Floor And Ceiling Recurrence

Floors and ceilings usually do not matter when solving recurrences.

Floor and ceiling recurrence.

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. N c np. 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 example in the following example see example here. T n c n 2 lg n 2. Here pg 2 exercise 4 1 1 is an example where ceiling is ignored. 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. When a recurrence contains floor and ceiling functions the math can become especially complicated. N has always an exact solution of the form f. The problem i m having is dealing with t n that have either ceilings or floors.

They end up using the guess. In fact in clrs pg 88 its mentioned that. For ceiling and. 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.

I gather from public opinion that this is somewhat fishy. 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. 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. Here pg 2 exercise 4 1 1 is an example where ceiling is ignored.

Example from clrs chapter 4 pg 83 where floor is neglected. 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.

For example we can ignore oors and ceilings when solving our recurrences as they usually do not a ect the nal guess. Floors and ceilings usually do not matter when solving. Log 2 n. Let s restrict the values of x with some inequalities to get rid of these pesky functions.

N d f. I m currently using substitution method to solve recurrences. Often it helps to assume that the recurrence is defined only on exact powers of a number. The j programming language a follow on to apl that is designed to use standard keyboard symbols uses.