Overview

By preprocessing the time required for each interval, we can efficiently compute the tour completion time for each group.

Analysis

Each group must visit all areas within the specified interval \([L_j, R_j]\), and the total time required is
$\( T_{L_j} + T_{L_j+1} + \cdots + T_{R_j} \)\( If we compute this naively each time, it takes \)O(N)\( in the worst case, resulting in \)O(MN)$ overall, which will TLE due to the large constraints.

Therefore, we consider using prefix sums to compute interval sums in \(O(1)\).
Define the prefix sum array \(prefix\_sum\) as follows:
$\( prefix\_sum[i] = T_1 + T_2 + \cdots + T_i \)\( Then, the sum over the interval \)[L, R]\( can be computed as: \)\( \text{sum}(L, R) = prefix\_sum[R] - prefix\_sum[L - 1] \)\( This allows us to compute the interval sum for each query in \)O(1)\(, resulting in \)O(N + M)$ overall.

Algorithm

1. Read the visit time \(T_i\) for each area and construct the prefix sum array \(prefix\_sum\).
2. For each group, compute the total time for the interval \([L_j, R_j]\) from the prefix sum array, in addition to the arrival time \(S_j\).
3. The tour completion time for each group is the arrival time plus the total time.

Complexity

• Time complexity: \(O(N + M)\)
• Space complexity: \(O(N)\)

Implementation Notes

• When computing the prefix sum, prepending a 0 at the beginning eliminates the need for special-case handling when \(prefix\_sum[L - 1]\) would otherwise not exist.

• Using a method that reads all input at once and splits it (sys.stdin.read) enables faster input processing.

  Source Code

import sys
from itertools import accumulate

def main():
    input = sys.stdin.read
    data = input().split()
    
    N = int(data[0])
    M = int(data[1])
    
    T = list(map(int, data[2:2+N]))
    
    # 累積和を計算 (1-indexed)
    prefix_sum = [0] + list(accumulate(T))
    
    idx = 2 + N
    results = []
    
    for _ in range(M):
        S = int(data[idx])
        L = int(data[idx+1])
        R = int(data[idx+2])
        idx += 3
        
        # 区間 [L, R] の合計時間 = prefix_sum[R] - prefix_sum[L-1]
        total_time = prefix_sum[R] - prefix_sum[L-1]
        finish_time = S + total_time
        results.append(finish_time)
    
    print('\n'.join(map(str, results)))

if __name__ == "__main__":
    main()

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