Similar statements (3)

Consider two infinite horizontal lines $A$ and $B$ , separated $\ell$ units apart. The line $A$ has $m$ points at the abscissae $a_1, \dots, a_m$ . The line $B$ has $n$ points at the abscissae $b_1, \dots, b_n$ . Given $p$ different indices $i_1, \dots, i_p$ choosen from $\{1 \dots m\}$ , and $p$ different indices $j_1, \dots, j_p$ choosen from $\{1 \dots n\}$ , define $d_k$ as the Euclidean distance between $a_{i_k}$ and $b_{j_k}$ , that is, $d_k = \sqrt{(a_{i_k} - b_{j_k})^2 + \ell^2} \enspace .$

You are given $\ell$ , $p$ , and the points in $A$ and in $B$ . Pick $i_1, \dots, i_p$ and $j_1, \dots, j_p$ in order to

minimize $\min_{k=1..p} \enspace d_k$

Input

Input consists of several cases, each one with only integer numbers. Every case begins with four strictly positive numbers $\ell$ , $p$ , $m$ and $n$ . Follow $a_1 \le a_2 \le \dots \le a_{m-1} \le a_m$ . Follow $b_1 \le b_2 \le \dots \le b_{n-1} \le b_n$ . Assume $\ell \le 10^6$ , $p \le \min(m, n)$ , and that the absolute value of each abscissa is at most $10^6$ .

Additionally, assume that $m$ and $n$ are at most $10^5$ .

Output

For every case, print the result with four digits after the decimal point. If you use the long double type, the input cases have no precision issues.

Problem information

Author: Salvador Roura

Generation: 2026-01-25T10:09:04.522Z