Overview

This problem asks you to find the number of lit rooms among \(M\) meeting rooms after performing \(Q\) range toggle operations on \(N\) rooms.

Analysis

Problems with the Naive Approach

A simple approach would be to prepare an array of length \(N\) and flip the lighting state within the range for each operation. However, since \(N\) can be as large as \(10^9\), we cannot even allocate such an array. Additionally, updating up to \(N\) elements per operation would result in a time complexity of \(O(NQ)\), causing TLE.

Key Observations

1. The parity of toggle count matters: The final lighting state of a room is determined by the parity of how many times that room was toggled. If toggled an odd number of times, it’s lit; if even, it’s off.

2. We only need to check \(M\) rooms: Ultimately, we only need the states of \(M\) meeting rooms. Since \(M \leq 2 \times 10^5\), it’s sufficient to determine each meeting room individually.

3. Applying the difference array technique (imos method): An operation on range \([L, R]\) can be recorded as events: \(+1\) at \(L\) and \(-1\) at \(R+1\). By computing the prefix sum, we can determine the toggle count at each position.

Solution Strategy

Using the concept of coordinate compression, we only manage the points where events occur (\(L\) or \(R+1\)). This allows us to efficiently construct a list of lit intervals.

Algorithm

1. Create events: For each operation \([L_i, R_i]\), add a \(+1\) event at position \(L_i\) and a \(-1\) event at position \(R_i + 1\).

2. Sort and process events: Sort events by position and compute the prefix sum from left to right. Intervals where the prefix sum is odd are lit.

3. Create list of lit intervals: Add intervals \([start, end]\) where the prefix sum is odd to the list.

4. Check each meeting room: Use binary search to determine whether meeting room number \(b\) is contained in a lit interval.

Concrete Example

For \(Q = 2\) with operations \([1, 5]\) and \([3, 7]\): - Events: \((1, +1), (6, -1), (3, +1), (8, -1)\) - After sorting: \((1, +1), (3, +1), (6, -1), (8, -1)\) - Prefix sum progression: - Positions \(1\)–\(2\): prefix sum \(1\) (odd) → lit - Positions \(3\)–\(5\): prefix sum \(2\) (even) → off - Positions \(6\)–\(7\): prefix sum \(1\) (odd) → lit

Complexity

• Time complexity: \(O(Q \log Q + M \log Q)\)
  • Sorting events: \(O(Q \log Q)\)
  • Binary search for each meeting room: \(O(M \log Q)\)
• Space complexity: \(O(Q + M)\)
  • Storing events and lit intervals: \(O(Q)\)
  • Meeting room list: \(O(M)\)

Implementation Notes

• Be careful with interval endpoints: For operation \([L, R]\), record \(-1\) at position \(R+1\). This ensures that positions up to \(R\) are affected by the operation.
• How to use binary search: For meeting room \(b\), the relevant intervals are those with start position at most \(b\). Use bisect_right(starts, b) - 1 to get the index of the corresponding interval, then check if that interval’s end is at least \(b\).
• Handling empty event lists: To correctly handle cases with no operations or no lit intervals, we guard with if intervals:.

Source Code

import sys
from bisect import bisect_left, bisect_right

def main():
    input_data = sys.stdin.read().split()
    idx = 0
    N = int(input_data[idx]); idx += 1
    M = int(input_data[idx]); idx += 1
    Q = int(input_data[idx]); idx += 1
    
    B = [int(input_data[idx + i]) for i in range(M)]
    idx += M
    
    operations = []
    for _ in range(Q):
        L = int(input_data[idx]); idx += 1
        R = int(input_data[idx]); idx += 1
        operations.append((L, R))
    
    # Collect event points for coordinate compression
    # For each operation [L, R], create +1 event at L and -1 event at R+1
    events = []
    for L, R in operations:
        events.append((L, 1))
        events.append((R + 1, -1))
    
    # Sort events
    events.sort()
    
    # Compute prefix sum to find toggle count for each interval
    # As a result, intervals with odd toggle count are lit
    
    # Process event points together
    toggle_count = 0
    # Create list of (start position, end position, parity of toggle count)
    intervals = []  # (start, end) for lit intervals
    
    prev_pos = None
    for pos, delta in events:
        if prev_pos is not None and prev_pos < pos:
            if toggle_count % 2 == 1:
                intervals.append((prev_pos, pos - 1))
        toggle_count += delta
        prev_pos = pos
    
    # Check whether each meeting room is lit
    # intervals is a list of lit intervals
    # For each meeting room b, use binary search to find if there's an interval containing b
    
    # intervals are sorted by start position (should already be sorted)
    # For each interval (start, end), if start <= b <= end then it's lit
    
    count = 0
    if intervals:
        starts = [interval[0] for interval in intervals]
        
        for b in B:
            # Find the interval with the largest start that is <= b
            idx_interval = bisect_right(starts, b) - 1
            if idx_interval >= 0:
                start, end = intervals[idx_interval]
                if start <= b <= end:
                    count += 1
    
    print(count)

if __name__ == "__main__":
    main()

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