School of Computing National University of Singapore CS2010 Data Structures and Algorithms 2 Semester 2, AY 2015/16. Tutorial 2 (Answers)

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1 chool of Computing National University of ingapore C10 Data tructures and lgorithms emester, Y /1 Tutorial (nswers) Feb, 1 (Week ) BT and Priority Queue/eaps Q1) Trace the delete() code for a BT for the cases in Figure 1, where node is to be deleted. Follow the tracing approach we used in Tutorial 1. 1 public void delete ( Key key ){ root = delete (root, key ); 3 5 private Node delete ( Node x, Key key ) { if ( x == null ) 7 return null ; 8 int cmp = key. compareto (x. key ); 9 if ( cmp < 0) 10 x. left = delete (x.left, key ); 11 else if ( cmp > 0) x. right = delete (x. right, key ); 13 1 if ( x. right == null ) return x. left ; 1 if ( x. left == null ) 17 return x. right ; 18 Node t = x; 19 x = min (t. right ); x. right = deletemin (t. right ); 1 x. left = t. left ; 3 x. count = size (x. left ) + size (x. right ) + 1; return x; nswer: ee Figure for the tracing of deletion of the first case, and Figure 3 for the last case. Q) nswer the following questions about max-heaps: 1. If you had a sorted array, how quickly could you build a max-heap in terms of O(n)? nswer: sorted array in a descending order satisfies the max-heap property. s a result, the fastest way to convert a sorted array in an ascending order to a max-heap would be to reverse the order of elements in the array. eversing the order of elements in an array can be done using the following algorithm. This algorithm has an O(n) time complexity as the algorithm needs to go through all elements in the array. 1

2 Figure 1: Delete node from the trees. 1 public void reverserray ( Key arr []) { for ( int start =1, end = arr. length -1; start < end ; start ++, end - -) { 3 Key temp = arr [ start ]; arr [ start ] = arr [ end ]; 5 arr [ end ] = temp ; 7. Consider you are already given a valid max-heap (all children are smaller than their parent). Would swapping the children of a node in a max-heap still result in a valid max-heap? nswer: No, consider the following example: 3. how the resulting binary max-heap by inserting the following elements:,,,,,, 1. nswer: Figure shows the steps that are taken to insert each element.. For a max-heap, to access the top element is O(1). ssuming you have access to the heap implementation, what would be the big O to find the second largest element in the heap? nswer: The largest element in a max-heap is the root. One of the root s two children is the second largest element in the heap. Consequently as shown in the following code, in order to find the second largest element, the root s two children need to be compared and the larger child should be returned. The time complexity for this code is O(1). 1 Public Key getecondlargest ( Key arr []) { if(n <) 3 return null ; if(n <3) 5 return arr []; 7 if( less ( arr [], arr [3]) ) 8 return arr [3]; 9 else 10 return arr []; 11 Q3) ketch a non-recursive algorithm to verify that an array of size N is a valid max-heap. For example, you are given an array, of size N, verify (yes or no) if it is a valid max-heap. ssume

3 public void delete(key key){ root = delete(root, key); if (cmp < 0) null x.left = delete(x.left, key); return x; if (cmp < 0) if (x.right == null) null return x.left; if (x.left == null) return x.right; Node t = x; x = min(t.right); x.right = deletemin(t.right); x.left = t.left; return x; key key == x x null Figure : Delete node, first case. our indexing starts from 1 is the same in the notes. What is the time complexity? nswer: One possible solution could be as follows. The time complexity for this algorithm is O(n). 1 public bool ismaxeap ( Key arr []) { for ( int i=n; i >1; i - -){ 3 if( less ( arr [i/], arr [i])) return false ; 5 return true ; 7 Q) ssume we have a constant stream of integers arriving from some data stream. ketch an algorithm that maintains the top K largest integers from the data stream? 3

4 public void delete(key key){ root = delete(root, key); if (cmp < 0) x.left = delete(x.left, key); return x; if (cmp < 0) if (x.right == null) null return x.left; if (x.left == null) return x.right; Node t = x; x = min(t.right); x.right = deletemin(t.right); x.left = t.left; return x; key key == x x x == min deletemin Figure 3: Delete node, last case. 1 1 swap < 17 violates the maxheap property Figure : n example of a heap for which swaping the children of a node will violate the max-heap property.

5 Max-heap Figure 5: Max-heap (left) to Min-heap Min-heap (right). nswer: One possible solution could be: 1 public Class KLargestItemseap { 3 private MinPQ mineap ; 5 public void keepklargestitems ( Key item ){ if( mineap. size () < K) 7 mineap. insert ( item ); 8 9 If( less ( mineap. min (),item )){ 10 mineap. delmin (); 11 mineap. insert ( item ); 13 1 Q5) ssume you have an Integer max-heap, but you want it to act as a min-heap. owever, you are not allowed to rewrite the max-heap code. Devise a strategy to turn the max-heap into a min-heap. (Clear your mind, this is either the easiest or hardest question in this tutorial!) nswer: tore values in the integer max-heap after negating them, consequently, negate the values again after removing them from the heap to get back the original values. ee Figure 5 for an example. 5

6 Figure : This figure shows insertion of each element to the heap Insert() Insert() 1 1 ince k=1 it does not go into the ince less(,) is false it does not go into the 3 3 Insert() ince less(,) is false it does not go into the Insert() 1 st turn: ince less(,) is right it goes into the while loop nd turn: ince less(,) is false it does not go into the Insert() 1 st turn: ince less(,) is right it goes into the Insert() 1 st turn: ince less(,) is right it goes into the nd turn: ince less(,) is false it does not go into the nd turn: ince less(,) is false it does not go into the Insert(1) 1 st turn: ince less(,) is right it goes into the 1 1