COMP 103 RECAP-TODAY. Priority Queues and Heaps. Queues and Priority Queues 3 Queues: Oldest out first

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1 COMP 0 Priority Queues and Heaps RECAP RECAP-TODAY Tree Structures (in particular Binary Search Trees (BST)) BSTs idea nice way to implement a Set, Bag, or Map TODAY Priority Queue = variation on Queue best first Heap idea nice way to implement Priority Queue Reading: Chapter (section. -.) 0-T Lecture Marcus Frean, Lindsay Groves, Peter Andreae and Thomas Kuehne, VUW Lindsay Groves School of Engineering and Computer Science, Victoria University of Wellington Queues and Priority Queues Queues: Oldest out first Bob Jack Jim Ian Priority Queues: Best out first Bob Jack Jim Emergency room, calls, job scheduling in factory, etc... Operating system process scheduling Ian Sometimes a lower number means higher priority Graph/network algorithms: route planning, find shortest/cheapest path Need to: Store items, along with their priorities Add items (enqueue/offer) Remove items (dequeue/poll) Test for empty One idea: Like a set with a find min (or max) operation. Use an array of queues, with one queue for each priority value Ok if priorities are discrete (integers) and range is small What is cost of operations? Ex: Implement a priority queue like this What if we can t use this approach? Can we adapt other data structures we know about?

2 Unsorted list (array or linked list): 6 Sorted list (array or linked list): Fast to enqueue ( offer ): O() Just add at the back so constant time Fast to dequeue ( poll ): Just remove from the front O() Slow to dequeue ( poll ): O(n) Have to search for highest priority item Slow to enqueue ( offer ): O(n) Have to search for insertion point (and move rest to make space in case of array) Binary Search Tree Fast to enqueue & dequeue (O(log n)) Just insert and delete as usual [if balanced] But, not cheap to keep balanced And always deleting the smallest/largest will make it unbalanced quickly! : Ideas? 8 A BST is always fully sorted supports ascending order through in-order traversal Do we need this for Priority Queues? what if we enqueue 000 elements but only need the first three? can we exploit that we are always just interested in a maximum element? is there a lazy sorting strategy?

3 : POT idea Abandon idea of total order effort may never pay off (almost) no need to worry about lower structure as long as we keep the priority item at the top Only aim for partial order Partially Ordered Tree Fast to enqueue ( offer ): O(log n) Fast to dequeue ( poll ): O(log n) Easy to keep balanced Fast to construct from unordered list! Partially Ordered Trees 0 Binary Search Tree Binary tree All in left subtree < parent, All in right subtree parent 6 Partially Ordered Tree Keep highest priority element at the root Binary tree children parent, Order of children not important 6 Add: version Start at top and navigate downwards Choose arbitrary subtree, and insert when you reach a node with less than two children 6 Fly Partially Ordered Tree: Removing Remove: version Remove top element and return it Need to replace with next larger element pull up largest child and recurse That s all good, but: tree may become unbalanced! can we do better? 6

4 Add (draft) start at top and navigate downwards to the right level bubble up to correct position by swapping Fly 6 Add insert next to bottom-rightmost bubble up to correct position by swapping how can we find rightmost position in bottom row? 6 Remove (better) replace root by bottom-rightmost node sink down to correct position by swapping keeps tree balanced and! 6 6 Remove (better) replace root by bottom-rightmost node sink down to correct position by swapping keeps tree balanced and! 6

5 Support required for locating bottom-rightmost node [add & remove (II)] child parent [ bubble up for add] parent child [ sink down for remove] 6 8 Answer Use a Heap! A, partially ordered, binary tree = every level full, except bottom, where nodes are to the left Implemented in an array using breadth-first order 6 How can we achieve this? Heap Bottom right node is last element used Recording its position gives tree size as well We can compute the index of parent and children of a node: the children of node i are at (i+) and (i+) the parent of node i is at (i-)/ If they exist! 0 Heaps What are the costs of the operations? How can we create an initial heap? Can we do better than just inserting each item? wow: no gaps!