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| 1 | +# Maximize Distance to Closest Person |
| 2 | + |
| 3 | +## Problem Description |
| 4 | + |
| 5 | +You are given an array representing a row of seats where `seats[i] = 1` |
| 6 | +represents a person sitting in the `i`th seat, and `seats[i] = 0` represents |
| 7 | +that the `i`th seat is empty (0-indexed). |
| 8 | + |
| 9 | +There is at least one empty seat, and at least one person sitting. |
| 10 | + |
| 11 | +Alex wants to sit in the seat such that the distance between him and the |
| 12 | +closest person to him is maximized. |
| 13 | + |
| 14 | +Return that maximum distance to the closest person. |
| 15 | + |
| 16 | +**Example 1:** |
| 17 | + |
| 18 | +* Input: `seats = [1, 0, 0, 0, 1, 0, 1]` |
| 19 | +* Output: `2` |
| 20 | +* Explanation: |
| 21 | + |
| 22 | + If Alex sits in the second open seat (i.e. `seats[2]`), then the closest person |
| 23 | + has distance 2. |
| 24 | + If Alex sits in any other open seat, the closest person has distance 1. |
| 25 | + Thus, the maximum distance to the closest person is 2. |
| 26 | + |
| 27 | +**Example 2:** |
| 28 | + |
| 29 | +* Input: `seats = [1, 0, 0, 0]` |
| 30 | +* Output: `3` |
| 31 | +* Explanation: |
| 32 | + |
| 33 | + If Alex sits in the last seat (i.e. `seats[3]`), the closest person is 3 seats away. |
| 34 | + This is the maximum distance possible, so the answer is 3. |
| 35 | + |
| 36 | +**Example 3:** |
| 37 | + |
| 38 | +* Input: `seats = [0, 1]` |
| 39 | +* Output: `1` |
| 40 | + |
| 41 | +**Constraints:** |
| 42 | + |
| 43 | +* `2 <= seats.length <= 2 * 10^4` |
| 44 | +* `seats[i]` is `0` or `1`. |
| 45 | +* At least one seat is empty. |
| 46 | +* At least one seat is occupied. |
| 47 | + |
| 48 | + |
| 49 | +## Solution |
| 50 | + |
| 51 | +```python |
| 52 | +def max_dist_to_closest(seats: list[int]) -> int: |
| 53 | + """ |
| 54 | + Calculate the maximum distance to the closest occupied seat in a row. |
| 55 | + |
| 56 | + The function finds the maximum possible distance a person can have to the nearest |
| 57 | + occupied seat when choosing a seat in a row represented by a binary list, |
| 58 | + where 1 represents an occupied seat and 0 represents an empty seat. |
| 59 | + |
| 60 | + :param seats: A list of integers where 0 represents an empty seat and 1 represents |
| 61 | + an occupied seat. The list must contain at least one occupied seat. |
| 62 | + :return: The maximum distance to the closest occupied seat. The distance is measured |
| 63 | + as the number of seats between two positions (inclusive of endpoints). |
| 64 | + """ |
| 65 | + length = len(seats) |
| 66 | + left = None |
| 67 | + result = 1 |
| 68 | + |
| 69 | + for index in range(length): |
| 70 | + if seats[index] == 0: |
| 71 | + if left is None: |
| 72 | + left = index |
| 73 | + if (left == 0) or (index == length - 1): |
| 74 | + result = max(result, index - left + 1) |
| 75 | + else: |
| 76 | + result = max(result, (index - left + 2) // 2) |
| 77 | + else: |
| 78 | + left = None |
| 79 | + return result |
| 80 | +``` |
| 81 | + |
| 82 | +* **Time Complexity:** $O(n)$ |
| 83 | +* **Space Complexity:** $O(1)$ |
| 84 | + |
| 85 | +## Explanation of the Solution |
| 86 | + |
| 87 | +### Key Insights |
| 88 | + |
| 89 | +1. **Empty Seats Between Occupied Seats:** |
| 90 | + * The optimal position is the middle of the longest block of empty seats between two 1s. |
| 91 | +2. **Edge Cases:** |
| 92 | + * If the first or last seat is empty, the farthest position is at that edge (no need to sit in the middle). |
| 93 | + |
| 94 | +### Approach |
| 95 | + |
| 96 | +1. **Track Empty Seat Blocks Efficiently:** |
| 97 | + * Use a single pointer left to mark the start of a block of empty seats. |
| 98 | + * If the block touches the start (`left=0`) or end (`index=last seat`) of the row, the distance is the full block length `(index - left + 1)`. |
| 99 | + * For internal blocks, the maximum distance is `(index - left + 2) // 2` (middle position). |
| 100 | +2. **Update Result Dynamically:** |
| 101 | + * When encountering an occupied seat (`1`), reset `left` to `None` (indicating no current empty block). |
| 102 | + * For empty seats (`0`), update left if starting a new block, then compute the current maximum distance. |
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