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Data structures and algorithms lab10


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  • 1. DATA STRUCTURES AND ALGORITHMS LAB 10 Bianca Tesila FILS, April 2014
  • 2. OBJECTIVES  Heap  Heap Sort
  • 3. HEAP: INTRODUCTION  What is a heap - a data structure that:  is stored as an array in memory  can be represented/viewed as an almost complete binary tree  respects the heap property
  • 4. HEAP: INTRODUCTION  What is an almost complete binary tree - a binary tree such that:  all the leaves are at the bottom level or the bottom 2 levels  all the leaves are in the leftmost possible positions  all the levels are completely filled with nodes (except for the bottom level)
  • 5. HEAP: INTRODUCTION The heap property:  Max heaps  For each node (except for the root): A[i] <= A[parent(i)]  Conclusion: The root (A[1]) contains the largest element  Min heaps  For each node (except the root): A[i] >= A[parent(i)]  Conclusion: The root (A[1]) contains the smallest element !! Heap Sort uses max heaps
  • 6. HEAP: DATA STRUCTURE  Store the elements in an array - the advantage of this method over using the usual pointers and nodes is that there is no wasting of space due to storing two pointer fields in each node.  Start by taking the nodes level by level from the top down, left to right. Then, store them in an array.
  • 7. HEAP: DATA STRUCTURE  An array A[0..n] can be represented as a binary tree:  A[0] –root of the tree  Parent of A[i] = parent(i) = A[(i-1)/2]  Left child of A[i] = left(i) = A[2*i+1]  Right child of A[i] = right(i) = A[2*(i+ 1)]  Height of the heap: Θ(log n)  Number of nodes from the root to the farthest leaf
  • 8. HEAP: BASIC OPERATIONS Insert a new element:  add it as the last element in the heap  however, the heap property may be broken  if the added element is larger than its parent, you need to find its correct position in the heap => filter up Filter up an element:  compare it with its parent  if the parent is smaller than it, stop  else, swap it with the parent and continue
  • 9. HEAP: BASIC OPERATIONS Insert E in this heap 1. Place E in the next available position: 2. Filter up E:
  • 10. HEAP: BASIC OPERATIONS Delete ( extract min/max):  simply remove A[0] (the root element)  however, the heap is broken as it has no root element  need to find a new root  move the last element in the heap as the new root  now, the heap property is lost (almost certainly) => need to filter down Filter down an element:  compare it to its children  if it is larger than both children, stop  else, swap it with the largest child and continue
  • 11. HEAP: BASIC OPERATIONS Remove C from this heap:
  • 12. HEAP SORT  convert the array you want to sort into a heap  remove the root item (the smallest), readjust the remaining items into a heap, and place the removed item at the end of the heap (array)  remove the new item in the root (the second smallest), readjust the heap, and place the removed item in the next to the last position and so on
  • 13. HEAP SORT Example: HeapSort.pdf
  • 14. HEAP !! Exercise: Implement the functions for returning the parent and the childs of a given node, using the following signatures: template <typename T> int Heap<T>::parent(int index) { // TODO } template <typename T> int Heap<T>::leftSubtree(int index) { // TODO } template <typename T> int Heap<T>::rightSubtree(int index) { // TODO } !! Use heap.h
  • 15. HEAP SORT !! Exercise: Implement the heapsort and test it for the following array: 25, 17, 36, 2, 3, 100, 1, 19, 17 !! Use heap.h
  • 16. HOMEWORK Finish all the lab assignments.