Floor And Ceiling Recurrence

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.

Floor and ceiling recurrence. I came across places where floors and ceilings are neglected while solving recurrences. When a recurrence contains floor and ceiling functions the math can become especially complicated. If we want an exact solution for values of n that are not powers of 2 then we have to be precise about this. For ceiling and.

For example we can ignore oors and ceilings when solving our recurrences as they usually do not a ect the nal guess. In fact in clrs pg 88 its mentioned that. Floors and ceilings usually do not matter when solving. 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. As a direct proof of a solution to a recurrence. The discontinuities inherent in floor and ceiling functions make this nontrivial. N d f.

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. Often it helps to assume that the recurrence is defined only on exact powers of a number. They end up using the guess. Log 2 n.

I m currently using substitution method to solve recurrences. Example from clrs chapter 4 pg 83 where floor is neglected. In fact in clrs pg 88 its mentioned that. The problem i m having is dealing with t n that have either ceilings or floors.

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. In our example if we had assumed that n 4 k for some integer k the floor functions could have been conveniently omitted. N has always an exact solution of the form f. T n c n 2 lg n 2.

I gather from public opinion that this is somewhat fishy. For example in the following example see example here. The j programming language a follow on to apl that is designed to use standard keyboard symbols uses. I came across places where floors and ceilings are neglected while solving recurrences.

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. 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. Let s restrict the values of x with some inequalities to get rid of these pesky functions. Floors and ceilings usually do not matter when solving recurrences.