# How to generate random numbers with each random number having a difference of at least x with all other elements?

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I know this goes against the definition of random numbers, but still I require this for my project. For instance, I want to generate an array with 5 random elements in `range(0, 200)`.

Now, I want each of the elements to have a difference of at least 15 between them. So the random array should look something like this:

``[15, 45, 99, 132, 199] ``

I can generate random numbers using numpy:

``np.random.uniform(low=0, high=200, size=5) ``

However, I am not able to keep a consistent difference of at least 15.

It would be nice if the question showed more effort towards solving the problem (i.e. from the Stack Overflow Tour: "Don't ask about... Questions you haven't tried to find an answer for (show your work!)"), but sometimes a question triggers an itch you just have to scratch...

Here's one way you could do it, written as the function `random_spaced`:

``import numpy as np   def random_spaced(low, high, delta, n, size=None):     """     Choose n random values between low and high, with minimum spacing delta.      If size is None, one sample is returned.     Set size=m (an integer) to return m samples.      The values in each sample returned by random_spaced are in increasing     order.     """     empty_space = high - low - (n-1)*delta     if empty_space < 0:         raise ValueError("not possible")      if size is None:         u = np.random.rand(n)     else:         u = np.random.rand(size, n)     x = empty_space * np.sort(u, axis=-1)     return low + x + delta * np.arange(n) ``

For example,

``In : random_spaced(0, 200, 15, 5) Out: array([ 30.3524969 ,  97.4773284 , 140.38221631, 161.9276264 , 189.3404236 ])  In : random_spaced(0, 200, 15, 5) Out: array([ 81.01616136, 103.11710522, 118.98018499, 141.68196775, 169.02965952]) ``

The `size` argument lets you generate more than one sample at a time:

``In : random_spaced(0, 200, 15, 5, size=3) Out:  array([[ 52.62401348,  80.04494534,  96.21983265, 138.68552066, 178.14784825],        [  7.57714106,  33.05818556,  62.59831316,  81.86507168, 180.30946733],        [ 24.16367913,  40.37480075,  86.71321297, 148.24263974, 195.89405713]]) ``

This code generates a histogram for each component using 100000 samples, and plots the corresponding theoretical marginal PDFs of each component:

``import matplotlib.pyplot as plt from scipy.stats import beta  low = 0 high = 200 delta = 15  n = 5 s = random_spaced(low, high, delta, n, size=100000)  for k in range(s.shape):     plt.hist(s[:, k], bins=100, density=True, alpha=0.25) plt.title("Normalized marginal histograms and marginal PDFs") plt.grid(alpha=0.2)  # Plot the PDFs of the marginal distributions of each component. # These are beta distributions. for k in range(n):     left = low + k*delta     right = high - (n - k - 1)*delta     xx = np.linspace(left, right, 400)     yy = beta.pdf(xx, k + 1, n - k, loc=left, scale=right - left)     plt.plot(xx, yy, 'k--', linewidth=1, alpha=0.25)     if n > 1:         # Mark the mode with a dot.         mode0 = k/(n-1)         mode = (right-left)*mode0 + left         plt.plot(mode, beta.pdf(mode, k + 1, n - k, loc=left, scale=right - left),                  'k.', alpha=0.25)  plt.show() ``

Here's the plot that it generates: As can be seen in the plot, the marginal distributions are beta distributions. The modes of the marginal distributions correspond to the positions of `n` evenly spaced points on the interval `[low, high]`.

By fiddling with how `u` is generated in `random_spaced`, distributions with different marginals can be generated (an old version of this answer had an example), but the distribution that `random_spaced` currently generates seems to be a natural choice. As mentioned above, the modes of the marginals occur in "meaningful" positions. Moreover, in the trivial case where `n` is 1, the distribution simplifies to the uniform distribution on [`low`, `high`].