Floor And Ceiling Recurrence

Floors and ceilings usually do not matter when solving.

Floor and ceiling recurrence.

Let s restrict the values of x with some inequalities to get rid of these pesky functions. They end up using the guess. The j programming language a follow on to apl that is designed to use standard keyboard symbols uses. The problem i m having is dealing with t n that have either ceilings or floors.

Here pg 2 exercise 4 1 1 is an example where ceiling is ignored. For example we can ignore oors and ceilings when solving our recurrences as they usually do not a ect the nal guess. I came across places where floors and ceilings are neglected while solving recurrences. I gather from public opinion that this is somewhat fishy.

Here pg 2 exercise 4 1 1 is an example where ceiling is ignored. Log 2 n. In fact in clrs pg 88 its mentioned that. T n c n 2 lg n 2.

N c np. N d f. If we want an exact solution for values of n that are not powers of 2 then we have to be precise about this. A recurrence or recurrence relation defines an infinite sequence by describing how to calculate the n th element of the sequence given the values of smaller elements as in.

Example from clrs chapter 4 pg 83 where floor is neglected. In fact in clrs pg 88 its mentioned that. The discontinuities inherent in floor and ceiling functions make this nontrivial. When a recurrence contains floor and ceiling functions the math can become especially complicated.

The ceiling function is usually denoted by ceil x or less commonly ceiling x in non apl computer languages that have a notation for this function. I have a recurrence equation that would be very easy to solve without ceil and floor functions but i can t solve them exactly including floor and ceil. As a direct proof of a solution to a recurrence. If we are only using recursion trees to generate guesses and not prove anything we can tolerate a certain amount of sloppiness in our analysis.

Begin align k 1 0 k n n 1 k left left lceil frac n 2 right rceil right k left left lfloor frac n 2 right rfloor right qquad n in mathbb n end align. In our example if we had assumed that n 4 k for some integer k the floor functions could have been conveniently omitted. For ceiling and. I m currently using substitution method to solve recurrences.

I came across places where floors and ceilings are neglected while solving recurrences. One of the main goals of this paper is to show that the bdc recurrence 1 1 under very general conditions on g. Example from clrs chapter 4 pg 83 where floor is neglected. Often it helps to assume that the recurrence is defined only on exact powers of a number.