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lecture14 - COMP 250 Winter 2010 14 recurrences 1 Feb 8...

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Unformatted text preview: COMP 250 Winter 2010 14 - recurrences 1 Feb 8, 2010 Recurrences For algorithms that involve for loops, it is straightforward to write the number of operation t ( n ) as a polynomial, whose the degree depends on the number of nested loops. For example, if we have k nested for loops, each of which run n times, then these loops take time t ( n ) proportional to n k . For recursive algorithms, it is sometimes less obvious how to express t ( n ), and some work needs to be done. Today, we will look at a number of examples. Example 1: Factorial Let t ( n ) be the time it takes to compute n ! . You should have an intuition that t ( n ) is O ( n ). This is easy to see if you use an iterative algorithm to compute n ! since we have a for loop which we iterate n times. What about if we use a recursive algorithm? Algorithm: Factorial(n) if n > then return n * Factorial ( n- 1) end if The recursive algorithm for computing n ! involves a method call and return and a multiplication. These operations can each be done in constant time. Moreover, each method call reduces the problem from size n to size n- 1. This suggests a relationship: t ( n ) = c + t ( n- 1) namely the time it takes to compute n ! is some constant plus the time it takes to compute ( n-...
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lecture14 - COMP 250 Winter 2010 14 recurrences 1 Feb 8...

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