Heaps, Heap Sort and Priority Queues CMPT 438 Algorithms Heaps, Heap Sort and Priority Queues

A Job Scheduling Application The key is the priority of the jobs in the queue The job with the highest priority needs to be executed next Operations Insert, remove maximum Data structures Priority queues Arrays Ordered Unordered

PQ Implementations & Cost Worst-case asymptotic costs for a PQ with N items Insert Remove max unordered array ? ? ordered array ? ? Can we implement both operations efficiently?

Background on Trees Def: Binary tree = structure composed of a finite set of nodes that : Is composed of three disjoint sets of nodes: a root node, a left subtree and a right subtree 4 root Right subtree Left subtree 1 3 2 16 9 10 14 8

Special Types of Trees Full binary tree 2 14 8 1 16 7 4 3 9 10 12 Def: Full binary tree = a binary tree in which each node is either a leaf or has degree (number of children) exactly 2. Def: Complete binary tree = a binary tree in which all leaves have the same depth and all internal nodes have degree 2. Complete binary tree 2 1 16 4 3 9 10

The Heap Data Structure Def: A heap is a nearly complete binary tree with the following two properties: Structural property: all levels are full, except possibly the last one, which is filled from left to right Order (heap) property: for any node x Parent(x) ≥ x Largest element in the heap? 8 7 4 It doesn’t matter that 4 in level 1 is smaller than 5 in level 2 Largest element in a heap 5 2 Heap

Definitions Height of a node = the number of edges on a longest simple path from the node down to a leaf Depth of a node = the length of a path from the root to the node Height of tree = lgn, for a heap of n elements Height of root = 3 4 1 3 Height of (2)= 1 2 16 9 10 Depth of (10)= 2 Height of a heap 14 8

Array Representation of Heaps A heap can be stored as an array A. Root of tree Parent(i) {} Left(i) {} right(i) {}

Array Representation of Heaps A heap can be stored as an array A. Root of tree is A[1] Parent(i) { return i/2; } Left(i) { return 2*i; } right(i) { return 2*i + 1; } The elements in the subarray A[(n/2+1) .. n] are leaves The root is the maximum element of the heap

Heap Types Max-heaps (largest element at root), have the max-heap property: for all nodes i, excluding the root: A[PARENT(i)] ≥ A[i] Min-heaps (smallest element at root), have the min-heap property: A[PARENT(i)] ≤ A[i]

Operations on Heaps Maintain the max-heap property MAX-HEAPIFY Create a max-heap from an unordered array BUILD-MAX-HEAP Sort an array in place HEAPSORT Priority queue operations

Max-Heapify()

Max-Heapify()

Max-Heapify()

Max-Heapify()

Max-Heapify() Running time?

Max-Heapify()

Building a Heap 8 2 4 14 7 1 16 10 9 3 5 6 2 14 8 1 16 7 4 3 9 10 4 1 3 2 16 9 10 14 8 7

Example: A 4 1 3 2 16 9 10 14 8 7 i = 5 i = 4 i = 3 2 14 8 1 16 7 4 3 9 10 5 6 2 14 8 1 16 7 4 3 9 10 5 6 14 2 8 1 16 7 4 3 9 10 5 6 i = 2 i = 1 14 2 8 1 16 7 4 10 9 3 5 6 14 2 8 16 7 1 4 10 9 3 5 6 8 2 4 14 7 1 16 10 9 3 5 6

Building a Heap Convert an array A[1 … n] into a max-heap (n = length[A]) The elements in the subarray A[(n/2+1) .. n] are leaves Apply MAX-HEAPIFY on elements between 1 and n/2 2 14 8 1 16 7 4 3 9 10 5 6 A: 4 1 3 2 16 9 10 14 8 7

Building a Heap Convert an array A[1 … n] into a max-heap (n = length[A]) The elements in the subarray A[(n/2+1) .. n] are leaves Apply MAX-HEAPIFY on elements between 1 and n/2 Alg: BUILD-MAX-HEAP(A) n = length[A] for i ← n/2 downto 1 do MAX-HEAPIFY(A, i) 2 14 8 1 16 7 4 3 9 10 5 6 A: 4 1 3 2 16 9 10 14 8 7

Running Time of BUILD MAX HEAP HEAPIFY takes O(h)  the cost of HEAPIFY on a node i is proportional to the height of the node i in the tree Height Level No. of nodes h0 = 3 (lgn) i = 0 20 h1 = 2 i = 1 21 h2 = 1 i = 2 22 h3 = 0 i = 3 (lgn) 23 hi = h – i height of the heap rooted at level i ni = 2i number of nodes at level i

Running Time of BUILD MAX HEAP Running time of BUILD-MAX-HEAP: T(n) = O(n) Cost of HEAPIFY at level i  number of nodes at that level Replace the values of ni and hi computed before Multiply by 2h both at the nominator and denominator and write 2i as Change variables: k = h - i The sum above is smaller than the sum of all elements to  and h = lgn The sum above is smaller than 2

Heapsort Given BuildHeap(), an in-place sorting algorithm can be easily constructed:

Heapsort Given BuildHeap(), an in-place sorting algorithm can be easily constructed:

Heapsort Given BuildHeap(), an in-place sorting algorithm can be easily constructed:

An Example for HEAPSORT

An Example for HEAPSORT