# lecture10 - CS 473: Algorithms Chandra Chekuri...

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Unformatted text preview: CS 473: Algorithms Chandra Chekuri [email protected] 3228 Siebel Center University of Illinois, Urbana-Champaign Fall 2009 Chekuri CS473ug Fibonacci Numbers Part I Introduction to Dynamic Programming Chekuri CS473ug Fibonacci Numbers Recursion Reduction: reduce one problem to another Recursion: a special case of reduction reduce problem to a smaller instance of itself self-reduction Chekuri CS473ug Fibonacci Numbers Recursion Reduction: reduce one problem to another Recursion: a special case of reduction reduce problem to a smaller instance of itself self-reduction Problem instance of size n is reduced to one or more instances of size n- 1 or less. For termination, problem instances of small size are solved by some other method as base cases Chekuri CS473ug Fibonacci Numbers Recursion in Algorithm Design Tail Recursion: problem reduced to a single recursive call after some work. Easy to convert algorithm into iterative or greedy algorithms. Examples: Interval scheduling, MST algorithms, etc. Divide and Conquer: problem reduced to multiple independent sub-problems that are solved separately. Conquer step puts together solution for bigger problem. Dynamic Programming: problem reduced to multiple (typically) dependent or overlapping sub-problems. Use memoization to avoid recomputation of common solutions leading to iterative bottom-up algorithm. Chekuri CS473ug Fibonacci Numbers Fibonacci Numbers Fibonacci numbers defined by recurrence: F ( n ) = F ( n- 1) + F ( n- 2) and F (0) = 0 , F (1) = 1 . These numbers have many interesting and amazing properties. A journal The Fibonacci Quarterly ! F ( n ) = ( φ n- (1- φ ) n ) / √ 5 where φ is the golden ratio (1 + √ 5) / 2 ’ 1 . 618. lim n →∞ F ( n + 1) / F ( n ) = φ Chekuri CS473ug Fibonacci Numbers Fibonacci Numbers Fibonacci numbers defined by recurrence: F ( n ) = F ( n- 1) + F ( n- 2) and F (0) = 0 , F (1) = 1 . These numbers have many interesting and amazing properties. A journal The Fibonacci Quarterly ! F ( n ) = ( φ n- (1- φ ) n ) / √ 5 where φ is the golden ratio (1 + √ 5) / 2 ’ 1 . 618. lim n →∞ F ( n + 1) / F ( n ) = φ Question: Given n , compute F ( n ). Chekuri CS473ug Fibonacci Numbers Recursive Algorithm for Fibonacci Numbers Fib(n): if (n = 0) return 0 else if (n = 1) return 1 else return Fib(n-1) + Fib(n-2) Chekuri CS473ug Fibonacci Numbers Recursive Algorithm for Fibonacci Numbers Fib(n): if (n = 0) return 0 else if (n = 1) return 1 else return Fib(n-1) + Fib(n-2) Running time? Let T ( n ) be the number of additions in Fib(n). Chekuri CS473ug Fibonacci Numbers Recursive Algorithm for Fibonacci Numbers Fib(n): if (n = 0) return 0 else if (n = 1) return 1 else return Fib(n-1) + Fib(n-2) Running time? Let T ( n ) be the number of additions in Fib(n)....
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## This note was uploaded on 01/22/2012 for the course CS 573 taught by Professor Chekuri,c during the Fall '08 term at University of Illinois, Urbana Champaign.

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lecture10 - CS 473: Algorithms Chandra Chekuri...

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