priorityQueue-ADTLPO.li Stores a collection of pairs ( item , - PowerPoint PPT Presentation

Priority Queue ! Heaps

priorityQueue-ADTLPO.li Stores a collection of pairs ( item , priority ) are from - Priorities ordered set some ( For simplicity , we use priorities " ) " highest priority from with 0 , 1,2 , 0 . . . - Main operations : - insert ( item , priority ) adds item with priority priority - mind . extract item with least priority removes 4. returns ) - update ( item , priority ) item to priority changes priority of

RootedBmaryTreeter or 2 children . 0 = every node has - proof . where every = proper BT - perfect same death . has the t.nl structure to implement a data We want • efficient PQS . n ) time for all operations . Eeg . 040g . We ( again ) will a particular kind of tree use

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RootedBmaryTreeter or 2 children . 0 = every node has - proof . where every = proper BT - perfect same death . has the on f. 1 Complete Binary Tree * * in the * has 2 level - order children traversal F- A complete binary tree of height h is a binary tree of height hi 1 ; 2 with 2d nodes at depth d , for every 0 ← dah 3. level order traversal visits every internal node before any leaf * * 4. every internal node is proper 4 except perhaps the last , which may have just a left child . . :* , F. 157,53 ✓ F. IF , t.sk#osITNjho7Ibbh.Jbb µ #b A µ sink ✓ 4) * G) × ✓

Binary Heap Data Structure ÷ " shape invariant - a complete binary tree ] - with vertices labelled by keys from ftp.iorities-st.keyldtkeiylpareutlvD-YIIIaut ordered set some , " for every node V ← . f- Eg ① / - ⑥ ② ③ ⑤ ' I ④ £1.0 { to ④ ② to ✓ ⑤ ⑨ × - This is the basic D. S . for implementing PQS ( Binary min - cheap ) . .

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Heap Extract - Min . - ssmo%¥ Consider : ⇒ §yj these heap we must replace the root with the smaller of its children : ① 8%8 ? f- 10 12 ok ! Not 0k

Heap Extract - Min . - To remove the litem with the ) smallest key from the heap : remove the root 1. " last leaf " 2. replace the root with the , as to maintain the shape invariant . so restore the order invariant by calling 4. - down ( root ) percolate Percolate - down more work than percolate - up , is both at because it must look children see what to do ( and the children to or may not exist ) may

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Heap Insert ! Extract - min Complexity of - 0( log a) to Extract - min take time Claim : Insert of size for heaps n . Recall : A perfect binary tree of height h has 2h " - I nodes . " the structure of the tree on h ( or II. By induction . ' we have It : If h=o then . V = 1 - I B¥s nodes assume the perfect : Consider some H70 and III. 2h " ii. des . - I binary tree of height h has + , 21h " ) s has nodes . b. t . of height htt - I . II show the ! p The tree is :} A DI¥ " = 21 " " I 1 modesty - it 2h11 + I so it has 2h " = 2. 2h11

sizeboundsoncompletebinarytrees.IE very complete binary tree with height hand in nodes satisfies : ⇐ zhtly 2h ± in Smallest : Left : I p.IT . of height h - I p.b.tv of height h . 21h - htt = 2h # nodes . It , = 2h In , we have : So log . 2h login I E login h h= Ollogn ) ⇒ Heap insert 4 extract min take time 0( log a) -

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Array-BasedBinarytteapImplemeutatimllse.co this embedding of binary tree complete a of size - n array site in a n : ⑥ ith node in ← ↳ ⑤ T o ③ /h ④ level - order traversal ⑤ ④ ¥ ④ ④ It 4 ith array element & 2 it 2 nodes 2it I • Children of node i are is node KE14 • Parent of node i 1AX 0*14*18%11 www.

Array-BasedBinarytteapImplemeutatimllse.co this embedding of binary tree complete a of size - n array site in a n : ⑥ ith node in ← ↳ ⑤ T o ③ /h ④ level - order traversal ⑤ ⑤ ④ ¥ ④ ④ It 4 ith array element ¢ 2 it 2 nodes 2it I • Children of node i are is node kindly • Parent of node i I was × Growing 4 Shrinking the tree in the array embedding is easy

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makingatteapfromase.to n keys and want to make a heap with them . Suppose you have can be done in time 04 log n ) , with n inserts . . Clearly in time 0 (a) . • Claim : the following alg does it . make - heap ( T ) S n keys XT is a complete bit . with . a) { = tht -1 down to for ( i call percolate on node i - down } 3

Howdoesmake-heapwork.7.hn/2J-l last internal node is the - the algorithm does a percolate - down at each bottom - up node , working internal . ( percolate a tree into heap - down makes a the only node violating the order if root ) is the property 0 • I ✓ \ ] N4 5TG " " 9 ' 8 16 " = I 15 - I = Leith Lnfhj - I

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