3. Priority Queues. ADT Stack : LIFO. ADT Queue : FIFO. ADT Priority Queue : pick the element with the lowest (or highest) priority.

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1 3. Priority Queues 3. Priority Queues ADT Stack : LIFO. ADT Queue : FIFO. ADT Priority Queue : pick the element with the lowest (or highest) priority. Malek Mouhoub, CS340 Winter

2 3. Priority Queues 3. Priority Queues 3.1 Model 3.2 Implementation 3.3 Binary Heap 3.4 d-heaps 3.5 Min-Max Heap Malek Mouhoub, CS340 Winter

3 3.2 Implementation 3.2 Implementation 1st case : use a linked list insert at the front of the list : O(1) deletemin requires traversing the list : O(N) Malek Mouhoub, CS340 Winter

4 3.2 Implementation 2nd case : use a sorted linked list insertion requires O(N) but only O(1) for deletemin. Malek Mouhoub, CS340 Winter

5 3.2 Implementation 3rd case : use a binary search tree O(log N) average running time for both insertion and deletion. Elements are deleted at each time from the left subtree which affects the balance of the tree. The binary search tree supports many operations that are not required by priority queue. Malek Mouhoub, CS340 Winter

6 3.3 Binary Heap A binary heap is a complete binary tree : a tree that is completely filled, with possible exception of the bottom level, which is filled from left to right. Can easily be represented by an array (and an integer representing the current heap size) instead of a linked list. For any element in array position i : the left child is in position 2i, the right child is in position 2i + 1, and the parent is in position [i/2]. The operations required to traverse the tree are very simple and are performed quickly (because of the heap-order property). Malek Mouhoub, CS340 Winter

7 I 3.3 Binary Heap A B C D E F G H J A B C D E F G H I J Malek Mouhoub, CS340 Winter

8 Heap-Order Property The smallest element should be at the root. Any node should be smaller than all of its descendants (any subtree should also be a heap). Heap-order property : For every node X, the key in the parent of X is smaller than (or equal to) the key in X, with the exception of the root (which has no parent). Malek Mouhoub, CS340 Winter

9 Two complete trees (only the left is a heap) Malek Mouhoub, CS340 Winter

10 Basic Heap Operations Insert 1. Create a hole in the next available location. 2. Place the element to be inserted in the hole if it does not violate the heap order. 3. Otherwise slide the element that is in the hole s parent node into the hole, thus bubbling the hole up toward the root. Continue the process until the element can be placed in the hole. Malek Mouhoub, CS340 Winter

11 bubbling the hole up Insert Malek Mouhoub, CS340 Winter

12 Insert 14 Malek Mouhoub, CS340 Winter

13 Insert /* Inserting item into the priority queue, maintaining the heap order. Duplicates are not allowed */ template <Class Comparable> void BinaryHeap<Comparable>::insert(const Comparable & x) { if (isfull()) throw Overflow(); // Percolate up int hole = ++currentsize; for ( ;hole > 1 && x < array[hole/2]; hole /= 2) array[hole] = array[hole/2]; 10 array[hole] = x; } Malek Mouhoub, CS340 Winter

14 deletemin 1. Create a hole at the root. 2. The last element X in the heap must move somewhere in the heap. 3. Slide the smaller of the hole s children into the hole, thus pushing the hole down one level. Repeat this step until X can be placed in the hole. Malek Mouhoub, CS340 Winter

15 deletemin Creation of the hole at the root Malek Mouhoub, CS340 Winter

16 Malek Mouhoub, CS340 Winter

17 Malek Mouhoub, CS340 Winter

18 Other heap operations A heap has very little ordering information, so there is no way to find any particular element without a linear scan through the entire heap. Assuming that the position of every element is known by some other method, the following operations run in logarithmic worst-case time. decreasekey(p, ) : lowers the value of the item at position p by a positive amount. A percolate up is required sometimes to maintain the heap order. increasekey(p, ) : increases the value of the item at position p by a positive amount. A percolate down is required sometimes to maintain the heap order. remove : removes the node at position p from the heap. This is done by performing decreasekey(p, ) and then performing deletemin(). Malek Mouhoub, CS340 Winter

19 Other heap operations Buildheap Takes as input N items and places them into an empty heap. N successive inserts O(N) average case but O(N log N) worst-case. Malek Mouhoub, CS340 Winter

20 Other heap operations Buildheap /** Establish heap order property from an arbitrary arrangement of items. Runs in linear time. */ template <class Comparable> void BinaryHeap<Comparable>::buildHeap() { for (int i = currentsize / 2; i > 0; i ) } perlocatedown(i); Malek Mouhoub, CS340 Winter

21 Applications of Priority Queues : The selection Problem Given a list of N elements, which can be totally ordered, and an integer k, find the kth largest element. Algorithm 1A : read the elements into an array and sort them, returning the appropriate element. assuming a simple sorting algorithm, the running time is O(N 2 ). Algorithm 1B : 1. read k elements into an array and sort them. The smallest of these is in the kth position. 2. Process the remaining elements one by one. As an element arrives, it is compared with the kth element in the array. If it is larger, then the kth element is removed, and the new element is placed in the correct place among the remaining k 1 elements. O(N k) running time. Why? If k = N/2 then both algorithms are O(N 2 ). k is known as the median in this case. The following algorithms run in O(N log N) in the extreme case of k = N/2. Malek Mouhoub, CS340 Winter

22 Algorithm 6A Algorithm for finding the kth smallest element 1. Read N elements into an array. 2. Apply the buildheap algorithm to this array. 3. Perform k deletemin operations. The last element extracted from the heap is the answer. Complexity : O(N + k log N) in the worst case. k = O(N/ log N) O(N) k = N/2 O(N log N) For large values of k : O(k log N) k = N O(N log N) (Idea of the heapsort). By changing the heap-order property, we will solve the problem of finding the kth largest element. Malek Mouhoub, CS340 Winter

23 Algorithm 6B Find the kth largest element 1. Same idea as algorithm 1B. 2. At any point in time, maintain a set S of the k largest elements. 3. After the first k elements are read, when a new element is read it is compared with the kth largest element, which we denote by S k (S k is the smallest element in S). If the new element is larger, then it replaces S k in S. 4. At the end of the input, we find the smallest element in S and return it as the answer. O(k + (N k) log k) = O(N log k) in the worst case. Why? Malek Mouhoub, CS340 Winter

24 3.4 d-heaps 3.4 d-heaps A d-heap is much shallower than a binary heap : Insert : O(log d N) DeleteMin : O(d log d N) X > X > X > X d subtrees Malek Mouhoub, CS340 Winter

25 3.5 Min-Max Heap 3.5 Min-Max Heap In a Min-Max Heap, the heap-order property is that : for any node, X, at even depth, the element stored at X is smaller than the parent but larger than the grandparent (where this makes sense), and for any node X at odd depth, the element stored at X is larger than the parent but smaller than the grandparent. Malek Mouhoub, CS340 Winter

26 3.5 Min-Max Heap Malek Mouhoub, CS340 Winter

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