Uniformly generating random pairs of different integers

Task:

• Generate a pair of random numbers (i,j) (order doesn't matter: (i,j) is equivalent to (j,i)).
• The pair must consist of two distinct values: i != j
• Pairs must be uniformly distributed. In other words, the probability is the same for all the possible pairs.
• Do this at a constant time.

1st attempt

Constant time? YES. Uniformly distributed? NO

x = np.random.randint(low=0, high=10 - 1)
y = np.random.randint(low=x + 1, high=10)

Visualizing the samples (by disregarding the order) :

Samples visualization (not uniformly distributed)

You can easily the effect of restricting the y to be larger than x, meaning higher pairs have higher probability (opacity here expresses density).

2nd attempt

Constant time? NO. Uniformly distributed? YES

x = np.random.randint(low=0, high=nbr_values)
y = np.random.randint(low=0, high=nbr_values)

while x == y:
  y = np.random.randint(low=0, high=nbr_values)

Visualizing the samples:

Samples visualization (uniformly distributed)

PS: it's not a homework, I'm experimenting with stochastic optimization techniques that use random neighbor generation using a swap operation.