[racket-dev] floor-log/base, ceiling-log/base, from Neil Toronto's recent commit
Thank you for this neat example. It is good for ho contracts and, if you don't mind, I may use it with attribution of course, in HtDP/2e. -- Matthias
On Feb 1, 2012, at 7:58 PM, John Boyle wrote:
> I happened to observe this commit from today by Neil Toronto:
>
> http://git.racket-lang.org/plt/commitdiff/47fcdd4916a2d33ee5c28eb833397ce1d2a515e2
>
> I may have some useful input on this, having dealt with similar problems myself.
>
> The problem: Given b and x, you want to find the largest integer n such that b^n <= x (or, for the ceiling problem, the smallest integer n such that b^n ≥ x). This can profitably be viewed as a case of a more general problem: given a monotonically increasing but otherwise opaque function f, find the largest integer n such that f(n) ≤ x. The previous algorithm to solve this can now be viewed as a linear search, taking n steps. This took too long, and Neil Toronto replaced it with an algorithm that depended on floating-point numbers; but I hate to see that happen, because floats will screw up when the numbers get too large; careful reasoning may prove that floats will be sufficiently accurate for a given application, but it would be nice to have an exact algorithm so that one didn't have to do that.
>
> I come here to present such an algorithm. Simply put: Use a binary search rather than a linear search. How do we do binary search on "the nonnegative integers"? Well, start with 0 and 1 as your bounds, and keep doubling the 1 until you get something that's actually too high; then you have a real lower and upper bound. This will take log(n) steps to find the upper bound, and then a further log(n) steps to tighten the bounds to a single integer. Thus, this takes roughly 2*log(n) steps, where each step involves calculating (expt b n) plus a comparison. It might be possible to do slightly better by not treating b^n as a black box (e.g. by using old results of b^n to compute new ones rather than calling "expt" from scratch, or by using "integer-length" plus some arithmetic to get some good initial bounds), but I suspect this is good enough and further complexity is not worth it.
>
> Note that this algorithm should work perfectly with any nonnegative rational arguments [other than 0 for b or x, and 1 for b], and should give as good an answer as any with floating-point arguments. Also, "integer-finverse", as I called it, might be useful for many other "floor"-type computations with exact numbers (I have so used it myself in an ad hoc Arc math library).
>
> ;Finds n such that n >= 0 and f(n) <= x < f(n+1) in about 2*log(n) steps.
> ;Assumes f is monotonically increasing.
> (define (integer-finverse f x)
> (define upper-bound
> (let loop ((n 1))
> (if (> (f n) x)
> n
> (loop (* n 2)))))
> (define (search a b) ;b is too big, a is not too big
> (let ((d (- b a)))
> (if (= d 1)
> a
> (let ((m (+ a (quotient d 2))))
> (if (> (f m) x)
> (search a m)
> (search m b))))))
> (search (quotient upper-bound 2) upper-bound))
>
> (define (floor-log/base b x)
> (cond ((< b 1) (- (ceiling-log/base (/ b) x)))
> ((= b 1) (error "Base shouldn't be 1."))
> ((< x 1) (- (ceiling-log/base b (/ x))))
> (else (integer-finverse (lambda (n) (expt b n)) x))))
>
> (define (ceiling-log/base b x)
> (cond ((< b 1) (- (floor-log/base (/ b) x)))
> ((= b 1) (error "Base shouldn't be 1."))
> ((< x 1) (- (floor-log/base b (/ x))))
> (else
> (let ((u (floor-log/base b x)))
> (if (= x (expt b u))
> u
> (+ u 1))))))
>
>
> Testing:
>
> > (floor-log/base 10 3)
> 0
> > (floor-log/base 3 10)
> 2
> > (ceiling-log/base 3 10)
> 3
> > (ceiling-log/base 1/3 10)
> -2
> > (floor-log/base 1/3 10)
> -3
> > (floor-log/base 2/3 10)
> -6
> > (ceiling-log/base 2/3 10)
> -5
> > (exact->inexact (expt 3/2 6))
> 11.390625
> > (exact->inexact (expt 3/2 5))
> 7.59375
>
> I might add, by the way, that I'm inclined to expect the base to be a second, optional argument (perhaps defaulting to 10) to a function called simply "floor-log" or "ceiling-log". I don't know if that fits with desired conventions, though.
> --John Boyle
> Science is what we understand well enough to explain to a computer. Art is everything else we do. --Knuth
>
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