# lect_22 - Recurrences Margaret M Fleck 15 March 2010 This...

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Recurrences Margaret M. Fleck 15 March 2010 This lecture does more examples of unrolling recurrences and shows how to use recursion trees to analyze divide-and-conquer recurrences. We’ll em- phasize finding big-O solutions rather than exact solutions. This material is in section 7.1 and 7.3 of Rosen. However, Rosen takes a slightly more abstract/general approach to divide-and-conquer recurrences in 7.3, so look at that section only if you are curious. 1 Announcements There’s a quiz coming Wednesday (17 March). Study materials are available on the web. There has been a rash of laptop thefts in/near Siebel. Keep careful watch on your laptop. Also, I need to close the door to my office on short errands (e.g. to pick up a printout). So, if you find my door closed, don’t immediately conclude I’m far away. 2 Recap Last class, we saw how to solve a recurrence using a technique called “un- rolling.” Here’s another example. Suppose that you deposit \$10,000 and your bank gives you 11% interest each year (which isn’t very likely, is it?). The function M for how much money you’ll have in n years is given by M (0) = 10000 M ( n ) = 1 . 11 M ( n - 1). 1

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Unrolling this recurrence, we get M ( n ) = 1 . 11 M ( n - 1) = 1 . 11(1 . 11 M ( n - 2)) = 1 . 11(1 . 11(1 . 11 M ( n - 3))) ... = (1 . 11) n M (0) = (1 . 11) n (10 , 000) M (30) = 228 , 922 . 97 3 A harder example Let’s do a more complex example: T (1) = 1 T ( n ) = 2 T ( n - 1) + 3 A note on terminology: If we call this a “recursive definition,” we usually refer to T (1) = 1 as the “base case.” If we call this a “recurrence relation,” we usually call T (1) = 1 the “initial condition.” This is just another of
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