Program to Find Out the Number of Moves to Reach the Finish Line in Python

In this problem, we have a car starting at position 0 with speed 1 on a one-dimensional road. We need to find the minimum number of moves to reach a target position using two possible operations:

• Acceleration: position += speed and speed *= 2

• Reverse: speed = -1 if speed > 0, otherwise speed = 1

For example, if the target is 10, the minimum number of moves required is 7.

Algorithm Approach

We use a depth-first search (DFS) with memoization to explore all possible move combinations. The algorithm considers different sequences of acceleration and reverse operations to find the optimal path.

Implementation

class Solution:
    def solve(self, target):
        self.ans = int(1e9)
        hi = 1
        while (1 << hi) < target:
            hi += 1
        self.dfs(hi, 0, 0, 0, target)
        return self.ans
    
    def dfs(self, digit, cost, pos, neg, target):
        tot = cost + max(2 * (pos - 1), 2 * neg - 1)
        if tot >= self.ans:
            return
        if target == 0:
            self.ans = min(self.ans, tot)
            return
        step = (1 << digit) - 1
        if step * 2 < abs(target):
            return
        self.dfs(digit - 1, cost, pos, neg, target)
        self.dfs(digit - 1, cost + digit, pos + 1, neg, target - step)
        self.dfs(digit - 1, cost + digit * 2, pos + 2, neg, target - step * 2)
        self.dfs(digit - 1, cost + digit, pos, neg + 1, target + step)
        self.dfs(digit - 1, cost + digit * 2, pos, neg + 2, target + step * 2)

# Test the solution
ob = Solution()
result = ob.solve(10)
print(f"Minimum moves to reach target 10: {result}")

Minimum moves to reach target 10: 7

How It Works

The algorithm works by:

• Starting Point: Calculate the initial digit value based on the target

• DFS Exploration: Recursively explore different move combinations

• Pruning: Skip paths that exceed the current best solution

• Optimization: Track positive and negative moves to minimize total cost

Example Walkthrough

For target = 10, one optimal sequence might be:

def trace_moves(target):
    print(f"Finding moves to reach target: {target}")
    print("Position: 0, Speed: 1, Moves: 0")
    
    # This is a simplified trace for demonstration
    moves = [
        "Accelerate: pos=1, speed=2",
        "Accelerate: pos=3, speed=4", 
        "Accelerate: pos=7, speed=8",
        "Accelerate: pos=15, speed=16",
        "Reverse: pos=15, speed=-1",
        "Accelerate: pos=14, speed=-2",
        "Reverse: pos=14, speed=1",
        "Accelerate: pos=15, speed=2",
        "Reverse: pos=15, speed=-1",
        "Accelerate: pos=14, speed=-2",
        "Accelerate: pos=12, speed=-4",
        "Reverse: pos=12, speed=1",
        "Accelerate: pos=13, speed=2",
        "Reverse: pos=13, speed=-1",
        "Accelerate: pos=12, speed=-2",
        "Reverse: pos=12, speed=1",
        "Accelerate: pos=13, speed=2",
        "Reverse: pos=13, speed=-1",
        "Accelerate: pos=12, speed=-2",
        "Reverse: pos=12, speed=1",
        "Final: pos=10"
    ]
    
    print("This demonstrates the complexity of finding optimal moves")

trace_moves(10)

Finding moves to reach target: 10
Position: 0, Speed: 1, Moves: 0
This demonstrates the complexity of finding optimal moves

Conclusion

This algorithm uses DFS with pruning to find the minimum moves needed to reach a target position. The key insight is exploring all possible combinations of acceleration and reverse operations while avoiding redundant calculations through intelligent pruning.