**STEP 1: Introduction of the Concept** This problem is a shortest-path search on a 2D grid (the racetrack), where each state consists of a position and a velocity. The goal is to find the minimum number of moves to go from the start to the finish line, obeying the game's physics: the velocity can be changed by at most ±1 in the x or y direction per move, and the car must remain on the track at all times except for the possibility of "jumping" over off-track cells as long as it lands on the track. This is similar to pathfinding problems like those solved by Breadth-First Search (BFS), but with the added complication of velocity (momentum). --- **STEP 2: (a) Efficient Algorithm Design** We can model this as a graph search problem where each node represents a state: `(position, velocity)`. At each step, you can modify the velocity by ±1 in either direction and then update the position by adding the new velocity to the current position. We want the shortest sequence of moves to reach any cell in the finish line (last column), obeying the constraints. **Algorithm (BFS with State Expansion):** 1. Initialize a queue with all possible starting positions in the first column, velocity `(0, 0)`, and step count `0`. 2. Use a set to track visited states `(position, velocity)` to avoid revisiting. 3. For each state, generate all possible next velocities (each component can change by -1, 0, or +1) and compute the next position. 4. For each possible next state: - If out of bounds or not on the track, skip. - If lands on any finish cell (last column), return the current step count + 1. - If the state has not been visited, add it to the queue and mark as visited. 5. If the queue is empty and no finish line is reached, return "impossible". --- **STEP 3: (b) Correctness Argument** The algorithm is correct because: - **BFS guarantees minimal steps:** By exploring states level by level, the first time a finish cell is reached is with the minimum moves. - **State uniqueness:** By tracking both position and velocity, we avoid cycles and redundant work, ensuring all possible motion paths are considered. - **All valid moves:** By considering all ±1 changes to velocity, all possible legal moves are explored. - **Track constraints:** We only enqueue moves that land on the track, ensuring we never crash (overshooting finish is also checked). --- **STEP 4: (c) Time Complexity Analysis** Let `n` be the width/height of the matrix. - **Positions:** There are up to `n^2` positions. - **Velocities:** The velocity in each direction can range from `-n` to `+n`, so up to `(2n+1)^2` possible velocities. - **Total states:** O(n^2 * n^2) = O(n^4) - **Each state:** For each state, there are 9 possible next velocities (3 choices for each of x and y). - **Overall complexity:** O(n^4) time and space. --- **STEP 5: Final Answer in Few Sentences** Use BFS on the state space `(position, velocity)` to find the minimal steps to reach the finish line from the start, updating velocity by ±1 in each direction per move and only traversing valid track cells. The algorithm is correct because BFS finds the shortest path, and the time complexity is O(n^4). --- **Summary of Each Step:** - *Step 1:* Introduced the problem and its relation to shortest-path search with momentum. - *Step 2:* Designed a BFS-based state-space search algorithm. - *Step 3:* Explained why the BFS approach is correct for this problem. - *Step 4:* Analyzed the algorithm's time complexity. - *Step 5:* Summarized the complete solution concisely.