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# PQ - CSE 4101/5101 Prof Andy Mirzaian Priority Queues...

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CSE 4101/5101 Priority Queues Prof. Andy Mirzaian

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TOPICS Ø Priority Queues Ø Leftist Heaps Ø Skew Heaps Ø Binomial Heaps Ø Fibonacci Heaps Ø Recent Developments 2
References: [CLRS 2nd edition] chapters 19, 20 or [CLRS 3rd edition] chapter 19 & Problem 19-2 (pp:527-529) Lecture Notes 4, 5 AAW animations 3

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Priority Queues 4
Basic Priority Queues § Each item x has an associated priority denoted key[x]. § Item priorities are not necessarily distinct and may be time varying. § A Priority Queue Q is a set of prioritized data items that supports the following basic priority queue operations: Insert(x,Q): insert (new) item x into Q. (Duplicate priorities allowed.) DeleteMin(Q): remove and return the minimum key item from Q. § Notes: 1. Priority Queues do not support the Dictionary Search operation. 2. DeleteMin for min-PQ: the lower the key the higher the priority. 3. DeleteMax for max-PQ: the higher the key the higher the priority. 4. More PQ operations shortly. § Example: An ordinary queue can be viewed as a priority queue where the priority of an item is its insertion time. 5

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HEAP § Heap Ordered Tree: Let T be a tree that holds one item per node. T is (min-) heap-ordered iff nodes x root(T): key[x] key[parent(x)]. 10 40 14 28 18 36 -  super-root § [ Note: Any subtree of a heap-ordered tree is heap-ordered.] § Heap: a forest of one or more node-disjoint heap-ordered trees. 20 40 22 60 24 32 56 38 56 76 46 52 6
Some Applications § Sorting and Selection. § Scheduling processes with priorities. § Priority driven discrete event simulation. § Many Graph and Network Flow algorithms, e.g., Prim’s Minimum Spanning Tree algorithm, Dijkstra’s Shortest Paths algorithm, Max Flow, Min-Cost Flow, Weighted Matching, … § Many problems in Computational Geometry, e.g., plane sweep (when not all events are known in advance). 7

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More Heap Operations Mergeable Heap Operations: MakeHeap(e): generate and return a heap that contains the single item e with a given key key[e]. Insert(e,H): insert (new) item e, with its key key[e], into heap H. FindMin(H): return the minimum key item from heap H. DeleteMin(H): remove and return the minimum key item from heap H. Union(H1, H2): return the heap that results from replacing heaps H1 and H2 by their disjoint union H1  H2. (This destroys H1 and H2.) DecreaseKey(x,K,H): Given access to node x of heap H with key[x] > K, decrease Key[x] to K and update heap H. Delete(x,H): Given access to node x of heap H, remove the item at node x from Heap H. 8
Beyond Standard Heap Williams[1964]: Array based binary heap (for HeapSort). See [CLRS ch 6]. Time complexities of mergeable heap operations on this structure are: O(1) MakeHeap(e), FindMin(H). O(log n) Insert(e,H), DeleteMin(H) (n = |H|) O(n) Union(H1, H2) (n = |H1| + |H2|) Insert(e,H) H’  MakeHeap(e) H  Union(H, H’) end r L R DeleteMin(H) r  root[H] if r = nil then return error MinKey  Key[r] root[H]  Union(left[r], right[r]) return MinKey end Insert and DeleteMin via Union in a pointer -based (binary tree) heap: Can we improve Union in order to improve both Insert & DeleteMin?

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