Problem of the Day  To get into club, must get past bouncer  Recorded correct responses of those before you  Guard said five & response was four  Guard said three & response was five  Guard said twelve & response was six  Guard said six & response was three  Guard now says to you ten, what do you say?

Problem of the Day  To get into club, must get past bouncer  Recorded correct responses of those before you  Guard said five & response was four  Guard said three & response was five  Guard said twelve & response was six  Guard said six & response was three  Guard now says to you ten, what do you say? Three!

CSC 212 – Data Structures

Priority Queue ADT  Prioritizes Entry s using their keys  For Entry s with equal priorities, order not specified  Priority given to each value when added to PQ  Normally, the priority not changeable while in PQ  Access single Entry : one with the lowest priority  Returns Entry using min() or removeMin()  Location are imaginary – only smallest matters

Heaps  Binary-tree based PQ implementation  Still structured using parent-child relationship  At most 2 children & 1 parent for each node in tree  Heaps must also satisfy 2 additional properties  Parent at least as important as its children  Structure must form a complete binary tree

BinaryTree   Picturing Linked BinaryTree B C A D   BACD BinaryTree root size  4

Legal CompleteBinaryTree  ADT which extends BinaryTree  Add & remove methods defined plus those inherited  For this ADT, trees must maintain specific shape  Fill lowest level first, then can start new level below it Illegal

CompleteBinaryTree  ADT which extends BinaryTree  Add & remove methods defined plus those inherited  For this ADT, trees must maintain specific shape  Fill lowest level first, then can start new level below it  Lowest level must be filled in from left-to-right Legal Illegal

What Is Purpose of a Heap?  Root has critical Entry that we always access  Entry at root always has smallest priority in Heap  O(1) access time in min() without any real effort  CompleteBinaryTree makes insert() easy  Create leftmost child on lowest level  When a level completes, start next one  Useful when:

Upheap  Insertion may violate heap-order property  Upheap immediately after adding new Entry  Goes from new node to restore heap’s order  Compare priority of node & its parent  If out of order, swap node's Entry s  Continue upheaping from parent node  Stop only when either case occurs:  Found properly ordered node & parent  Binary tree's root is reached

6 insert() in a Heap

6 1 Start your upheaping!

1 6 Upheaping Must Continue

2 6 Upheaping Sounds Icky

2 6 Check If We Should Continue

2 6 Stop At The Root

2 6 insert() Once Again

2 6 Upheaping Begins Anew

2 6 Maintain Heap Order Property

2 6 We Are Done With This Upheap!

Removing From a Heap  removeMin() must kill Entry at heap’s root  For a complete tree, must remove last node added  How to reconcile these two different demands?  Removed node's Entry moved to the root  Then remove node from the complete tree  Heap's order preserved by going down

Removing From a Heap  removeMin() must kill Entry at heap’s root  For a complete tree, must remove last node added  How to reconcile these two different demands?  Removed node's Entry moved to the root  Then remove node from the complete tree  Heap's order preserved by going down

Downheap  Restores heap’s order during removeMin()  Downheap work starts at root  Swap with smallest child, if at least out-of-order  Downheaping continues with old smallest child  Stop at leaf or when node is legal

5 Before removeMin() is called

5 Move Last Entry Up To Root

5 Compare Parent W/Smaller Child 9 2 7

5 Continue Downheaping W/Node 2 9 7

5 Swap If Out Of Order 2 7 9

5 Check If We Should Continue 2 7 9

5 Stop When We Reach a Leaf 2 7 9

Implementation Excuses  upheap & downheap travel height of tree  O (log n ) running time for each of these  Serves as bound for adding & removing from PQ  What drawbacks does heap have?  PriorityQueue can be faster using Sequence  Only for specific mix of operations, however  PriorityQueue using heaps are fastest overall

What Is Purpose of a Heap?  Root has critical Entry that we always access  Entry at root always has smallest priority in Heap  O(1) access time in min() without any real effort  CompleteBinaryTree makes insert() easy  Create leftmost child on lowest level  When a level completes, start next one  Useful when:  Will eventually add & remove everything  Equally (large) numbers of adds & removes

Final Exam Schedule  Lab Mastery Exam is: Tues., Dec. 14 th from 2:45PM – 3:45PM in OM 119  Final Exam is: Fri., Dec. 17 th from 8AM – 10AM in OM 200