lec0406 - CS 173 Discrete Mathematical Structures Cinda...

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Unformatted text preview: CS 173: Discrete Mathematical Structures Cinda Heeren [email protected] Siebel Center, rm 2213 Office Hours: BY APPOINTMENT Cs173 - Spring 2004 CS 173 Announcements Exam #2… Cs173 - Spring 2004 CS 173 Recurrences Where did you see this before? T(n) = 2T(n/2) + n T(1) = c How about this one? T(n) = 2T(n-1) + 1 T(1) = c And this one? f(n) = f(n-1) + f(n-2) f(0) = 1, f(1) = 1 Merge Sort Towers of Hanoi Fibonacci #s Cs173 - Spring 2004 CS 173 Recurrences A train is defined to be an engine, followed by cars of two different kinds…long and short. Long cars are 2 units long, and short ones are 1. Examples: How many trains of length n are there? t n = t n-1 + t n-2 t = 1 t 1 = 1 We still don’t really know. Cs173 - Spring 2004 CS 173 Recurrences How many bit strings do not have 2 consecutive 0s? Examples: 01111010, 10, 10101 s n = s n-1 + s n-2 s 1 = 2 s 2 = 3 Cs173 - Spring 2004 CS 173 Recurrences A computer system considers a string of decimal digits a valid codeword if it contains an even number of 0s. Find a recurrence relation for a n , the number of valid n-digit codewords. Total # of codewords is sum of those starting with 0 and those starting with something else. How many start with 0? How many start with something else? 10 n-1 - a n-1 9a n-1 a n = 10 n-1 - a n-1 + 9a n-1 = 10 n-1 - 8a n-1 a 1 = 9 Cs173 - Spring 2004 CS 173 Recurrences We’ve seen a variety. Techniques for solving depend on the particular form of the recurrence. Linear Recurrences with Constant Coefficients a n = c 1 a n-1 + c 2 a n-2 + … + c k a n-k + f(n) Ex. a n = 5a n-1 - 6a n-2 , a 0 = 0, a 1 = 1 “order k” Given any k consecutive values, a unique solution exists....
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  • Spring '08
  • [email protected]
  • Recurrence relation, Discrete Mathematical Structures, Siebel Center, Cinda Heeren

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lec0406 - CS 173 Discrete Mathematical Structures Cinda...

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