1. 1 Priority Queues A priority queue is an ADT where: –Each element has an associated priority –Efficient extraction of the highest-priority element is supported. Typical operations of a priority queue are: –Enqueue (insert an element) –Dequeue (remove the highest-priority element) –Max (return the highest-priority element) –Meld (join two priority queues into one)

2. 2 Priority Queues: Applications Scheduling –Examples: Printer scheduling. –Jobs sent to a printer by faculty have a higher priority than jobs sent by students Process scheduling. –Several processes require CPU time. Priority may be based on various factors: »Shortest jobs have higher priority »Jobs with a shorter remaining time have higher priority »Jobs that have waited the longest have higher priority »etc.

3. 3 Priority Queues: Applications Discrete-event simulations –Simulations of systems modeled as collections of entities that interact with one another through events that occur at discrete times. The priority assigned to an event is its start time. –Example: PA1, bank simulation

4. 4 Binary Heaps Binary heap = –a binary tree that is –complete every level except possibly the bottom one is completely filled and the leaves in the bottom level are as far left as possible –satisfies the heap property: the key stored in every node is greater than or equal to the keys stored in its children –this is called a max-heap. If the key at each node is smaller than or equal to the keys of its children, then we have a min-heap.

5. 5 Binary Heaps Since heaps are complete trees, they can easily be stored in arrays: 12 18 8 9 1075 143 1812981075143 123456789 index 1 2 3 4 5 6 7 8 9 10 given index i of a node, the index of its parent is  i / 2  the indices of its children are 2i and 2i+1

6. 6 Binary Heaps Is every array a heap? –Not necessarily. But every sorted array is a max-heap, if sorted in descending order. min-heap, if sorted in ascending order. What is the height of a heap? –O(lgn) where n is the number of elements in the heap. Where in a max-heap is the smallest element? –It is always a leaf but we cannot know exactly where it is located. What are the maximum and minimum numbers of elements in a heap of height h? –max #elements: 2 h+1 -1 –min #elements: 2 h

7. 7 Binary Heaps: Heapify In the discussion that follows, "heap" refers to a max-heap unless otherwise noted. Every time we extract or insert an element, the heap property may be violated. Heapify: Given node i of heap that potentially violates the heap property, fix it. Input: Array A storing a heap, index i Preconditions: The children of node i are heaps. Algorithm: while (i < heapsize) { max_index = max(A[i], A[2i], A[2i+1]) if (max_index == i) return; swap(A, i, max_index); // swap contents at indices i, max_index i = max_index; // Since we moved A[i] to A[max_index], the heap // property may now be violated there. }

8. 8 Binary Heaps: Heapify Running time of Heapify: –O(lgn) because we may have to travel all the way from the root to a leaf, fixing the heap property. Can you come up with an array that causes Heapify to be called recursively on every node on a path from the root to a leaf? –Any array sorted in ascending order.

9. 9 Binary Heaps: MakeHeap Given a random array, create a heap How? –Algorithm sketch: Elements A[  n/2  +1] through A[n] are leaves, so they can be considered one-element heaps. Call Heapify in a bottom-up manner starting at index  n/2 . –Running time : easy analysis: O(nlgn) more exact analysis (proof omitted): O(n)

10. 10 Binary Heaps: MakeHeap Why does the loop run in decreasing order? If it ran in increasing order, would we still get a heap? MakeHeap: Given random array A, turn it into a heap Input: Array A Output: Array A as a heap Algorithm: heapsize(A) = length(A) for i =  length(A)/2  downto 1 Heapify(A, i) endfor

11. 11 Binary Heaps: Insert New elements are always inserted at the end (last element of the array) This may violate the heap property: –Travel up the tree looking for an appropriate position for the new node (one that maintains the heap property). As you go, copy (shift) down the values of the nodes you pass, to make room HeapInsert (A, key) i = ++heapsize(A) while i >1 and A[parent(i)]<key A[i] = A[parent(i)] i = parent(i) A[i] = key Running time: O(lgn) since we may end up traveling from a leaf all the way to the root.

12. 12 Binary Heaps: ExtractMax The maximum element is always at the root. Extract the maximum by replacing it with the last element of the array and then Heapifying. ExtractMax (A) max = A[1] A[1] = A[heapsize(A)] heapsize(A) = heapsize(A)-1 Heapify(A,1) return max Running time: O(lgn) since we may end up traveling from the root all the way to a leaf.

13. 13 Binary Heaps: Sorting Idea: –Since the maximum element is at the root, we can move it to the last position in the array, and decrement the size of the heap. –The root will now contain the new maximum –Repeat... Heapsort(A) Input: Array A Output: Array A in ascending order. Algorithm: Build-heap(A) for i = length(A) downto 1 swap(A[i], A[1]) heapsize = heapsize – 1 Heapify(A, i) Running time: O(nlgn) since the loop is executed n times and Heapify takes O(lgn)

14. 14 Priority queues in the STL #include int main() { priority_queue, greater > q; int nums[10] = {10,6,8,1,5,3,7,2,9,4}; for (int i =0; i < 10; ++i) q.push(nums[i]); std::cout << q.size() << "\n"; while (!q.empty() ) { std::cout << q.top() << "\n"; q.pop(); } return 0; } based on a vector higher values = lower priority

15. 15 Heaps in the STL #include int main () { vector nums; for (int i=1; i<11; i++) nums.push_back(i); make_heap(nums.begin(), nums.end()) ; sort_heap(nums.begin(), nums.end()) ; nums.push_back(32); make_heap(nums.begin(), nums.end()); // remake heap pop_heap(nums.begin(), nums.end()); // extraxt max return 0; }