Problem of the Day

Problem of the Day • At low tide, a ship docks in the harbor • Ship is 9’ above the water line when it docks • Over the side hangs a rope ladder w/ rungs 1’ apart • Tide rises at a rate of 9” per hour After 6 hours what length of ladder will still be above water?

Problem of the Day • At low tide, a ship docks in the harbor • Ship is 9’ above the water line when it docks • Over the side hangs a rope ladder w/ rungs 1’ apart • Tide rises at a rate of 9” per hour After 6 hours what length of ladder will still be above water? 9’ – the ladder will rise with the boat!

CSC 212 – Data Structures Lecture 39:Implementing Heaps (& Trees & BinaryTrees)

Priority Queue • Priority queue uses strict ordering of data • Values assigned priority when added to the queue • Priorities used to process in completely biased order • Return element only from find() & remove()

PriorityQueueOperations

Elements in a PriorityQueue • PriorityQueues use more to hold data • Details not specified, implementations may differ • Data stored has 2 items defining how it is used • PQ will only use priority – the importance of element • Element important data program actually cares about

Is PriorityQueue Linear? • PriorityQueueyet another Collection • Prioritize each element contained in the collection • PQ is organized from lowest to highest priority • Implementation not required: this could be ADT • Often use Heap (it is faster) & order is theoretical

Heaps • Binary-tree implementation with add & remove • 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’s value smaller than its children’s values • Structure must form a complete binary tree 2 5 9 7 6

Complete Binary Tree • Specific way to organize a BinaryTree • Add & remove location defined so can be discussed • For this idea, 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 2 2 5 9 5 9 7 6 7 6

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

Reheapify Down • removeMin()removes & returns root’s element • Mustremove last node to remain complete tree • Last added node’s element swapped with root’s • Then remove node from the complete tree • Reheapify process restores heap’s order • Starts at root and works down heap • If out-of-order, swap with smaller child • Stop at leaf or when node is legal

D C B A Picturing LinkedBinaryTree B A C      D

 D C B A Picturing LinkedHeap B A C      D

 D C B A Nodes in a LinkedHeap B A C      D

 References in a LinkedHeap B B A C      D A C D

Trees Recursion • Trees are recursive structure • Subtree defined by a node • C is root of this subtree • B root of this subtree • D root of this subtree • Recursive methods common • Can be used going up tree • Required going down the tree • Often easiest way to define actions A D B C E G H F K I J

Writing Recursive Methods • Common pattern for recursive methods in Java: public static intfactorial(int i) { if (i <= 1) { return 1; } else {intnextI = i – 1;int result = factorial(nextI); return i * result; } }

Writing Recursive Methods Base case: Start with check for base case • Common pattern for recursive methods in Java: public static intfactorial(inti) { if (i <= 1) { return 1;} else {intnextI = i – 1;int result = factorial(nextI); return i * result; } }

Writing Recursive Methods Base case: Solution is simple • Common pattern for recursive methods in Java: public static intfactorial(inti) {if (i <= 1) { return 1; } else {intnextI = i – 1;int result = factorial(nextI); return i * result; } }

Writing Recursive Methods • Recursive Step: • Take 1 step to solution • Common pattern for recursive methods in Java: public static intfactorial(inti) { if (i <= 1) { return 1; } else {intnextI = i – 1;int result = factorial(nextI); return i * result; } }

Writing Recursive Methods • Recursive Step: • Take 1 step to solution • Make 1 or more recursive calls • Common pattern for recursive methods in Java: public static intfactorial(inti) { if (i <= 1) { return 1; } else {intnextI = i – 1;int result = factorial(nextI);return i * result; } }

Writing Recursive Methods • Recursive Step: • Take 1 step to solution • Make 1 or more recursive calls • (Simple process computes result) • Common pattern for recursive methods in Java: public static intfactorial(inti) { if (i <= 1) { return 1; } else {intnextI = i – 1;int result = factorial(nextI);return i * result; } }

Your Turn • Get into your groups and complete activity

For Next Lecture • Week #12 assignment due tomorrow • Programming Assignment #2due ??? • Will send out if enough reviews received by 5PM today • BinaryTree, Heap, & PriorityQueue Quiz Wed.

Problem of the Day