Ladder Range
July 25, 2017
We represent the ladder as two vectors, left and right, both the same length, containing the y coordinates of the endpoints, sorted in ascending order. Here’s some sample data, giving the y coordinates in ascending order, bounded by 0 at the bottom and 100 at the top:
(define left '(0 1 4 9 16 25 36 49 64 81 100)) (define right '(0 3 14 15 23 24 25 49 57 92 100))
We have 11 rungs on the ladder, including top and bottom, and we need the slope and intercept of each line. Normally the formula for computing the slope of a line given two points on the line is (y2 − y1) / (x2 − x1), but since the x coordinates of our ladder are 0 and 1, the slope reduces to the height of the right end of the rung minus the height of the left end of the rung. The y-intercept of the rung is even easier to compute: since the left side of the ladder is the y axis, the intercept is just the height of the left end of the rung:
(define slopes
(map (lambda (left right) (- right left))
left right))
(define intercepts left)
> slopes (0 2 10 6 7 -1 -11 0 -7 11 0) > intercepts (0 1 4 9 16 25 36 49 64 81 100)
To determine if a target point (x, y) is above or below a line, we compute the y-value of the line at the target x, then compare to the target y:
(define (lt? idx x y)
(let ((slope (list-ref slopes idx))
(intercept (list-ref intercepts idx)))
(positive? (- (+ (* slope x) intercept) y))))
For instance, target point (0.5, 50) is between lines 7 and 8 (the rung from 49 to 49 and the rung from 64 to 57):
> (lt? 7 0.5 50) #f > (lt? 8 0.5 50) #t
Now we can find the desired interval by binary search:
(define (ladder x y)
(when (or (lt? 0 x y) (not (lt? 10 x y)))
(error 'ladder "out of bounds"))
(let loop ((lo 0) (hi (- (length left) 1)))
(let ((mid (quotient (+ lo hi) 2)))
(cond ((= (- hi lo) 1) (values lo hi))
((lt? mid x y) (loop lo mid))
(else (loop mid hi))))))
Here are some examples:
> (ladder 0.5 50) 7 8 > (ladder 0.25 25) 5 6
You can run the program at http://ideone.com/QP8BGS.
Binary search is where I tend to make the relevant assumptions explicit as comments, as below (in Julia). Otherwise I keep getting something off by one or some condition accidentally reversed.
I had two typos in my test code on the first run. I had also written the test invocations to expect zero-based indexes, so the correct one-based indexes were reported as off-by one on that first run :)
My function returns only the lower-rung index. The corresponding upper-rung index is the next integer.
module Ladder export findlower function findlower(ladder, x, y) n = length(ladder) # (x, y) is above (or on) rung 1 # (x, y) is below rung n b, e = 1, n while b + 1 < e # (x, y) is above (or on) rung b # (x, y) is below rung e m = b + div(e - b, 2) # b < m < e # (m is half-way between b and e) k, c = ladder[m] if y < k * x + c # (x, y) is below rung m e = m else # (x, y) is above (or on) rung m b = m end end # b + 1 == e # (x, y) is above (or on) rung b # (x, y) is below rung e b end end using Ladder # Praxis test slopes and intercepts but as Float64 ladder = collect(zip((0., 2., 10., 6., 7., -1., -11., 0., -7., 11., 0.), (0., 1., 4., 9., 16., 25., 36., 49., 64., 81., 100.))) println("Testing Praxis points (but Julia indexing is 1-based):") println("expecting 8, observing ", findlower(ladder, 0.5, 50.)) println("expecting 6, observing ", findlower(ladder, 0.25, 25.)) println() println("Testing a point on the third rung:") k, c = ladder[3] println("expecting 3, observing ", findlower(ladder, 0.2, k * 0.2 + c))In Python using the bisect module for the search. As bisect needs a list, the class LazyRungs is used, that behaves like a list and only calculates the y-value when needed.
from bisect import bisect_left as bisect class LazyRungs(object): 'rungs is a list of tuples (left, right)' def __init__(self, rungs, x): self.rungs = rungs self.x = x def __len__(self): return len(self.rungs) def __getitem__(self, i): (left, right), x = rungs[i], self.x return left * (1-x) + right * x def ladder(x, y, rungs): 'return 2 bracketing rungs (a, b), such that a < y <= b' i = bisect(LazyRungs(rungs, x), y) # i is the index such that rungs[i] >= y if not 0 < i < len(rungs): raise ValueError("(x,y) outside range") return i-1, i left = map(int, '0 1 4 9 16 25 36 49 64 81 100'.split()) right = map(int, '0 3 14 15 23 24 25 49 57 92 100'.split()) rungs = list(zip(left, right)) print(ladder(0.5, 50, rungs)) # (7, 8) print(ladder(0.25, 25, rungs)) # (5, 6)