Floor And Ceiling Recurrence

For example we can ignore oors and ceilings when solving our recurrences as they usually do not a ect the nal guess.

Floor and ceiling recurrence.

T n c n 2 lg n 2. 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. The problem i m having is dealing with t n that have either ceilings or floors. The discontinuities inherent in floor and ceiling functions make this nontrivial.

I m currently using substitution method to solve recurrences. N has always an exact solution of the form f. Let s restrict the values of x with some inequalities to get rid of these pesky functions. As a direct proof of a solution to a recurrence.

N c np. Example from clrs chapter 4 pg 83 where floor is neglected. Log 2 n. When a recurrence contains floor and ceiling functions the math can become especially complicated.

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. 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. N d f. Here pg 2 exercise 4 1 1 is an example where ceiling is ignored.

One of the main goals of this paper is to show that the bdc recurrence 1 1 under very general conditions on g. The j programming language a follow on to apl that is designed to use standard keyboard symbols uses. Floors and ceilings usually do not matter when solving. I came across places where floors and ceilings are neglected while solving recurrences.

For ceiling and. I came across places where floors and ceilings are neglected while solving recurrences. In our example if we had assumed that n 4 k for some integer k the floor functions could have been conveniently omitted. 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.

If we want an exact solution for values of n that are not powers of 2 then we have to be precise about this. 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. In fact in clrs pg 88 its mentioned that. In fact in clrs pg 88 its mentioned that.

They end up using the guess. Often it helps to assume that the recurrence is defined only on exact powers of a number. I gather from public opinion that this is somewhat fishy. Example from clrs chapter 4 pg 83 where floor is neglected.