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More on the Cost of Computation

More on the Cost of Computation. Examples from Last Time. add m n (on naturals): m recursive calls append l_m l_n: m recursive calls Cross l_m l_n: m*n recursive calls map f l_n: n recursive calls, and calls to f find-root: n logn recursive calls

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More on the Cost of Computation

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  1. More on the Cost of Computation

  2. Examples from Last Time • add m n (on naturals): m recursive calls • append l_m l_n: m recursive calls • Cross l_m l_n: m*n recursive calls • map f l_n: n recursive calls, and calls to f • find-root: n logn recursive calls • insert l_n: upto n recursive calls to insert • insert-sort l_n: upto n*n/2 calls to insert • maxi l_n: ? (Homework problem)

  3. Order of Complexity • Consider three algorithms • CostA(n) = 2*n3 + n2 + 1 • CostB(n) = 3*n2 + 10 • CostC(n) = 2n • Which one is better?

  4. Order of Complexity • Consider three algorithms • CostA(n) = 2*n3 + n2 + 1 • CostB(n) = 3*n2 + 10 • CostC(n) = 2n • One with smaller dominator is best • Can we formalize this notion?

  5. Order of Complexity • We'll say that "CostZ is order f (n) " if • CostZ(n+offset) < factor * f (n+offset) • More concisely: "CostZ is O (f (n))" • Examples: • CostA(n) = 2*n3 + n2 + 1 CostA is O(n3) • CostB(n) = 3*n2 + 10 CostB is O(n2) • CostC(n) = 2n CostC is O(2n) • Essentially in the theory and practice of "Algorithms", and in "Complexity theory".

  6. Famous "Complexity Classes" • O (1) constant-time (head, tail) • O (log n) logarithmic (binary search) • O (n) linear (vector multiplication) • O (n * log n) "n log n" (sorting) • O (n2) quadratic (matrix addition) • O (n3) cubic (matrix multiplication) • nO(1) polynomial (…many! …) • 2O(n) exponential (guess password) • Are there more?

  7. Plug for the lab: Vectors • Speaking of cost… • Anything ever bother you about lists? • Vectors • (build-vector N f) • (vector-ref V i) • (vector-length V) • (vector? e) • Home work problem on vectors

  8. "The Next Big Thing" • Say we want to "integrate" a sequence: • Given: [1,1,0,2,1] • Return: [1,2,2,4,5] • How can we implement this? • How fast is our program? • Can we do any better?

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