PPT-1 The Sorting Problem Input:

A sequence of n numbers a 1 a 2 a n Output A permutation reordering a 1 a 2 a n of the input sequence such that a 1 a 2 a. Insertion Sort. Insertion Sort. Sorting problem:. Given an array of N integers, rearrange them so that they are in increasing order.. Insertion sort. Brute-force sorting solution.. In each iteration . Review. What makes a problem . decidable. ?. 3 properties of an . efficient. algorithm?. What is the meaning of “. complete. ”, “. mechanistic. ”, and “. deterministic. ”?. Is the . Halting Problem . Really. . Big Files. Sorting Part 3. Using K Temporary Files. Given . N records in file F. M records will fit into internal memory. Use K temp files, where K = N / M. Create K sorted files from F, then merge them. Sorting. . and. . Order. . Statistics. – . Part. II. Lecture . 5. CIS 670. Comp 122. Sorting – Definitions. Input: . n. . records. , . R. 1. … . R. n. , from a . file. .. Each record . R. Polynomial time O(. n. k. ) input size n, k constant. Tractable problems solvable in polynomial time(Opposite Intractable). Ex: sorting, whether number is prime, shortest path between two vertices . 2. Motivation. Sells( bar, beer, price ) Bars( bar, . addr. ). Joe’s Bud 2.50 Joe’s Maple St.. Joe’s Miller 2.75 Sue’s River Rd.. Sue’s Bud 2.50. Sue’s Coors 3.00. . Query: Find all locations that sell beer for less than 2.75.. Keyang. He. Discrete Mathematics. Basic Concepts. Algorithm . – . a . specific set of instructions for carrying out a procedure or solving a problem, usually with the requirement that the procedure terminate at some point. Insertion Sort: . Θ. (n. 2. ). Merge Sort:. Θ. (. nlog. (n)). Heap Sort:. Θ. (. nlog. (n)). We seem to be stuck at . Θ. (. nlog. (n)). Hypothesis: . Every sorting algorithm requires . Ω. (. nlog. Algorithms . that Err Extremely Rarely. Oded. . Goldreich. Weizmann Institute of Science. Based on Joint work with . Avi. . Wigderson. Standard . Derandomization. Challenges. Given. a c. ircuit . Algorithms . that Err Extremely Rarely. Oded. . Goldreich. Weizmann Institute of Science. Based on Joint work with . Avi. . Wigderson. Standard . Derandomization. Challenges. Given. a c. ircuit . Bubble Sort . of an array. Inefficient --- . O ( N. 2. ). easy to code. , . hence unlikely to contain errors. Algorithm. for . outerloop. = 1 to N. for . innerloop. = 0 to N-2. if ( item[. David Woodruff. Carnegie Mellon University. Theme: Tight Upper and Lower Bounds. Number of comparisons to sort an array. Number of exchanges to sort an array. Number of comparisons needed to find the largest and second-largest elements in an array. Based on . M. . Sipser. , “Introduction to the Theory of Computation,” Second Edition, Thomson/Course Technology, 2006, Chapter 5.. Review. Recall the . halting problem. :. . HALT. TM. = { . . CS 165: Project in Algorithms . and Data Structures. Michael T. Goodrich. Some slides are from J. Miller, CSE 373, U. Washington. Why Sorting?. Practical application. People by last name. Countries by population.