Priority Queues and Heaps in Computer Science, Study notes of Computer Science

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Lecture 17 CS2110 Fall 2008

Priority Queues

and Heaps

Announcements

• Some changes to the CS major to be

announced soon

• 2111 no longer required for new majors
• A&S and Engr students may drop 2111 until

Nov 14 without transcript annotation

• If this affects you, please see me offline

Stacks and Queues as Lists

• Stack (LIFO) implemented as list
  • (^) insert() , extract() from front of list
• Queue (FIFO) implemented as list
  • (^) insert() on back of list, extract() from front of list
• All^ Bag^ operations are O(1) first 55 120 19 16 last

Priority Queue

• A^ Bag^ in which data items are^ Comparable
• lesser elements (as determined by

compareTo() ) have higher priority

• extract() returns the element with the

highest priority = least in the compareTo()

ordering

• break ties arbitrarily

java.util.PriorityQueue

boolean add(E e) {...} //insert an element (insert) void clear() {...} //remove all elements E peek() {...} //return min element without removing //(null if empty) E poll() {...} //remove min element (extract) //(null if empty) int size() {...}

Priority Queues as Lists

• Maintain as unordered list
  • (^) insert() puts new element at front – O(1)
  • (^) extract() must search the list – O(n)
• Maintain as ordered list
  • (^) insert() must search the list – O(n)
  • (^) extract() gets element at front – O(1)
• In either case, O(n 2

) to process n elements

Can we do better?

Heaps

• A heap is a concrete data structure that can

be used to implement priority queues

• Gives better complexity than either ordered

or unordered list implementation:

• (^) insert(): O(log n)
• (^) extract(): O(log n)
• O(n log n) to process n elements
• Do not confuse with heap memory , where

the Java virtual machine allocates space for

objects – different usage of the word heap

Heaps

• Binary tree with data at each node

• Satisfies the Heap Order Invariant :

The least (highest priority)

element of any subtree is found

at the root of that subtree

Examples of Heaps

• Ages of people in family tree
  • parent is always older than children, but you can have an uncle who is younger than you
• Salaries of employees of a company
  • bosses generally make more than subordinates, but a VP in one subdivision may make less than a Project Supervisor in a different subdivision

Balanced Heaps

Two restrictions:

1. Any node of depth < d – 1 has exactly 2

children, where d is the height of the tree

• implies that any two maximal paths (path from a root to a leaf) are of length d or d – 1, and the tree has at least 2 d nodes

2. All maximal paths of length d are to the left

of those of length d – 1

• Elements of the heap are stored in the array

in order, going across each level from left to

right, top to bottom

• The children of the node at array index n are

found at 2n + 1 and 2n + 2

• The parent of node n is found at (n – 1)/

Store in an ArrayList or Vector

0 1 2 (^3 4 5 ) 7 8 9 10 11 children of node n are found at 2n + 1 and 2n + 2

Store in an ArrayList or Vector

insert()

insert()