Module 2: Priority Queues
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1 Module 2: Priority Queues CS 240 Data Structures and Data Management T. Biedl K. Lanctot M. Sepehri S. Wild Based on lecture notes by many previous cs240 instructors David R. Cheriton School of Computer Science, University of Waterloo Winter 2018 version :23 Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
2 Outline 1 Priority Queues Abstract Data Types ADT Priority Queue Binary Heaps Operations in Binary Heaps PQ-Sort and Heapsort Intro for the Selection Problem Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter 2018
3 Outline 1 Priority Queues Abstract Data Types ADT Priority Queue Binary Heaps Operations in Binary Heaps PQ-Sort and Heapsort Intro for the Selection Problem Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter 2018
4 Abstract Data Types Abstract Data Type (ADT): A description of information and a collection of operations on that information. The information is accessed only through the operations. We can have various realizations of an ADT, which specify: How the information is stored (data structure) How the operations are performed (algorithms) Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
5 Stack ADT Stack: an ADT consisting of a collection of items with operations: push: inserting an item pop: removing the most recently inserted item Items are removed in LIFO (last-in first-out) order. We can have extra operations: size, isempty, and top Applications: Addresses of recently visited sites in a Web browser, procedure calls Realizations of Stack ADT using arrays using linked lists Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
6 Queue ADT Queue: an ADT consisting of a collection of items with operations: enqueue: inserting an item dequeue: removing the least recently inserted item Items are removed in FIFO (first-in first-out) order. Items enter the queue at the rear and are removed from the front. We can have extra operations: size, isempty, and front Realizations of Queue ADT using (circular) arrays using linked lists Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
7 Outline 1 Priority Queues Abstract Data Types ADT Priority Queue Binary Heaps Operations in Binary Heaps PQ-Sort and Heapsort Intro for the Selection Problem Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter 2018
8 Priority Queue ADT Priority Queue: An ADT consisting of a collection of items (each having a priority) with operations insert: inserting an item tagged with a priority deletemax: removing the item of highest priority deletemax is also called extractmax. The priority is also called key. The above definition is for a maximum-oriented priority queue. A minimum-oriented priority queue is defined in the natural way, by replacing the operation deletemax by deletemin. Applications: typical todo list, simulation systems Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
9 Using a Priority Queue to Sort PQ-Sort(A[0..n 1]) 1. initialize PQ to an empty priority queue 2. for i 0 to n 1 do 3. create an item x with key A[i] 4. PQ.insert(x) 5. for i 0 to n 1 do 6. A[n 1 i] key(pq.deletemax()) Note: Run-time depends on how we implement the priority queue. Sometimes written as: O(n + n insert + n deletemax) Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
10 Realizations of Priority Queues Attempt 1: Use unsorted arrays Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
11 Realizations of Priority Queues Attempt 1: Use unsorted arrays insert: O(1) deletemax: O(n) Note: We assume dynamic arrays, i. e., expand by doubling as needed. (Amortized over all insertions this takes O(1) extra time.) Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
12 Realizations of Priority Queues Attempt 1: Use unsorted arrays insert: O(1) deletemax: O(n) Note: We assume dynamic arrays, i. e., expand by doubling as needed. (Amortized over all insertions this takes O(1) extra time.) Using unsorted linked lists is identical. This realization used for sorting yields selection sort. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
13 Realizations of Priority Queues Attempt 1: Use unsorted arrays insert: O(1) deletemax: O(n) Note: We assume dynamic arrays, i. e., expand by doubling as needed. (Amortized over all insertions this takes O(1) extra time.) Using unsorted linked lists is identical. This realization used for sorting yields selection sort. Attempt 2: Use sorted arrays Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
14 Realizations of Priority Queues Attempt 1: Use unsorted arrays insert: O(1) deletemax: O(n) Note: We assume dynamic arrays, i. e., expand by doubling as needed. (Amortized over all insertions this takes O(1) extra time.) Using unsorted linked lists is identical. This realization used for sorting yields selection sort. Attempt 2: Use sorted arrays insert: O(n) deletemax: O(1) Using sorted linked-lists is identical. This realization used for sorting yields insertion sort. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
15 Outline 1 Priority Queues Abstract Data Types ADT Priority Queue Binary Heaps Operations in Binary Heaps PQ-Sort and Heapsort Intro for the Selection Problem Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter 2018
16 Third Realization: Heaps A (binary) heap is a certain type of binary tree. Recall binary trees: A binary tree is either empty, or consists of three parts: a node and two binary trees (left subtree and right subtree). Terminology: root, leaf, parent, child, level, sibling, ancestor, descendant, etc. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
17 Example Heap Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
18 Heaps Definition A max-heap is a binary tree with the following two properties: 1 Structural Property: All the levels of a heap are completely filled, except (possibly) for the last level. The filled items in the last level are left-justified. 2 Heap-order Property: For any node i, the key of parent of i is larger than or equal to key of i. A min-heap is the same, but with opposite order property. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
19 Heaps Definition A max-heap is a binary tree with the following two properties: 1 Structural Property: All the levels of a heap are completely filled, except (possibly) for the last level. The filled items in the last level are left-justified. 2 Heap-order Property: For any node i, the key of parent of i is larger than or equal to key of i. A min-heap is the same, but with opposite order property. Lemma: Height of a heap with n nodes is Θ(log n). Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
20 Storing Heaps in Arrays Heaps should not be stored as binary trees! Let H be a heap of n items and let A be an array of size n. Store root in A[0] and continue with elements level-by-level from top to bottom, in each level left-to-right. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
21 Storing Heaps in Arrays Heaps should not be stored as binary trees! Let H be a heap of n items and let A be an array of size n. Store root in A[0] and continue with elements level-by-level from top to bottom, in each level left-to-right. A[0] 50 A[1] A[2] A[3] A[4] A[5] 8 10 A[6] A[7] A[8] Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
22 Heaps in Arrays Navigation It is easy to navigate the heap using this array representation: the root node is A[0] the left child of A[i] (if it exists) is A[2i + 1], the right child of A[i] (if it exists) is A[2i + 2], the parent of A[i] (i 0) is A[ i 1 2 ] Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
23 Heaps in Arrays Navigation It is easy to navigate the heap using this array representation: the root node is A[0] the left child of A[i] (if it exists) is A[2i + 1], the right child of A[i] (if it exists) is A[2i + 2], the parent of A[i] (i 0) is A[ i 1 2 ] Should hide implementation details using helper-functions! Allow to pass node v (identified by array-index) functions root(), parent(v), etc. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
24 Outline 1 Priority Queues Abstract Data Types ADT Priority Queue Binary Heaps Operations in Binary Heaps PQ-Sort and Heapsort Intro for the Selection Problem Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter 2018
25 Insertion in Heaps Place the new key at the first free leaf The heap-order property might be violated: perform a bubble-up: Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
26 Insertion in Heaps Place the new key at the first free leaf The heap-order property might be violated: perform a bubble-up: bubble-up(v) v: a node of the heap 1. while parent(v) exists and key(parent(v)) < key(v) do 2. swap v and parent(v) 3. v parent(v) The new item bubbles up until it reaches its correct place in the heap. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
27 Insertion in Heaps Place the new key at the first free leaf The heap-order property might be violated: perform a bubble-up: bubble-up(v) v: a node of the heap 1. while parent(v) exists and key(parent(v)) < key(v) do 2. swap v and parent(v) 3. v parent(v) The new item bubbles up until it reaches its correct place in the heap. Time: O(height of heap) = O(log n). Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
28 bubble-up example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
29 bubble-up example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
30 bubble-up example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
31 bubble-up example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
32 deletemax in Heaps The maximum item of a heap is just the root node. We replace root by the last leaf (last leaf is taken out). The heap-order property might be violated: perform a bubble-down: Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
33 deletemax in Heaps The maximum item of a heap is just the root node. We replace root by the last leaf (last leaf is taken out). The heap-order property might be violated: perform a bubble-down: bubble-down(v) v: a node of the heap 1. while v is not a leaf do 2. u child of v with largest key 3. if key(u) > key(v) then 4. swap v and u 5. v u 6. else 7. break Time: O(height of heap) = O(log n). Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
34 bubble-down example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
35 bubble-down example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
36 bubble-down example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
37 bubble-down example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
38 Priority Queue Realization Using Heaps Store items in priority queue in array A and keep track of size We assume existence of accessor functions root() and last() insert(x) 1. increase size 2. v last() 3. create item with priority x and assign it to v 4. bubble-up(v) deletemax() 1. x key(root()) 2. swap(root(),last()) 3. decrease size 4. bubble-down(root()) 5. return (x) Insert and deletemax: O(log n) Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
39 Outline 1 Priority Queues Abstract Data Types ADT Priority Queue Binary Heaps Operations in Binary Heaps PQ-Sort and Heapsort Intro for the Selection Problem Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter 2018
40 Sorting using heaps Heaps are one implementation of priority queues. PQ-Sort using Heaps takes O(n log n) time. PQ-SortWithHeaps(A) 1. initialize H to an empty heap 2. for i 0 to n 1 do 3. H.insert(A[i]) 4. for i 0 to n 1 do 5. A[n 1 i] H.deleteMax() Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
41 Sorting using heaps Heaps are one implementation of priority queues. PQ-Sort using Heaps takes O(n log n) time. PQ-SortWithHeaps(A) 1. initialize H to an empty heap 2. for i 0 to n 1 do 3. H.insert(A[i]) 4. for i 0 to n 1 do 5. A[n 1 i] H.deleteMax() Can improve this with two simple tricks: 1 Heaps can be built faster if we know all input in advance. 2 Can use the same array for input and heap. (O(1) additional space.) Heapsort Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
42 Building Heaps by Bubble-up Problem statement: Given n items (in A[0 n 1]) build a heap containing all of them. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
43 Building Heaps by Bubble-up Problem statement: Given n items (in A[0 n 1]) build a heap containing all of them. Solution 1: Start with an empty heap and insert items one at a time: simpleheapbuilding(a) A: an array 1. initialize H as an empty heap 2. for i 0 to size(a) 1 do 3. H.insert(A[i]) Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
44 Building Heaps by Bubble-up Problem statement: Given n items (in A[0 n 1]) build a heap containing all of them. Solution 1: Start with an empty heap and insert items one at a time: simpleheapbuilding(a) A: an array 1. initialize H as an empty heap 2. for i 0 to size(a) 1 do 3. H.insert(A[i]) This corresponds to doing bubble-ups Worst-case running time: Θ(n log n). Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
45 Building Heaps by Bubble-down Problem statement: Given n items (in A[0 n 1]) build a heap containing all of them. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
46 Building Heaps by Bubble-down Problem statement: Given n items (in A[0 n 1]) build a heap containing all of them. Solution 2: Using bubble-downs instead: heapify(a) A: an array 1. n size(a) 1 2. use A as the array storing the heap 3. // start at parent(last()), which has index (n 2)/2 4. for i (n 2)/2 downto 0 do 5. let v be the node corresponding to array-index i 6. bubble-down(v) Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
47 Building Heaps by Bubble-down Problem statement: Given n items (in A[0 n 1]) build a heap containing all of them. Solution 2: Using bubble-downs instead: heapify(a) A: an array 1. n size(a) 1 2. use A as the array storing the heap 3. // start at parent(last()), which has index (n 2)/2 4. for i (n 2)/2 downto 0 do 5. let v be the node corresponding to array-index i 6. bubble-down(v) A careful analysis yields a worst-case complexity of Θ(n). A heap can be built in linear time. Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
48 heapify example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
49 heapify example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
50 heapify example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
51 heapify example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
52 heapify example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
53 heapify example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
54 heapify example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
55 heapify example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
56 heapify example Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
57 HeapSort Idea: PQ-Sort with heaps. But: Use same input-array A for storing heap. Need heapdeletemax: Same as deletemax except pass array A of heap as parameter HeapSort(A) 1. heapify(a) 2. for i 0 to n 1 do 3. A[n 1 i] heapdeletemax(a) Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
58 Outline 1 Priority Queues Abstract Data Types ADT Priority Queue Binary Heaps Operations in Binary Heaps PQ-Sort and Heapsort Intro for the Selection Problem Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter 2018
59 Selection Problem Statement: The kth-max problem asks to find the kth largest item in an array A of n numbers. Solution 1: Make k passes through the array, deleting the maximum number each time. Complexity: Θ(kn). Solution 2: First sort the numbers. Then return the kth largest number. Complexity: Θ(n log n). Solution 3: Scan the array and maintain the k largest numbers seen so far in a min-heap Complexity: Θ(n log k). Solution 4: Make a max-heap by calling heapify(a). Call deletemax(a) k times. Complexity: Θ(n + k log n). Biedl, Lanctot, Sepehri, Wild (SCS, UW) CS240 Module 2 Winter / 23
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