Algorithmic Analysis

The core requirement involves accumulating the totals of every contiguous segment of length $m$ within a sequence of $n$ integers. A straightforward nested loop approach computes each window independently, yielding $O(n \cdot m)$ operations. With constraints reaching $10^6$, this quadratic scaling triggers timeout errors. Linear optimization is mandatory.

Prefix Sum Strategy

Precomputing cumulative frequencies transforms range queries into constant-time lookups. By constructing an auxiliary array where index $k$ stores the sum of all preceding elements, any subarray total becomes the difference between two boundary values. Specifically, if $S$ represents the prefix table, the aggregate of a block starting at index $i$ and spanning $m$ positions equals $S[i + m] - S[i]$. Iterating through valid bounds and accumulating these deltas produces the target metric. When $m$ exceeds $n$, the result defaults to zero due to insufficinet elements for a complete window.

Implementation Structure

High-performance C++ implementations leverage standard libray containers and optimized input handling. Disabling default synchronization streams accelerates data ingestion. Zero-based indexing aligns with native array memory layouts, reducing off-by-one errors during offset calculations. Separating data parsing, table population, and summation phases enhances maintainability and debug visibility.

#include <iostream>
#include <vector>
#include <stdexcept>

int main() {
    std::ios::sync_with_stdio(false);
    std::cin.tie(nullptr);

    int n = 0, m = 0;
    if (!(std::cin >> n >> m)) return 0;

    if (m > n) {
        std::cout << 0 << '\n';
        return 0;
    }

    std::vector<long long> sequence(n);
    for (auto &val : sequence) {
        std::cin >> val;
    }

    std::vector<long long> cumulative(n + 1, 0);
    for (int idx = 0; idx < n; ++idx) {
        cumulative[idx + 1] = cumulative[idx] + sequence[idx];
    }

    long long grand_total = 0;
    for (int start = 0; start <= n - m; ++start) {
        grand_total += cumulative[start + m] - cumulative[start];
    }

    std::cout << grand_total << '\n';
    return 0;
}

Performance Metrics

Processing scales linearly relative to input size. Table initialization requires one traversal, while aggregation demands another pass covering $n - m + 1$ iterations. Memory footprint grows proportionally to store the prefix records, necessitating 64-bit signed integers to cap arithmetic overflow when multiplying potential window counts by maximum element mganitudes ($10^8$).