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Introduction to Algorithms 6.046J/18.401J/SMA5503 Lecture 2 Prof. Erik Demaine

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Solving recurrences The analysis of merge sort from Lecture 1 required us to solve a recurrence. Recurrences are like solving integrals, differential equations, etc. o Learn a few tricks. Lecture 3: Applications of recurrences.

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Substitution method The most general method: 1. Guess the form of the solution. 2. Verify by induction. 3. Solve for constants. Example: T(n) = 4T(n/2) + n [Assume that T(1) = Θ(1).] Guess O( n 3). (Prove O and separately.) Assume that T(k) ck 3 for k < n. Prove T(n) cn 3 by induction.

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Example of substitution T(n) = 4T (n/2) + n 4c (n/2) 3 + n = (c/2) n 3 + n = cn 3 – ((c/2) n 3 –n) desired – residual cn 3 desired whenever (c/2) n 3 – n 0, for example, if c 2 and n 1. residual

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Example (continued) We must also handle the initial conditions, that is, ground the induction with base cases. Base: T(n) = Θ(1) for all n < n 0, where n 0 is a suitable constant. For 1 n < n 0, we have Θ(1) cn 3, if we pick c big enough. ============================ This bound is not tight!

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A tighter upper bound? We shall prove that T(n) = O(n2). Assume that T(k) ck2 for k < n:: T(n) = 4T(n/2) + n 4 cn 2 + n = cn2 – (- n) [desired – residual] cn 2 for no choice of c > 0. Lose!

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A tighter upper bound! IDEA: Strengthen the inductive hypothesis. Subtract a low-order term. Inductive hypothesis: T(k) c1k2 – c2k for k < n. T(n) = 4T(n/2) + n 4(c 1 (n/2) 2 –c2(n/2) + n = c 1 n 2 –2 c 2 n – (c2n – n) c1 n 2 –c2n if c2 < 1 Pick c1 big enough to handle the initial conditions.

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Recursion-tree method A recursion tree models the costs (time) of a recursive execution of an algorithm. The recursion tree method is good for generating guesses for the substitution method. The recursion-tree method can be unreliable, just like any method that uses ellipses ( … ). The recursion-tree method promotes intuition, however.

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Example of recursion tree Solve T(n) = T(n/4) + T(n/2) + n 2:

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Example of recursion tree Solve T(n) = T(n/4) + T(n/2) + n 2: T(n)

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Example of recursion tree Solve T(n) = T(n/4) + T(n/2) + n2: N2 T(n/4) T(n/2)

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Example of recursion tree Solve T(n) = T(n/4) + T(n/2) + n2: n 2 (n/4) 2 (n/2) 2 T(n/16) T(n/8) T(n/8) T(n/4)

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Example of recursion tree Solve T(n) = T(n/4) + T(n/2) + n2: N2 (n/4) 2 (n/2) 2 (n/16) 2 (n/8) 2 (n/8) 2 (n/4) 2 Θ(1)

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Example of recursion tree Solve T(n) = T(n/4) + T(n/2) + n2: n 2 n 2 (n/4)2 (n/2)2 (n/16)2 (n/8)2 (n/8)2 (n/4)2 Θ(1)

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Example of recursion tree Solve T(n) = T(n/4) + T(n/2) + n2: n 2 (n/4)2 (n/2)2 5/16*n2 (n/16)2 (n/8)2 (n/8)2 (n/4)2 Θ(1)

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Example of recursion tree Solve T(n) = T(n/4) + T(n/2) + n2: n 2 (n/4)2 (n/2)2 5/16*n2 (n/16)2 (n/8)2 (n/8)2 (n/4)2 25/256*n2 Θ(1)

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Example of recursion tree Solve T(n) = T(n/4) + T(n/2) + n2: n 2 (n/4)2 (n/2)2 5/16*n2 (n/16)2 (n/8)2 (n/8)2 (n/4)2 25/256*n2 Θ(1) Total = n 2(1+5/16+ (5/16) 2+ (5/16) 3+…) =Θ( n 2)geometric series

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The master method The master method applies to recurrences of the form T(n) = aT(n/b) + f (n), where a 1, b > 1, and f is asymptotically positive.

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Three common cases Compare f (n) with n logba: 1. f (n) =O( n logba – ε)for some constantε> 0. f (n) grows polynomially slower than n logba (by an n ε factor). Solution: T(n) = Θ( n logba). 2.f (n)=Θ( n logba l g kn) for some constant k 0. f (n) and n logba grow at similar rates. Solution: T(n) = Θ( n logba l g k+1n).

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Three common cases (cont.) Compare f (n) with nlogba: 3.f (n) = ( n logba +ε) for some constantε> 0. f (n) grows polynomially faster than nlogba (by an nε factor), and f (n) satisfies the regularity condition that a f (n/b) c f (n) for some constant c < 1. Solution: T(n) = Θ( f (n)).

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Examples Ex. T(n) = 4T(n/2) + n a = 4, b = 2 n logba = n 2; f (n) = n. CASE 1: f (n) = O( n 2 – ε) for ε = 1. T(n) = Θ( n 2). Ex. T(n) = 4T(n/2) + n 2 a = 4, b = 2 n logba = n 2; f (n) = n 2. CASE 2: f (n) = Θ( n 2 lg 0 n ), that is, k = 0. T(n) = Θ( n 2 lgn ).

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Examples Ex. T(n) = 4T(n/2) + n 3 a = 4, b = 2 n logba = n 2; f (n) = n 3. CASE 3: f (n) = ( n 2 + ε) for ε = 1 and 4 (cn/2) 3 cn 3 (reg. cond.) for c = 1/2. T(n) = Θ( n 3). Ex. T(n) = 4T(n/2) + n 2/lgn a = 4, b = 2 n logba = n 2; f (n) = n 2/lgn. Master method does not apply. In particular, for every constant ε > 0, we have n ε= ω(lgn).

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General method (Akra-Bazzi) Let p be the unique solution to Then, the answers are the same as for the master method, but with n p instead of n logba. (Akra and Bazzi also prove an even more general result.)

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Idea of master theorem

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Conclusion Next time: applying the master method. For proof of master theorem, see CLRS.

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Appendix: geometric series

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Solving Recurrences with the Substitution Method.

Solving Recurrences with the Substitution Method.

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