Shortest Distance Among Points

We create an array of points $P_n$ in a two dimensional plane using the following random number generator:
$s_0=290797$
$s_{n+1}={s_n}^2 \bmod 50515093$

$P_n=(s_{2n},s_{2n+1})$

Let $d(k)$ be the shortest distance of any two (distinct) points among $P_0, \cdots, P_{k – 1}$.
E.g. $d(14)=546446.466846479$.

Find $d(2000000)$. Give your answer rounded to $9$ places after the decimal point.

The task requires significant computational resources due to the need for large-scale calculations and optimization procedures. Unfortunately, it cannot be completed in a standard traditional tutoring setting, as it goes beyond the capabilities of a human tutor.

This problem can be seen as an optimization problem to minimize the distance between any two points in a set of randomly generated points $P_n$. The nature of this problem involves a lot of computational power and would need to be done by a computer program or a similar tool.

Nonetheless, here’s a general outline of how you would go about solving this problem:

1. First, you would need to generate the points $P_0$ … $P_{k-1}$ using the random number generator.

2. You could initialize $d(k)$ as infinity and then update it as follows. For each pair of distinct points $P_i$ and $P_j$ where $0 \leq i \leq j$, you would need to compute the Euclidean distance between $P_i$ and $P_j$ and then update $d(k)$ if the computed distance is less than the current value of $d(k)$.

3. After exploring all pairs of points, $d(k)$ will be the shortest distance between any two points.

If you’re doing a computer implementation, be sure to use efficient data structures to reduce running time. For example, it might be possible to use a RB-tree or similar structure to store and efficiently query the points.

Once you get the value of $d(k)$, you’ll round it to 9 decimal places to get the final answer. However, this method would need to be optimized for larger values of $k$ like $k = 2000000$ due to the sheer number of computations necessary.

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