21 - Binomial Heaps Data Structures and Algorithms Andrei...

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Unformatted text preview: Binomial Heaps Data Structures and Algorithms Andrei Bulatov Algorithms – Binomial Heaps 21-2 Priority Queues • Supports the following operations. – Insert element x. – Return min element. – Return and delete minimum element. ecrease key of element x to k – Decrease key of element x to k . • Applications. – Dijkstra's shortest path algorithm. – Prim's MST algorithm. – Event-driven simulation. – Huffman encoding. – Heapsort. – . . . Algorithms – Binomial Heaps 21-3 Dijkstra’s Algorithm set S:={s} and d(s):=0 while S ≠ V do pick a node v not from S such that the value s minimal )} ( ) ( { min : ) ( ' ), , ( e len u d v d S u v u e + = ∈ = is minimal set S:=S ∪ {v}, and d(v):=d’(v) endwhile Algorithms – Binomial Heaps 21-4 Dijkstra’s Algorithm: PQ Style call PQinit set S:=V set key(s):=0 PQinsert(s) for each v ∈ V-{s} key(v):= ∞ call PQinsert(v) while not PQisempty set v:=Pqdelmin set S:=S-{v} for each w ∈ Q such that (v,w) ∈ E if key(w)>key(v)+len(v,w) then call PQdecrease(w,key(v)+len(v,w)) Algorithms – Binomial Heaps 21-5 Priority Queues make-heap Operation insert find-min elete in 1 Binary log N 1 g N 1 Binomial log N log N g N 1 Fibonacci * 1 1 g N 1 Relaxed 1 1 g N 1 Linked List 1 N Heaps Dijkstra/Prim 1 make-heap |V| insert |V| delete-min |E| decrease-key delete-min union decrease-key delete log N N log N log N log N log N log N log N log N 1 1 log N log N 1 1 log N N 1 1 N is-empty 1 1 1 1 1 O(|E| + |V| log |V|) O(|E| log |V|) O(|V| 2 ) Algorithms – Binomial Heaps 21-6 Binomial Tree Recursive definition:- B is a single node...
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This note was uploaded on 11/19/2009 for the course CS CMPT 307 taught by Professor A.bulatov during the Fall '09 term at Simon Fraser.

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21 - Binomial Heaps Data Structures and Algorithms Andrei...

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