Formatted question description: https://leetcode.ca/all/818.html

# 818. Race Car

Hard

## Description

Your car starts at position 0 and speed +1 on an infinite number line. (Your car can go into negative positions.)

Your car drives automatically according to a sequence of instructions A (accelerate) and R (reverse).

When you get an instruction “A”, your car does the following: position += speed, speed *= 2.

When you get an instruction “R”, your car does the following: if your speed is positive then speed = -1, otherwise speed = 1. (Your position stays the same.)

For example, after commands “AAR”, your car goes to positions 0->1->3->3, and your speed goes to 1->2->4->-1.

Now for some target position, say the length of the shortest sequence of instructions to get there.

Example 1:

Input:

target = 3

Output: 2

Explanation:

The shortest instruction sequence is “AA”.

Example 2:

Input:

target = 6

Output: 5

Explanation:

The shortest instruction sequence is “AAARA”.

Note:

• 1 <= target <= 10000.

## Solution

Use dynamic programming. Create an array dp of length Math.max(target + 3, target * 2), where dp[i] represents the length of the shortest sequence of instructions to get to position i. Initialize dp = 0, dp = 1 and dp = 4.

If i is 2^k - 1 for some k, then dp[i] = k. Otherwise, suppose 2^(k - 1) <= i < 2^k for some positive integer k, it is possible to reach 2^(k - 1) - 1 first and then use instructions with “R” to reach i, or reach 2^k - 1 first and then use instructions with “R” to reach i. Calculate the minimum length accordingly.

class Solution {
public int racecar(int target) {
int[] dp = new int[Math.max(target + 3, target * 2)];
Arrays.fill(dp, Integer.MAX_VALUE);
dp = 0;
dp = 1;
dp = 4;
for (int i = 3; i <= target; i++) {
int bits = (int) (Math.log(i) / Math.log(2)) + 1;
if (i == (1 << bits) - 1)
dp[i] = bits;
else {
for (int j = 0; j < bits - 1; j++)
dp[i] = Math.min(dp[i], dp[i - (1 << (bits - 1)) + (1 << j)] + bits - 1 + j + 2);
if ((1 << bits) - 1 - i < i)
dp[i] = Math.min(dp[i], dp[(1 << bits) - 1 - i] + bits + 1);
}
}
return dp[target];
}
}