First Day Where You Have Been in All the Rooms

First Day Where You Have Been in All the Rooms

1. What is this problem about?

The First Day Where You Have Been in All the Rooms interview question is a complex state-based simulation. There are $n$ rooms. Each day, if you visit room $i$ for an odd number of times, you must visit nextVisit[i] (where $0 \leq nextVisit[i] \leq i$) the next day. If you visit room $i$ for an even number of times, you move to room $i+1$. You need to find the total number of days it takes to reach the last room.

2. Why is this asked in interviews?

ByteDance asks the First Day coding problem to test a candidate's mastery of Dynamic Programming. It is a difficult problem because the movements involve cycles (always going back to a previous room). It evaluation your ability to recognize that the "cost" to traverse a room is fixed and can be calculated using previous room costs.

3. Algorithmic pattern used

This problem follows the Dynamic Programming with Cumulative Sums pattern.

• DP State: dp[i] is the number of days needed to reach room $i$ from room 0.
• The Insight: To move from room $i$ to $i+1$, you must first reach $i$ (takes dp[i] days). On that day (odd visit), you are sent back to nextVisit[i]. To get back to $i$ again, you must spend the same amount of time it takes to go from nextVisit[i] to $i$ plus one day for the initial jump and one day for the final move.
• Transition: dp[i+1] = dp[i] + (dp[i] - dp[nextVisit[i]] + 1) + 1.
• Simplified: dp[i+1] = 2 * dp[i] - dp[nextVisit[i]] + 2.

4. Example explanation

nextVisit = [0, 0, 2].

1. dp[0] = 0.
2. dp[1] = 2*dp[0] - dp[0] + 2 = 2.
3. dp[2] = 2*dp[1] - dp[0] + 2 = 2*2 - 0 + 2 = 6. Result: 6 days to reach the last room.

5. Common mistakes candidates make

• Simulation: Trying to simulate every single day. The number of days can be huge ($10^9$), leading to TLE.
• Negative Modulo: In languages like Java or C++, (a - b) % mod can be negative. You must use (a - b + mod) % mod.
• Incorrect Recurrence: Failing to account for the "jump back" and "jump forward" days accurately.

6. Interview preparation tip

Problems with "odd/even visit" rules often imply that you must pass through a state twice to move forward. This doubling effect is a major hint for Dynamic Programming interview patterns.