Lecture19BigO - CS2603 Applied Logic for Hardware and...

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Unformatted text preview: CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 1 Computation Time and the Big O Lecture 19 — CS 2603 Applied Logic for Hardware and Software CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 2 Exponentiation the slow way … and the fast way Russian peasant’s algorithm – exponentiation, the fast way rp b n = if n == 0 then 1-- rp0 else if (odd n) then b ∗ (rp (b ∗ b) (n `div` 2))-- rp odd else (rp (b ∗ b) (n `div` 2))-- rp even ¡ Theorem (RPexp). ∀ n ∈ N .( ∀ b ≠ 0.((rp b n) = b n )) ¢ Note: There’s a similar algorithm for multiplication That’s the one the peasants were using No need for multiplication tables … and almost as fast For a few dollars more, you can do division similarly ¡ Proof — induction: P(n) ≡ ( ∀ b ≠ 0.((rp b n) = b n )) Base case: n = rp b = 1 rp0 = b 8 th grade algebra 8 `div` 3 = 2 3 rd grade quotient Exponentiation – the slow way pow b n = if n == 0 then 1 else b ∗ (pow b (n – 1)) Theorem: (pow b n) = b n CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 3 Inductive Cases — Russian Peasant Proof rp b n = if n == 0 then 1--rp 0 else if (odd n) then b ∗ (rp (b ∗ b) (n `div` 2))-- rp odd else (rp (b ∗ b) (n `div` 2))--rp even P(n) ≡ ( ∀ b ≠ 0.((rp b n) = b n )) Inductive case, odd exponent: n = 2 ∗ k + 1 > rp b n = rp b (2 ∗ k + 1) n = 2 ∗ k + 1 = b ∗ (rp (b ∗ b) ((2 ∗ k + 1) `div` 2)) rp odd = b ∗ (rp (b ∗ b) k) 3 nd grade arithmetic = b ∗ (b ∗ b) k P(k) = b ∗ (b 2 ) k 8 th grade algebra = b 2k+1 8 th grade algebra = b n n = 2 ∗ k + 1 Inductive case, even exponent: n = 2 ∗ k > … similar … qed CS2603 Applied Logic for Hardware and Software Rex Page – University of Oklahoma 4 Fast Peasants ¡ Russian peasants algorithm for exponentiation rp b n = if n == 0 then 1-- rp0} else if (odd n) then b ∗ (rp (b ∗ b) (n `div` 2))-- rp odd else (rp (b ∗ b) (n `div` 2))-- rp even ¡ Theorem (RPexp). ∀ n ∈ N .( ∀ b ≠ 0.((rp b n) = b n )) ¡ How long does it take rp to deliver?...
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Lecture19BigO - CS2603 Applied Logic for Hardware and...

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