Floor And Ceiling Recurrence

I m currently using substitution method to solve recurrences.

Floor and ceiling recurrence.

N c np. One of the main goals of this paper is to show that the bdc recurrence 1 1 under very general conditions on g. 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. T n c n 2 lg n 2.

When a recurrence contains floor and ceiling functions the math can become especially complicated. As a direct proof of a solution to a recurrence. Here pg 2 exercise 4 1 1 is an example where ceiling is ignored. Let s restrict the values of x with some inequalities to get rid of these pesky functions.

The discontinuities inherent in floor and ceiling functions make this nontrivial. In fact in clrs pg 88 its mentioned that. N has always an exact solution of the form f. In our example if we had assumed that n 4 k for some integer k the floor functions could have been conveniently omitted.

For example in the following example see example here. 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. I gather from public opinion that this is somewhat fishy. Floors and ceilings usually do not matter when solving recurrences.

The j programming language a follow on to apl that is designed to use standard keyboard symbols uses. They end up using the guess. If we want an exact solution for values of n that are not powers of 2 then we have to be precise about this. I came across places where floors and ceilings are neglected while solving recurrences.

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. For example we can ignore oors and ceilings when solving our recurrences as they usually do not a ect the nal guess. Here pg 2 exercise 4 1 1 is an example where ceiling is ignored. Often it helps to assume that the recurrence is defined only on exact powers of a number.

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 came across places where floors and ceilings are neglected while solving recurrences. Floors and ceilings usually do not matter when solving. 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.

Example from clrs chapter 4 pg 83 where floor is neglected. Log 2 n. For ceiling and. The problem i m having is dealing with t n that have either ceilings or floors.