Lecture: Priority Queue

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Presentation transcript:

Lecture: Priority Queue

Quiz Sample Is array a data structure? What is a data structure? What data structures are implemented by array? Priority queue (max --, min --). No! Why? It is a standard part of algorithm Stack, Queue, List, Heap, Max-heap, Min-heap, …

Quiz Sample Is Dial algorithm with running time O(m+nc) a polynomial-time algorithm, where c is the maximum arc length? Answer: No

Implementations With min-priority queue, Dijkstra algorithm can be implemented in time With Fibonacci heap, Dijkstra algorithm can be implemented in time With Radix heap, Dijkstra algorithm can be implemented

Contents Recall: Heap, a data structure Min-heap (a) Min-Heapify procedure (b) Building a min-heap Min-Priority Queue Implementation of Dijkstra’s Algorithm

Heap

A Data Structure Heap A heap is an array object that can be viewed as a nearly complete binary tree. 1 6 2 3 5 3 6 5 3 2 4 1 4 5 6 2 4 1 Tied with three procedures for finding Parent, finding left child, and finding Right child. All levels except last level are complete.

Min-Heap

Min-Heap

Min-Heapify Min-Heapify(A,i) is a subroutine. Input: When it is called, two subtrees rooted at Left(i) and Right(i) are min-heaps, but A[i] may not satisfy the min-heap property. Output: Min-Heapify(A,i) makes the subtree rooted at A[i] become a min-heap by letting A[i] “float down”.

14 4 7 4 7 14 12 8 11 12 8 11 14 8 7 2 1 14

Building a Min-Heap e.g., 4, 1, 3, 2, 16, 9, 10, 14, 8, 7.

4 1 3 9 10 2 16 8 7 14

4 1 3 9 10 2 7 8 16 14

4 1 3 9 10 2 7 14 8 16

4 1 3 9 10 2 7 14 8 16

4 1 3 9 10 2 7 14 8 16

1 4 3 9 10 2 7 14 8 16

1 2 3 9 10 4 7 14 8 16

Priority Queue

Priority Queue A priority queue is a data structure for maintaining a set of elements, each with an associated value, called a key. A min-priority queue supports the following operations: Minimum(S), Extract-Min(S), Increase-Key(S,x,k), Insert(S,x). Min-Heap can be used for implementing min-priority queue.

Input: 4, 1, 3, 2, 16, 9, 10, 14, 8, 7. Build a min-heap 1, 2, 3, 4, 7, 9, 10, 14, 8, 16.

16 2 3 10 4 7 9 14 8 16, 2, 3, 4, 7, 9, 10, 14, 8.

2 16 3 10 4 7 9 14 8 2, 16, 3, 4, 7, 9, 10, 14, 8.

2 4 3 10 7 9 16 14 8 2, 4, 3, 16, 7, 9, 10, 14, 8.

2 4 3 10 8 7 9 14 16 2, 4, 3, 8, 7, 9, 10, 14, 16.

1 2 3 9 10 4 7 14 8 16 1, 2, 3, 4, 7, 9, 10, 14, 8, 16.

1 2 3 9 10 4 7 14 1 16 1, 2, 3, 4, 7, 9, 10, 14, 1, 16.

1 2 3 9 10 1 7 14 4 16 1, 2, 3, 1, 7, 9, 10, 14, 4, 16.

1 1 3 9 10 2 7 14 4 16 1, 1, 3, 2, 7, 9, 10, 14, 4, 16.

1 3 6 9 10 4 7 14 8 16 1, 3, 6, 4, 7, 9, 10, 14, 8, 16.

1 3 6 9 10 4 7 14 8 16 +∞ 1, 3, 6, 4, 7, 9, 10, 14, 8, 16, +∞.

1 3 6 9 10 4 7 14 8 16 2 1, 3, 6, 4, 7, 9, 10, 14, 8, 16, 2.

1 2 6 9 10 4 3 14 8 16 7 1, 2, 6, 4, 3, 9, 10, 14, 8, 16, 7.

Implementation of Dijkstra’s Algorithm

Dijkstra’s Algorithm

Implementations With min-priority queue, Dijkstra algorithm can be implemented in time With Fibonacci heap, Dijkstra algorithm can be implemented in time With Radix heap, Dijkstra algorithm can be implemented

Thanks, end.