09AlgorithmEfficiencyAndSorting

09AlgorithmEfficiencyAndSorting - CS 240 Chapter 9...

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Unformatted text preview: CS 240 Chapter 9 Algorithm Efficiency & CS 240 Chapter 9 Algorithm Efficiency & 1Page 1 Chapter 9 Algorithm Efficiency & Sorting Programming efficiently has been important up to this point, but were now going to examine the methods that have been developed to determine just how efficient a program is. Measuring Algorithm Efficiency Example: Sorting Algorithms Example: Searching Algorithms CS 240 Chapter 9 Algorithm Efficiency & CS 240 Chapter 9 Algorithm Efficiency & 2Page 2 Function T(n) is said to be O(f(n)) if there are positive constants c and n0 such that T(n)  c f(n) for every n � n0. Example: n3 + 3n2 + 6n + 5 is O(n3). (Use c = 15 and n0 = 1.) Example: n2 + n logn is O(n2). (Use c = 2 and n0 = 1.) Time Complexity Terminology: Big-O g(n) r(n) ng nr r(n) is O(g(n)) since (1)g(n) exceeds r(n) for all n-values past ng g(n) is O(r(n)) since (3)r(n) exceeds g(n) for all n-values past nr CS 240 Chapter 9 Algorithm Efficiency & CS 240 Chapter 9 Algorithm Efficiency & 3Page 3 Both algorithms below have O(n3) time complexity. (In fact, the execution time for Algorithm A is n3 + n2 + n, and the execution time for Algorithm B is n3 + 101n2 + n.) Demonstrating The Big-O Concept 1,110 11,110 1,010,100 2,010,100 1,001,001,000 1,101,001,000 1,000,100,010,000 1,010,100,010,000 1,000,010,000,100,000 1,001,010,000,100,000 1,000,001,000,001,000,000 1,000,101,000,001,000,000 10 100 1,000 10,000 100,000 1,000,000 A B ALGORITHM Input Size n CS 240 Chapter 9 Algorithm Efficiency & CS 240 Chapter 9 Algorithm Efficiency & 4Page 4 Both algorithms below have O(n2) time complexity. (In fact, the execution time for Algorithm C is n2 + 2n + 3, and the execution time for Algorithm D is n2 + 1002n + 3.) A Second Big-O Demonstration 123 10,123 10,203 110,203 1,002,003 2,002,003 100,020,003 110,020,003 10,000,200,003 10,100,200,003 1,000,002,000,003 1,001,002,000,003 10 100 1,000 10,000 100,000 1,000,000 C D ALGORITHM Input Size n CS 240 Chapter 9 Algorithm Efficiency & CS 240 Chapter 9 Algorithm Efficiency & 5Page 5 Both algorithms below have O(nlogn) time complexity. (In fact, the execution time for Algorithm E is nlogn + 5n, and the execution time for Algorithm F is nlogn + 105n. Note that the linear term for Algorithm F will dominate until n = 2105.) One More, Rather Complex Big-O Demonstration 90 1,090 1,200 11,200 15,000 115,000 190,000 1,190,000 2,200,000 12,200,000 25,000,000 125,000,000 10 100 1,000 10,000 100,000 1,000,000 E F ALGORITHM Input Size n CS 240 Chapter 9 Algorithm Efficiency & CS 240 Chapter 9 Algorithm Efficiency &...
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This note was uploaded on 08/26/2009 for the course CS 240 taught by Professor Klein,s during the Spring '08 term at Southern Illinois University Edwardsville.

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09AlgorithmEfficiencyAndSorting - CS 240 Chapter 9...

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