lecture15

# lecture15 - COMP 250 Winter 2010 15 recurrences 2 Notation...

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COMP 250 Winter 2010 15 - recurrences 2 Feb 10, 2010 Notation: floor and ceiling (rounding) The recurrence equations we will work with have arguments that are positive integers. If we have a fractional number and we wish to round it down (floor) or up (ceiling) to the nearest integer, then we sometimes use the following notation: x is the largest integer that is less than or equal to x . ⌊⌋ is called the floor operator. x is the smallest integer that is greater than or equal to x . ⌈⌉ is called the ceiling operator. Example 4: converting decimal to binary Recall the recursive version of the algorithm for converting a decimal number n to binary (see lecture 11). What is the asymptotic running time of this algorithm? We can write a recurrence relation as follows: t ( n ) = 1 + t ( n/ 2 ) where the floor operator is there to remind ourselves that we are ignoring the fractional part if n is odd. The “floor” operator is a bit annoying, so we bound the recurrence as follows. Let k = log n , where log n log 2 n (i.e 2 is the default base of log in this course). That is, k

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