P3638 [APIO2013] Robots

Engineers of VRI (Voltron Robotics Institute) built $n$ robots. Any two compatible robots standing in the same cell can merge into a composite robot. We number the robots from $1$ to $n$ ($n \le 9$). Two robots are compatible if their numbers are consecutive, in which case they can merge into a composite robot. Initially, each of the $n$ robots has a unique number. A composite robot formed by two or more robots carries two numbers: the minimum and maximum among the numbers of all robots that compose it. For example, robot $2$ can merge only with robot $1$ or robot $3$. If robot $2$ merges with robot $3$, they form the composite robot $2$–$3$. If the composite robot $2$–$3$ merges with the composite robot $4$–$6$, they form the composite robot $2$–$6$. Once all robots have merged, they form the composite robot $1$–$n$. The engineers placed these $n$ robots in a closed room bounded by walls. The room is divided into $w \times h$ grid cells. Some cells contain obstacles that robots cannot pass through or stop on; other cells allow multiple robots to stop in or pass through. At any time, a robot occupies exactly one cell. Initially, all robots are in distinct cells. These original robots do not move on their own. They only move when an engineer pushes them along the $x$-axis or $y$-axis. After being pushed, a robot continues moving straight in that direction until it hits an obstacle or a wall, where it stops. After stopping, it checks whether there are robots in the current cell that it can merge with; if so, it merges and continues checking until no further merge is possible. Engineers may push robots only in four directions: left, right, up, and down. Also, while a robot is still moving, no other robot may be pushed; therefore, at any moment, at most one robot is moving in the room. To help robots turn, the engineers placed turners in some cells. Specifically, there are clockwise turners (right-turners) and counterclockwise turners (left-turners). A clockwise turner makes a robot that arrives at that cell turn $90^\circ$ clockwise; a counterclockwise turner makes it turn $90^\circ$ counterclockwise. Given the initial state of the room, compute the minimum number of pushes needed to merge all $n$ robots (if possible).

Your program must read from standard input. The first line contains three integers $n$, $w$, and $h$, separated by spaces. Each of the next $h$ lines describes the initial state of the room and contains $w$ characters. Each of these $w \times h$ characters represents one cell of the room, with the following meanings: - '1' to '9': there is a robot in this cell with that number. - 'x': there is an obstacle in this cell. - 'A': there is a counterclockwise turner in this cell. - 'C': there is a clockwise turner in this cell. - '.': this cell is empty.

Your program must write to standard output. Output a single integer: the minimum number of pushes required. If it is impossible to merge all robots, output -1.

Step 1: Push robot 3 to the right. After it hits a turner, it will continue moving upward until it hits the wall and stops. Step 2: Push robot 4 upward. After it hits the wall, it stops and merges with robot 3 to form the composite robot 3–4. Step 3: Push robot 2 upward. After it hits a turner, it moves left. Because there is a wall to the left, it stays in place. Step 4: Push robot 2 to the right. Since it is on a turner, it will move upward until it hits the wall and stops, then merges with robot 1 to form the composite robot 1–2. Step 5: Push the composite robot 3–4 to the left. After it hits the wall, it stops and merges with the composite robot 1–2 to form the composite robot 1–4. We will use the following four classes of input test cases to test your program. 1. (10 points) $n = 2$, $w \le 10$, $h \le 10$, with no turners. 2. (20 points) $n = 2$, $w \le 10$, $h \le 10$. 3. (30 points) $n \le 9$, $w \le 300$, $h \le 300$. 4. (40 points) $n \le 9$, $w \le 500$, $h \le 500$. Translated by ChatGPT 5