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Efficiency (Chapter 2)

Efficiency (Chapter 2). Efficiency of Algorithms. Difficult to get a precise measure of the performance of an algorithm or program Can characterize a program by how the execution time or memory requirements increase as a function of increasing input size Big-O notation

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Efficiency (Chapter 2)

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  1. Efficiency (Chapter 2)

  2. Efficiency of Algorithms • Difficult to get a precise measure of the performance of an algorithm or program • Can characterize a program by how the execution time or memory requirements increase as a function of increasing input size • Big-O notation • A simple way to determine the big-O of an algorithm or program is to look at the loops and to see whether the loops are nested

  3. Counting Executions • Generally, • A statement counts as “1” • Sequential statements add: • Statement; statement 2 • Loops multiply based on how many times they run

  4. Counting Example • Consider: • First time through outer loop, inner loop is executed n-1 times; next time n-2, and the last time once. • So we have • T(n) = 3(n – 1) + 3(n – 2) + … + 3 or • T(n) = 3(n – 1 + n – 2 + … + 1)

  5. Counting Example (continued) • We can reduce the expression in parentheses to: • n x (n – 1) 2 • So, T(n) = 1.5n2 – 1.5n • This polynomial is zero when n is 1. For values greater than 1, 1.5n2 is always greater than 1.5n2 – 1.5n • Therefore, we can use 1 for n0 and 1.5 for c to conclude that T(n) is O(n2)

  6. Big-O Growth

  7. Importance of Big-O • Doesn’t matter for small values of N • Shows how time grows with size • Regardless of comparative performance for small N, smaller big-O will eventually win • Technically, what we usually use when we say big-O, is really big-Theta

  8. Some Rules • If you have to read or write N items of data, your program is at least O(N) • Search programs range from O(log N) to O(N) • Sort programs range from O(N log N) to O(N2)

  9. Cases • Different data might take different times to run • Best case • Worst case • Average case • Example: consider sequential search…

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