具有距离条件的随机采样

Question

I want to sample two integers x and y at random from an interval [1,N] such that |xy| >= D, for some D < N. The code below (written in R) is what I have been using but it is terribly inefficient. Are there better methods for this sort of sampling? Thanks in adv.

N <- 100; D <- 10;

i <- sample(1:N, 2)

while ( abs( i[1] - i[2] ) < D ){
   i <- sort(sample(1:N, 2))
}

Answer 1

I guess the key is to realize that y is dependent on x (or the other way around). Here's the algorithm that should work in at most three steps:

1. sample x from [1:N]
2. sample y from [1:(x-D)] if (x-D) >= 1
   sample y from [x + D:N] if (x+D) <= N
3. If both conditions for y are met, choose one of the generated y  uniform at random

The idea is that once x was sampled, y needs to be in the range [1:(xD)] or [x+D:N] in order to satisfy |xy| >= D.

Examples:

N=100; D=10

a) x is close to N

1. x is sampled from 1:N as 95
2. to satisfy |x-y| >= D, y can be at most 85, so the range to sample y is [1:85]

b) x is close to 1

1. x is sampled from 1:N as 9
2. y must be at least 19, so the range to sample y is [19:N]

c) x is close to 50

1. x is sampled from 1:N as 45
2. y must be either at most 35, or at least 55, so the ranges to sample from are [1:35] and [55:N]

Answer 2

I would approach this by first randomly sampling a difference between the numbers is greater than or equal to D . In other words, we want to sample numbers between D and N-1 with replacement.

difference <- sample(D:(N-1), 20, replace = TRUE)

Now all we need to do is select our lower number by selecting a number between 1 and N - difference . We can do this using vapply .

lowerval <- vapply(N - difference, sample, numeric(1), 1)

Finally we get the upper value by adding the difference to the lower value.

upperval <- lowerval + difference