Question 1

You are given a sequence of integers $a$ of length $2n$. You have to split these $2n$ integers into $n$ pairs; each pair will represent the coordinates of a point on a plane. Each number from the sequence $a$ should become the $x$ or $y$ coordinate of exactly one point. Note that some points can be equal. After the points are formed, you have to choose a path $s$ that starts from one of these points, ends at one of these points, and visits all $n$ points at least once. The length of path $s$ is the sum of distances between all adjacent points on the path. In this problem, the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is defined as $|x_1 - x_2| + |y_1 - y_2|$. Your task is to form $n$ points and choose a path $s$ in such a way that the length of path $s$ is minimized. For example, if the given sequence is `15, 1, 10, 5`, then you can form points $(10, 1)$ and $(15, 5)$ and start the path $s$ from the first point and end it at the second point. Then the length of the path will be $|10 - 15| + |1 - 5| = 5 + 4 = 9$. It can be shown that this is the best possible. Based on the above data, answer the given subquestions. Figure from the original question paper You are given a sequence of integers $a$ of length $2n$. You have to split these $2n$ integers into $n$ pairs; each pair will represent the coordinates of a point on a plane. Each number from the sequence $a$ should become the $x$ or $y$ coordinate of exactly one point. Note that some points can be equal. After the points are formed, you have to choose a path $s$ that starts from one of these points, ends at one of these points, and visits all $n$ points at least once. The length of path $s$ is the sum of distances between all adjacent points on the path. In this problem, the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is defined as $|x_1 - x_2| + |y_1 - y_2|$. Your task is to form $n$ points and choose a path $s$ in such a way that the length of path $s$ is minimized. For example, if the given sequence is `15, 1, 10, 5`, then you can form points $(10, 1)$ and $(15, 5)$ and start the path $s$ from the first point and end it at the second point. Then the length of the path will be $|10 - 15| + |1 - 5| = 5 + 4 = 9$. It can be shown that this is the best possible. Based on the above data, answer the given subquestions. Figure from the original question paper You are given a sequence of integers $a$ of length $2n$. You have to split these $2n$ integers into $n$ pairs; each pair will represent the coordinates of a point on a plane. Each number from the sequence $a$ should become the $x$ or $y$ coordinate of exactly one point. Note that some points can be equal. After the points are formed, you have to choose a path $s$ that starts from one of these points, ends at one of these points, and visits all $n$ points at least once. The length of path $s$ is the sum of distances between all adjacent points on the path. In this problem, the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is defined as $|x_1 - x_2| + |y_1 - y_2|$. Your task is to form $n$ points and choose a path $s$ in such a way that the length of path $s$ is minimized. For example, if the given sequence is `15, 1, 10, 5`, then you can form points $(10, 1)$ and $(15, 5)$ and start the path $s$ from the first point and end it at the second point. Then the length of the path will be $|10 - 15| + |1 - 5| = 5 + 4 = 9$. It can be shown that this is the best possible. Based on the above data, answer the given subquestions. Figure from the original question paper