Priority Queues and Heaps

Priority Queues We already know a ”standard” queue: Elements can only be added to the start of the queue Elements can only be removed from the end of the queue Order of removal is FIFO (first in, first out) RHS – SWC

Priority Queues In a Priority Queue, each element is assigned a priority as well A priority is a numeric value – the lower the number, the higher the priority Elements are removed in order of their priority – no longer FIFO Can still add elements in any order RHS – SWC

Priority Queues public interface PriorityQueue<T> { void add(T element); T remove(); T peek(); } RHS – SWC

Priority Queues The interface to a priority queue is almost the same as to a regular queue However, elements in the queue must now be of the type Comparable (i.e. implement the interface) The remove method always returns the element with highest priority (lowest numeric value…) RHS – SWC

Priority Queues A Priority Queue is an abstract data structure, which can be implemented in various ways Linked list, removal will be O(n) Binary search tree, removal will usually be O(log(n)), but not always Heap, removal will always be O(log(n)) RHS – SWC

Heaps A Heap is a Binary Tree, but not a Binary Search Tree In order for a Binary Tree to be a Heap, two conditions must be fulfilled: A heap is almost complete; only nodes missing in bottom layer All nodes store values which are at most as large as the values stored in any descendant RHS – SWC

Heaps RHS – SWC

Heaps On a heap, only two operations are of interest: Insert a new element Remove the element with lowest value, i.e. the root element However, both of these operations must preserve the heap property of the tree RHS – SWC

Heaps - insertion 32 23 29 42 37 56 Find an empty slot in the tree RHS – SWC

Heaps - insertion 32 23 29 42 If the value in the parent is larger than the new value, swap the parent and the new slot (repeat this) 37 56 RHS – SWC

Heaps - insertion 32 23 29 32 37 56 42 Now insert the new value RHS – SWC

Heaps - insertion Since the heap is ”almost complete”, the number of layers in a tree with n nodes is at most (log(n) + 1) Each swap operation can be done in constant time, so insertion of an element has the run-time complexity O(log(n)) RHS – SWC

Heaps - removal Remove the root node (always smallest) 23 29 32 37 56 42 RHS – SWC

Heaps - removal 29 32 Move the last element into the root position 37 56 42 RHS – SWC

Heaps - removal If any child has a lower value, swap position with child with lowest value (repeat this) 42 29 32 37 56 RHS – SWC

Heaps - removal If any child has a lower value, swap position with child with lowest value (repeat this) 29 42 32 37 56 RHS – SWC

Heaps - removal The heap property has now been reestablished This is known as ”fixing the heap” 29 37 32 42 56 RHS – SWC

Heaps - removal Since the heap is ”almost complete”, the number of layers in a tree with n nodes is at most (log(n) + 1) Each swap operation can be done in constant time, so deletion of an element has the run-time complexity O(log(n)) RHS – SWC

Heaps - removal The important point is that for a heap, we are guaranteed O(log(n)) run time for insertion and deletion This cannot be guaranteed for a binary search tree A binary search tree can ”degenerate” into a linked list; a heap cannot RHS – SWC

Heaps - representation Due to the regularity of a heap, it can efficiently be stored in an array Root node is stored in position 1 (not 0) A node in position i has its children stored in position 2i and (2i + 1) A node in position i has its parent stored in position i/2 (integer division) When running out of space, double the size RHS – SWC

Heapsort A heap can be used for a quite efficient way of sorting an array of n objects Run-time complexity of O(nlog(n)) Does not use extra space Main steps Turn the array into a heap Repeatedly remove the root element (which has the smallest value) RHS – SWC

Heapsort In order to turn the array into a heap, we could just insert all the elements into a new heap However, we can do this without using an extra heap! We use the ”fix heap” procedure from the bottom in the tree and upwards RHS – SWC

Heapsort Why will this work…? Remember that the ”fix heap” procedure takes two ”subheaps” as input, plus a root node with a ”wrong” value If we work from the bottom and up, the input will always be like above RHS – SWC

Heapsort Fix heap here Fix heaps here Fix heaps here Trivially a heap RHS – SWC

Heapsort Now the tree is a heap Repeatedly remove the root from the heap, and fix the remaining heap We ”remove” the root by placing it at the end of the array, beyond the last element in the remaining heap RHS – SWC

Heapsort 23 29 32 37 56 42 23 29 32 37 56 42 RHS – SWC

Heapsort 29 32 37 56 42 29 32 37 56 42 23 RHS – SWC

Heapsort 29 37 32 42 56 29 37 32 42 56 23 RHS – SWC

Heapsort 37 32 42 56 37 32 42 56 29 23 RHS – SWC

Heapsort 32 37 56 42 32 37 56 42 29 23 RHS – SWC

Heapsort 37 56 42 37 56 42 32 29 23 RHS – SWC

Heapsort 37 42 56 37 42 56 32 29 23 RHS – SWC

Heapsort 42 56 42 56 37 32 29 23 RHS – SWC

Heapsort 42 56 42 56 37 32 29 23 RHS – SWC

Heapsort 56 56 42 37 32 29 23 RHS – SWC

Heapsort 56 56 42 37 32 29 23 RHS – SWC

Heapsort 56 42 37 32 29 23 RHS – SWC

Heapsort One minor issue – numbers are sorted in wrong order Could just reverse order, takes O(n) Or use max-heap Min-heap: All nodes store values at most as large as the values stored in any descendant Max-heap: All nodes store values at least as large as the values stored in any descendant RHS – SWC