First Search 1 Prim's Algorithm In Prim's algorithm the set A maintained by the algorithm forms a single tree The tree starts from an arbitrary root vertex and grows until it spans all the vertices in V At each step a light edge is added to the tree A which connects A to an isolated vertex of G A