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CS221: Algorithms and Data Structures Lecture #3 Mind Your Priority Queues Steve Wolfman 2014W1 1 Learning Goals • Provide examples of appropriate applications for priority queues. • Describe efficient implementations of priority queues. • Relate heaps to binary trees, binary search trees, and arrays. 2 It is not a goal for this term to be able to manipulate heaps by hand. Today’s Outline • Priority Queue ADT • Solutions So Far? • 10km View of Heaps 3 Back to Queues • Some applications – ordering CPU jobs – simulating events – picking the next search site when searching {a maze, a game tree, a room, the WWW} • Problems? – short jobs should go first – earliest (simulated time) events should go first – most promising sites should be searched first 4 Priority Queue ADT • Priority Queue operations – – – – – create destroy insert deleteMin is_empty G(9) insert F(7) E(5) D(100) A(4) B(6) deleteMin C(3) • Priority Queue property: for two elements in the queue, x and y, if x has a lower priority value than y, x will be deleted before y 5 Applications of the Priority Q • • • • • • • Call queues for help lines (or don’t you think so?) Your call will Hold jobs for a printer in order of length not be answered in the order it Simulate events (hold in order of time) was received. Sort numbers Store packets on network routers in order of urgency Select symbols for compression (hold in order of frequency) Anything greedy: an algorithm that makes the “locally best choice” at each step (hold in order of quality) 6 Naïve Priority Q Data Structures • Unsorted array (or linked list): – insert: – deleteMin: • Sorted array: – insert: – deleteMin: a. b. c. d. e. O(lg n) O(n) O(n lg n) O(n2) Something else 7 Today’s Outline • Priority Queue ADT • Solutions So Far? • 10km View of Heaps 8 How Can We Efficiently Implement a PQ? • Stack? • Queue? • Linked List – Singly, doubly, … Priority value is _______ insert is ________ • Array – Circular, resizing, … • BST • AVL Tree delete_min is _______ 9 AVL Tree as PQ How well does this work in terms of the number of priority values/data in the tree, n? Runtime of insert? Runtime of delete_min? 10 Today’s Outline • Priority Queue ADT • Solutions So Far? • 10km View of Heaps 11 Binary Heap Priority Q Data Structure Look! Invariants! • Heap-order property 2 – parent’s key is less than or equal to children’s keys – result: minimum is always at the top 4 5 • Structure property – “nearly complete tree” – result: depth is always O(log n); next open location always known 7 11 6 9 10 8 12 14 20 WARNING: this has NO SIMILARITY to the “heap” you hear about when people say “things you create with new go on the heap”. And, 12 this is a binary tree but is NOT a binary search tree. Nifty Storage Trick 2 0 1 2 4 5 3 4 7 6 7 11 6 5 10 8 8 9 12 14 20 9 11 10 0 1 2 3 4 5 6 7 8 2 4 5 7 6 10 8 11 9 9 10 11 12 12 14 20 So, we get the speed and “spatial locality” of arrays. 13 (Also the occasional expensive resizes of arrays.) DeleteMin pqueue.deleteMin() 2 2 ? 4 7 11 5 6 9 10 12 14 20 4 8 7 11 5 6 9 Invariants violated! DOOOM!!! 10 8 12 14 20 14 Percolate Down ? 4 4 7 5 6 ? 10 8 7 11 9 12 14 20 5 6 10 11 9 12 14 20 4 4 6 7 5 ? 8 10 11 9 12 14 20 6 8 7 5 12 10 11 9 20 14 20 8 15 Finally… 4 6 7 11 5 12 10 8 9 20 14 Algorithm intuition: move the rightmost, bottom element to the root; then, fix the invariant downward until stable. 16 Insert pqueue.insert(3) 2 2 4 7 11 5 6 9 10 12 14 20 4 8 7 11 5 6 9 10 12 14 20 Invariant violated! What will we do? Intuition: insert at the bottom-right; then, fix invariant upward until stable. 8 3 17 Insert Result 2 2 4 7 4 5 6 10 11 9 12 14 20 3 8 7 3 6 5 8 11 9 12 14 20 10 18 Closer Look at Creating Heaps To create a heap given a list of items: Create an empty heap. For each item: insert into heap. Time complexity? a. O(lg n) b. O(n) c. O(n lg n) d. O(n2) e. None of these 9, 4, 8, 1, 7, 2 3 5 10 3 12 11 19 A Better BuildHeap Floyd’s Method. Thank you, Floyd. 12 5 11 3 10 6 9 4 8 1 7 2 pretend it’s a heap and fix the heap-order property! 12 Invariant violated! Where can the order invariant be violated 3 in general? a. Anywhere b. Non-leaves 4 8 c. Non-roots 5 11 10 1 6 7 2 9 20 Build(this)Heap: fix from bottom to top 12 12 5 11 3 4 10 8 1 2 7 5 9 11 3 6 4 1 2 8 10 7 6 12 12 5 3 4 2 1 8 10 7 6 11 9 1 9 3 4 2 5 8 10 7 6 11 9 21 Finally… 1 3 4 2 5 12 8 10 7 6 9 11 runtime: 22 Build(any)Heap This is as many violations as we can get. How do we fix them? Let’s play colouring games! 23 “amortized analysis!” Other Uses for Trees (besides BSTs and heaps) • Family Trees • Organization Charts • Classification trees – what kind of flower is this? – is this mushroom poisonous? • File directory structure – folders, subfolders in Windows – directories, subdirectories in UNIX • Function call tree (i.e., a record of everything that 24 goes in the call stack) To Do • Read: KW Section 8.5 25 Coming Up • Sorting, sorting, and more sorting! 26 Tree Terminology Reference A root: the single node with no parent leaf: a node with no children B child: a node pointed to by me parent: the node that points to me D E sibling: another child of my parent ancestor: my parent or my parent’s ancestor descendent: my child or my child’s descendent subtree: a node and its descendents C F H G I J K L M N We sometimes use degenerate versions of these definitions that allow NULL as the empty tree. (This can be very handy for recursive base cases!) 27