CE 221 Data Structures and Algorithms

Transcription

1 CE 2 Data Structures and Algorithms Chapter 6: Priority Queues (Binary Heaps) Text: Read Weiss, Izmir University of Economics 1

2 A kind of queue Priority Queue (Heap) Dequeue gets element with the highest priority Priority is based on a comparable value (key) of each object (smaller value higher priority, or higher value higher priority) Example Applications: printer -> print (dequeue) the shortest document first operating system -> run (dequeue) the shortest job first normal queue -> dequeue the first enqueued element first Source: Muangsin / Weiss 2

3 Priority Queue (Heap) Operations deletemin Priority Queue insert insert (enqueue) deletemin (dequeue) smaller value higher priority Find / save the minimum element, delete it from structure and return it Source: Muangsin / Weiss 3

4 Implementation using Linked List Unsorted linked list insert takes O(1) time deletemin takes O(N) time Sorted linked list insert takes O(N) time deletemin takes O(1) time Source: Muangsin / Weiss 4

5 Implementation using Binary Search Tree insert takes O(log N) time on the average deletemin takes O(log N) time on the average support other operations that are not required by priority queue (for example, findmax) deletemin operations make the tree unbalanced Source: Muangsin / Weiss 5

6 Binary Heap Implementation Property 1: Structure Property Binary tree & completely filled (bottom level is filled from left to right) (complete binary tree) if height is h, size between 2 h (bottom level has only one node) and 2 h+1-1 A B C D E F G H I J Source: Muangsin / Weiss 6

7 Array Implementation of Binary Heap A B C D E F G left child is in position 2i H I J right child is in position 2i+1 parent is in position floor(i/2) (or simply integer division) A B C D E F G H I J Source: Muangsin / Weiss 7

8 Property 2: Heap Order Property (for Minimum Heap) Any node is smaller than (or equal to) all of its children (any subtree is a heap) Smallest element is at the root (findmin take O(1) time) Source: Muangsin / Weiss

9 13 Insert Create a hole in the next available location Move the hole up (swap with its parent) until data can be placed in the hole without violating the heap order property (called percolate up) Source: Muangsin / Weiss 9

10 Insert insert Percolate Up -> move the place to put 14 up (move its parent down) until its parent <= 14 Source: Muangsin / Weiss 10

11 Insert Source: Muangsin / Weiss 11

12 deletemin Create a hole at the root Move the hole down (swap with the smaller one of its children) until the last element of the heap can be placed in the hole without violating the heap order property (called percolate down) Source: Muangsin / Weiss 12

13 deletemin Percolate Down -> move the place to put 31 down (move its smaller child up) until its children >= 31 Source: Muangsin / Weiss 13

14 deletemin Source: Muangsin / Weiss 14

15 deletemin Source: Muangsin / Weiss 15

16 insert Running Time worst case: takes O(log N) time, moves an element from the bottom to the top on average: takes a constant time (2.607 comparisons), moves an element up levels deletemin worst case: takes O(log N) time on average: takes O(log N) time (element that is placed at the root is large, so it is percolated almost to the bottom) Source: Muangsin / Weiss 16

17 Implementation in Java - constructors public BinaryHeap( ) { this( DEFAULT_CAPACITY ); } public BinaryHeap( int capacity ) { currentsize = 0; array = (AnyType[]) new Comparable[ capacity + 1 ]; } Izmir University of Economics 17

18 Implementation in Java - utilities public boolean isempty( ) { return currentsize == 0; } public void makeempty( ) { currentsize = 0; } public AnyType findmin( ) { if( isempty( ) ) throw new UnderflowException( ); } return array[ 1 ]; private void enlargearray( int newsize ) { AnyType [] old = array; array = (AnyType []) new Comparable[ newsize ]; } for( int i = 0; i < old.length; i++ ) array[ i ] = old[ i ]; Izmir University of Economics 18

19 Implementation in Java - insert Izmir University of Economics

20 Implementation in Java - deletemin Izmir University of Economics 20

21 Implementation in Java - percolatedown Izmir University of Economics

22 Building a Heap Sometimes it is required to construct it from an initial collection of items O(NlogN) in the worst case. But insertions take O(1) on the average. Hence the question: is it possible to do any better? Izmir University of Economics 22

23 buildheap Algorithm General Algorithm Place the N items into the tree in any order, maintaining the structure property. Call buildheap Izmir University of Economics 23

24 buildheap Example - I initial heap after percolatedown(7) after percolatedown(6) after percolatedown(5) Izmir University of Economics 24

25 buildheap Example - II after percolatedown(4) after percolatedown(3) after percolatedown(2) after percolatedown(1) Izmir University of Economics 25

26 Complexity of buildheap The number of dashed lines must be bounded which can simply be done by computing the sum of the heights of all the nodes in the heap. Theorem: For a perfect binary tree of height h with N=2 h+1-1 nodes, this sum is 2 h+1-1-(h+1). Proof: S 2S h i 0 2S S 2 ( h i) 1( h) 2( h 1) 4( h 2) 8( h 3) 16( h 4) ( h) 4( h 1) 8( h 2) 16( h 3)... 2 i S h 1 2 h 1 ( h 1) number of nodes in a complete tree of height h is less than or equal to the the number of nodes in a perfect binary tree of the same height. Therefore, O(N) h 2 h 1 h (1) h 1 (1) Izmir University of Economics 26

27 Homework Assignments 6.2, 6.3, 6.4, 6.10.a, 6.38 You are requested to study and solve the exercises. Note that these are for you to practice only. You are not to deliver the results to me. Izmir University of Economics 27