Data Structures: Disjoint Sets and Graphs, Lecture notes of Computer Graphics

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CS 225

Data Structures

April 5 – Disjoint Sets Finale + Graphs

Wade Fagen-Ulmschneider, Craig Zilles

Disjoint Sets Find int DisjointSets::find(int i) { if ( arr_[i] < 0 ) { return i; } else { return find( arr[i] ); } } 1 2 3 4 void DisjointSets::unionBySize(int root1, int root2) { int newSize = arr_[root1] + arr_[root2]; // If arr_[root1] is less than (more negative), it is the larger set; // we union the smaller set, root2, with root1. if ( arr_[root1] < arr_[root2] ) { arr_[root2] = root1; arr_[root1] = newSize; } // Otherwise, do the opposite: else { arr_[root1] = root2; arr_[root2] = newSize; } } 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Disjoint Sets Analysis The iterated log function: The number of times you can take a log of a number. log(n) = 0 , n ≤ 1 1 + log(log(n)) , n > 1 What is lg( 65536 )*?

Disjoint Sets Analysis In an Disjoint Sets implemented with smart unions and path compression on find : Any sequence of m union and find operations result in the worse case running time of O( ____________ ), where n is the number of items in the Disjoint Sets.

• Constant time access to any element, given an index a[k] is accessed in O(1) time, no matter how large the array grows
• Cache-optimized Many modern systems cache or pre-fetch nearby memory values due the “Principle of Locality”. Therefore, arrays often perform faster than lists in identical operations. [0] [1] [2] [3] [4] [5] [6] [7] Array

• Efficient general search structure Searches on the sort property run in O(lg(n)) with Binary Search
• Inefficient insert/remove Elements must be inserted and removed at the location dictated by the sort property, resulting shifting the array in memory – an O(n) operation [0] [1] [2] [3] [4] [5] [6] [7] Array [0] [1] [2] [3] [4] [5] [6] [7] Sorted Array

• First In First Out (FIFO) ordering of data Maintains an arrival ordering of tasks, jobs, or data
• All ADT operations are constant time operations enqueue() and dequeue() both run in O(1) time [0] [1] [2] [3] [4] [5] [6] [7] Array [0] [1] [2] [3] [4] [5] [6] [7] Unsorted Array [0] [1] [2] [3] [4] [5] [6] [7] Queue (FIFO)

• Last In First Out (LIFO) ordering of data Maintains a “most recently added” list of data
• All ADT operations are constant time operations push() and pop() both run in O(1) time [0] [1] [2] [3] [4] [5] [6] [7] Array [0] [1] [2] [3] [4] [5] [6] [7] Unsorted Array [0] [1] [2] [3] [4] [5] [6] [7] Stack (LIFO)

In Review: Data Structures Array

**- Sorted Array

• Unsorted Array** - Stacks - Queues - Hashing - Heaps - Priority Queues - UpTrees - Disjoint Sets List - Doubly Linked List - Skip List - Trees - BTree - Binary Tree - Huffman Encoding - kd-Tree - AVL Tree Graphs

The Internet 2003 The OPTE Project (2003) Map of the entire internet; nodes are routers; edges are connections.

“Rush Hour” Solution Unknown Source Presented by Cinda Heeren, 2016

Wolfram|Alpha's "Personal Analytics“ for Facebook Generated: April 2013 using Wade Fagen-Ulmschneider’s Profile Data

Conflict-Free Final Exam Scheduling Graph Unknown Source Presented by Cinda Heeren, 2016

Class Hierarchy At University of Illinois Urbana-Champaign A. Mori, W. Fagen-Ulmschneider, C. Heeren Graph of every course at UIUC; nodes are courses, edges are prerequisites http://waf.cs.illinois.edu/discovery/class_hi erarchy_at_illinois/