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Program Efficiency

Program Efficiency. Interested in “order of magnitude” That is, how the work grows as n grows Where n is the “problem size” Usually we just look at the loops Since that’s where most work occurs Big-O notation O(f( n )) means work grows like about f( n ) Constants don’t matter.

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Program Efficiency

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  1. Program Efficiency • Interested in “order of magnitude” • That is, how the work grows as n grows • Where n is the “problem size” • Usually we just look at the loops • Since that’s where most work occurs • Big-O notation • O(f(n)) means work grows like about f(n) • Constants don’t matter

  2. Big-O Example 1 for(int i = 0; i < x.length; i++) { no loops here } • We say this is O(n) • Since it takes about x.length “work” • Depends on problem “size” • Size is always given as n

  3. Big-O Example 2 for(int i = 0; i < x.length; i++) { for(int j = 0; j < x.length; j++) { no loops here } } • It’s n “work” for each of n times thru outer loop • So, this is O(n2)

  4. Big-O Example 3 for(int i = 0; i < x.length; i++) { for(int j = i + 1; j < x.length; j++) { no loops here } } • When i = 0, we have n - 1 “work” • When i = 1, we have n - 2 “work”, etc. • So, (n - 1)+(n-2)+ … + 1 = n(n-1)/2, or O(n2)

  5. Program Efficiency • Big-O notation • O(1) constant time • O(n) linear time • O(n log n) log linear • O(n2) quadratic • O(n3) cubic • O(2n) exponential • O(n!) factorial • Polynomials are good, exponentiala are bad

  6. Big-O Example 4 for(int i = 1; i <= n; i++) { for(int j = 1; j <= n; j++) { for(int k = n; k >= 1; k--) { no loops here } } } • n  n n “work”, so this is O(n3)

  7. Big-O Example 5 for(int i = 0; i <= n; i++) { for(int j = 0; j < i * i; j++) { no loops here } } • 02 + 12 + 22 + … + n2 = n(n+1)(2n+1)/6 • So, O(n3)

  8. Big-O Example 6 for(int i = n; i >= 0; i -= 2) { no loops here } • About n/2 iterations • So, O(n)

  9. Big-O Example 7 for(int i = 0; i < n; i++) { for(int j = i; j > 0; j /= 2) { no loops here } } • 0 + 1 + 2 + 2 + 3 + 3 + 4 + 4 + … + log n + log n • Adds up to about (log n)((log n) + 1) • Where “log” is to the base 2 • This is O((log n)2)

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