Overview

This problem asks us to find, for each location \(i\), the sum of the absolute differences \(S_i\) between the fixed forecast temperature \(T_i\) and the actual temperatures over \(D\) days, and then output the maximum of these sums.

Analysis

This problem can be solved by simply calculating the given formula directly, which is a straightforward “simulation” (or brute-force) approach. No special algorithms or complex techniques are required.

For each location \(i\), we perform the following calculation:

\[S_i = \sum_{j=1}^{D} |A_{i,j} - T_i|\]

Then, the maximum value among \(S_1, S_2, \ldots, S_N\) is the answer.

The main point to be careful of is the scale of the data. The constraints state that \(N \times D \leq 10^6\), which means we need to process up to \(10^6\) elements to read and calculate all temperature data. While the number of elements itself is not extremely large, in Python, slow input reading can lead to a TLE (Time Limit Exceeded) error.

For example, calling input() line-by-line or performing inefficient conversions to int() can be too slow for \(10^6\) data points. Therefore, optimizing input processing is crucial.

Let’s look at a concrete example. Suppose the forecast temperature for location \(i\) is \(T_i = 10\), and the actual temperatures are \([12, 7, 15, 8]\). In this case,

\[S_i = |12-10| + |7-10| + |15-10| + |8-10| = 2 + 3 + 5 + 2 = 12\]

We just need to calculate this for all locations and find the maximum.

Algorithm

1. Read all inputs at once.
2. For each location \(i\), calculate \(S_i\) by summing the absolute differences between the forecast temperature \(T_i\) and the actual temperatures \(A_{i,j}\) for each day.
3. Keep track of and update the maximum value of \(S_i\).
4. Output the maximum value at the end.

The absolute value can be calculated by finding the difference \(v = A_{i,j} - T_i\), and if \(v < 0\), flipping its sign (\(v = -v\)). While you can use the built-in abs() function, manually branching with an if statement can sometimes be slightly faster.

Complexity

• Time Complexity: \(O(N \times D)\) (up to \(10^6\) based on constraints)
• Space Complexity: \(O(N \times D)\) (if storing all input data)

Implementation Points

• Fast input processing is the most important part. In Python, a common technique is to read the entire standard input at once using sys.stdin.buffer.read() and split it into tokens with split(). This avoids the overhead of reading line-by-line.

data = sys.stdin.buffer.read().split()

• We can access the loaded data by advancing an index pointer idx. After reading \(N\) and \(D\), we process the forecast temperatures \(T\), followed by the actual temperature data for each location sequentially.

• To optimize the absolute value calculations, we reduce the processing load by performing the int() conversion only when necessary. In the code, we convert to an integer right before summing, like int(data[j]) - ti.

• The maximum value can be found without creating an additional list by updating the answer with if s > ans: ans = s each time \(S_i\) is calculated for a location.

With these optimizations, we can comfortably process data of scale \(10^6\) within the time limit.

Source Code

import sys

def main():
    data = sys.stdin.buffer.read().split()
    idx = 0
    N = int(data[idx]); idx += 1
    D = int(data[idx]); idx += 1
    T = data[idx:idx+N]
    idx += N
    Ti = [int(x) for x in T]
    
    ans = 0
    for i in range(N):
        ti = Ti[i]
        s = 0
        base = idx
        for j in range(base, base + D):
            v = int(data[j]) - ti
            if v < 0:
                v = -v
            s += v
        idx += D
        if s > ans:
            ans = s
    
    print(ans)

main()

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