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Objectives

Heap. Objectives. Upon completion you will be able to: Define and implement heap structures Understand the operation and use of the heap ADT Design and implement selection applications using a heap Design and implement priority queues using a heap. Basic Concepts.

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Objectives

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  1. Heap Objectives • Upon completion you will be able to: • Define and implement heap structures • Understand the operation and use of the heap ADT • Design and implement selection applications using a heap • Design and implement priority queues using a heap Data Structures: A Pseudocode Approach with C, Second Edition

  2. Basic Concepts • A heap is a binary tree whose left and right subtrees have values less than their parents. We begin with a discussion of the basic heap structure and its two primary operations, reheap up and reheap down. • Definition • Maintenance Operations Data Structures: A Pseudocode Approach with C, Second Edition

  3. Basic Concepts • Definition • A heap is a binary tree structure with the following properties: • The tree is complete or nearly complete. • The key value of each node is greater than or equal to the key value in each of its descendents. • The root of a heap is guaranteed to hold the largest node in the tree; its subtrees contain data that have lesser values. • Heaps are often implemented in an array rather than a linked list. Data Structures: A Pseudocode Approach with C, Second Edition

  4. Sometimes this structure is called a max-heap. A min-heap is to create a heap which the key value in a node is less than the key values in all of its subtrees. Data Structures: A Pseudocode Approach with C, Second Edition

  5. Data Structures: A Pseudocode Approach with C, Second Edition

  6. Data Structures: A Pseudocode Approach with C, Second Edition

  7. The reheap up operation reorders a “broken” heap by floating the last element up the tree until it is in its correct location in the heap. Data Structures: A Pseudocode Approach with C, Second Edition

  8. Data Structures: A Pseudocode Approach with C, Second Edition

  9. Reheap Down Operation reorders a “broken” heap by pushing the root down the tree until it is in its correct position in the heap. Data Structures: A Pseudocode Approach with C, Second Edition

  10. Data Structures: A Pseudocode Approach with C, Second Edition

  11. Heap Implementation • Heaps are usually implemented in an array structure. In this section we discuss and develop five heap algorithms. • Reheap Up • Reheap Down • Build a Heap • Insert a Node into a Heap • Delete a Node from a Heap Data Structures: A Pseudocode Approach with C, Second Edition

  12. Heap Implementation The relationship between a node and its children is fixed and can be calculated as shown below: 1. For a node located at index i, its children are found at: a. Left child: 2i +1 b. Right child: 2i +2 2. The parent of a node located at index i is located at (i -1)/2 3. Given the index for a left child, j, its right sibling, if any, is found at j + 1. Conversely, given the index for a right child, k, its left sibling, which must exist, is found at k - 1. 4. Given the size, n, of a completed heap, the location of the first leaf is (n/2) 5. Given the location of the first leaf element, the location of the last nonleaf element is one less. Data Structures: A Pseudocode Approach with C, Second Edition

  13. Data Structures: A Pseudocode Approach with C, Second Edition

  14. Data Structures: A Pseudocode Approach with C, Second Edition

  15. Data Structures: A Pseudocode Approach with C, Second Edition

  16. (continued) Data Structures: A Pseudocode Approach with C, Second Edition

  17. Data Structures: A Pseudocode Approach with C, Second Edition

  18. Data Structures: A Pseudocode Approach with C, Second Edition

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  20. Data Structures: A Pseudocode Approach with C, Second Edition

  21. Data Structures: A Pseudocode Approach with C, Second Edition

  22. Data Structures: A Pseudocode Approach with C, Second Edition

  23. Data Structures: A Pseudocode Approach with C, Second Edition

  24. Data Structures: A Pseudocode Approach with C, Second Edition

  25. Data Structures: A Pseudocode Approach with C, Second Edition

  26. Data Structures: A Pseudocode Approach with C, Second Edition

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