CSE 310-Lecture4

# CSE 310-Lecture4 - C M L Click to edit Master subtitle...

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Unformatted text preview: C M L Click to edit Master subtitle style 10/22/10 C M L CSE 310 Algorithms and Data Structures Aviral Shrivastava Assistant Professor Computer Science and Engineering Arizona State University C M L 10/22/10 C M L Announcements • Assignment 2 is online – Due by Monday, June 21, 2010 • Assignment 3 is online – Due Monday, June 28, 2010 • Teaching Assistant – Ke Bai, [email protected] – BYENG 407 AA – Office Hours: 12:00 pm – 1:00 pm • No class on June 21, and June 23 • Yes Class on June 28 • Midterm on June 30 – Open book, open notes – closed computers C M L 10/22/10 C M L CSE 310 Roadmap • Induction • Recursion • Complexity Analysis • Sorting • Selection C M L 10/22/10 C M L Algorithm • An ' algorithm' is a method for solving a problem expressed as a finite sequence of instructions. – Proof of correctness: always give the right answer and never give a wrong answer – Proof of termination: always be completed in a finite number of steps, rather than in an infinite number C M L 10/22/10 C M L Power Function • T(n) = Time required to solve a problem of size n • Recurrence relations are used to determine the running time of recursive programs – recurrence relations themselves are recursive • T(0) = time to solve problem of size 0 – Base Case • T(n) = time to solve problem of size n long power(long x, long n) if (n == 0) return 1; else return x * power(x, n-1); T(0) = c1 for some constant c1 T(n) = c2 + T(n - 1) for some constant c2 T(n) = Ѳ(n) C M L 10/22/10 C M L Improved Power Algorithm long power(long x, long n) if (n==0) return 1; if ((n % 2) == 0) return power(x,n/2) * power(x,n/ 2); else return power(x,n/2) * power(x,n/ 2) * x; T(0) = c1 for some constant c1 T(n) = T(n/2) +c2 for some constant c2 T(n) = Ѳ(lgn) C M L 10/22/10 C M L Recurrences • The expression: is a recurrence . – Recurrence: an equation that describes a function in terms of its + = = 1 2 2 1 ) ( n c n T n c n T C M L 10/22/10 C M L Recurrence Examples =- + = ) 1 ( ) ( n n n s c n s - + = = ) 1 ( ) ( n n s n n n s + = = 1 2 2 1 ) ( n c n T n c n T + = = 1 1 ) ( n cn b n aT n c n T C M L 10/22/10 C M L Solving Recurrences • Substitution method • Iteration method • Master method C M L 10/22/10 C M L Solving Recurrences • The substitution method (CLR 4.1) – A.k.a. the “making a good guess method” – Guess the form of the answer, then use induction to find the constants and show that solution works – Examples: • T(n) = 2T(n/2) + & (n) & T(n) = &(n lg n) C...
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CSE 310-Lecture4 - C M L Click to edit Master subtitle...

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