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Topic: How to find a bounding line?
Replies: 39   Last Post: Jul 12, 2013 5:39 AM

 Messages: [ Previous | Next ]
 William Elliot Posts: 2,637 Registered: 1/8/12
Re: How to find a bounding line?
Posted: Jul 8, 2013 3:14 AM

On Sun, 7 Jul 2013, ols6000@sbcglobal.net wrote:

> I have a N points (x_i, y_i). I want to find a bounding line y = ax + b,
> such that y_i - y(x_i) >= 0. x_i and y_i are positive integers; a, b and y
> are real,and x is an integer.
> y_i = i, and x_{i+1} > x_i for all i<N
> N is very large (100,000 - 1,000,000), so I'm looking for a method that is O(N) (or better).

Why does x have to be an integer? Is't it a variable over the x-axis?

S = { (a1,1),.. (ak,k) }
for j = 1,.. k-1, 0 < aj < a_(j+1)
for j = 1,.. k, aj in N, y(aj) <= j
y(x) = rx + s; r,s in R

Find r,s in R for which sum(j=1,k) (j - y(aj)) is a minimum.
Since sum(j=1,k) (j - y(aj)) = k(k+1)/2 - sum(j=1,k) y(aj), the
problem becomes find r,s in R for which sum(j=1,k) y(aj) is a maximum.
with the constraints for j = 1,.. k, r.aj + s <= j.

s = sum(j=1,k) y(aj) = r.sum(j=1,k) aj + ks

Since r.a1 + s <= 1, to maximize s, set r.a1 + s = 1.
Thus s = 1 - r.a1. Let L1 be the horizontal line y = 1
and for j = 2,.. k draw a line Lj from (a1,1) to (aj,j).
Let r be the minimal slope of all the lines, L2,.. Lk.
Whence, y(x) = r(x - a1) + 1 is the line you want.

> An ideas on how this line could be found, i e, how to determine a and b
> efficiently from the N data points?

Date Subject Author
7/7/13 Woody
7/7/13 Scott Berg
7/7/13 Peter Percival
7/7/13 Woody
7/7/13 quasi
7/7/13 quasi
7/8/13 quasi
7/8/13 Woody
7/8/13 quasi
7/8/13 LudovicoVan
7/8/13 LudovicoVan
7/10/13 Woody
7/10/13 quasi
7/8/13 Leon Aigret
7/8/13 Woody
7/10/13 Leon Aigret
7/10/13 Leon Aigret
7/10/13 Woody
7/10/13 RGVickson@shaw.ca
7/10/13 Woody
7/10/13 quasi
7/7/13 quasi
7/7/13 quasi
7/7/13 quasi
7/8/13 William Elliot
7/8/13 Peter Percival
7/8/13 quasi
7/11/13 Woody
7/11/13 quasi
7/11/13 LudovicoVan
7/11/13 quasi
7/11/13 Leon Aigret
7/11/13 Woody
7/11/13 Leon Aigret
7/12/13 Woody
7/12/13 Leon Aigret
7/11/13 Woody
7/12/13 quasi
7/12/13 Woody
7/12/13 quasi