結果
| 問題 |
No.1079 まお
|
| ユーザー |
 qwewe
|
| 提出日時 | 2025-05-14 12:55:48 |
| 言語 | C++17 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 232 ms / 2,000 ms |
| コード長 | 9,333 bytes |
| コンパイル時間 | 1,108 ms |
| コンパイル使用メモリ | 103,880 KB |
| 実行使用メモリ | 20,864 KB |
| 最終ジャッジ日時 | 2025-05-14 12:57:22 |
| 合計ジャッジ時間 | 4,740 ms |
|
ジャッジサーバーID （参考情報） |
judge5 / judge1 |
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| ファイルパターン | 結果 |
|---|---|
| sample | AC * 2 |
| other | AC * 30 |
ソースコード
#include <iostream>
#include <vector>
#include <stack>
#include <map>
#include <vector>
#include <numeric>
#include <algorithm>
using namespace std;
// Structure to store indices and prefix sums for a specific value in the array A.
// This helps efficiently query the count and sum of indices within a given range [a, b].
struct IndexData {
vector<int> indices; // List of 1-based indices where the value appears
vector<long long> prefix_sums; // prefix_sums[k] = sum of indices[0]...indices[k]
// Build prefix sums after the indices vector is fully populated.
void build_prefix_sums() {
prefix_sums.resize(indices.size());
if (!indices.empty()) {
// Calculate prefix sums of the indices. Useful for range sum queries.
prefix_sums[0] = indices[0];
for (size_t i = 1; i < indices.size(); ++i) {
prefix_sums[i] = prefix_sums[i - 1] + indices[i];
}
}
}
// Query the count of indices and the sum of indices for this value within the range [a, b] (inclusive).
// Returns a pair {count, sum_of_indices}.
pair<long long, long long> query(int a, int b) {
// If there are no indices for this value, or the query range is invalid, return {0, 0}.
if (indices.empty() || a > b) {
return {0, 0};
}
// Use binary search (lower_bound and upper_bound) to find the range of indices within [a, b].
// lower_bound finds the first index >= a.
auto it_l = lower_bound(indices.begin(), indices.end(), a);
// upper_bound finds the first index > b.
auto it_r = upper_bound(indices.begin(), indices.end(), b);
// If it_l == it_r, it means no indices fall within the range [a, b].
if (it_l == it_r) {
return {0, 0};
}
// Calculate the 0-based indices in the `indices` vector corresponding to the range found.
int start_idx = distance(indices.begin(), it_l);
// it_r points one position past the last element <= b. So the index of the last element is it_r - 1.
int end_idx = distance(indices.begin(), it_r) - 1;
// This check is technically redundant if it_l != it_r, but added for robustness.
if (start_idx > end_idx) {
return {0, 0};
}
// Calculate the count of indices in the range.
long long count = end_idx - start_idx + 1;
// Calculate the sum of indices using the precomputed prefix sums.
long long sum_indices = prefix_sums[end_idx];
if (start_idx > 0) {
sum_indices -= prefix_sums[start_idx - 1];
}
return {count, sum_indices};
}
};
int main() {
// Use faster I/O operations.
ios_base::sync_with_stdio(false);
cin.tie(NULL);
int N; // Number of elements in the array A
long long K; // Target sum for the endpoints of subarray B
cin >> N >> K;
vector<long long> A(N + 1); // Use 1-based indexing for array A. Values can be up to 10^9.
map<long long, IndexData> val_indices; // Map from value to its IndexData structure.
// Read input array A and populate the map `val_indices`.
for (int i = 1; i <= N; ++i) {
cin >> A[i];
val_indices[A[i]].indices.push_back(i); // Store the 1-based index i.
}
// Build prefix sums for each value's index list. This is done after all indices are collected.
for (auto it = val_indices.begin(); it != val_indices.end(); ++it) {
it->second.build_prefix_sums();
}
// Compute Previous Less or Equal Element (PLEE) and Next Less or Equal Element (NLEE) indices.
// L[p] = largest index k < p such that A[k] <= A[p]. If none exists, L[p] = 0.
// R[p] = smallest index k > p such that A[k] <= A[p]. If none exists, R[p] = N + 1.
// These define the maximal range (L[p], R[p]) where A[p] can be the unique minimum.
vector<int> L(N + 1);
vector<int> R(N + 1);
// Compute L (PLEE) using a stack.
stack<int> st_plee;
for(int i=1; i<=N; ++i) {
// Pop elements from stack while the top element's value is strictly greater than A[i].
while(!st_plee.empty() && A[st_plee.top()] > A[i]) {
st_plee.pop();
}
// The PLEE is the index at the top of the stack, or 0 if the stack is empty.
L[i] = st_plee.empty() ? 0 : st_plee.top();
// Handle elements with equal value: if A[top] == A[i], pop the top element.
// This ensures that for subsequent elements, `i` is considered as the PLEE candidate instead of `st_plee.top()`,
// effectively keeping the largest index among equal values relevant for the PLEE definition.
if (!st_plee.empty() && A[st_plee.top()] == A[i]) {
st_plee.pop();
}
st_plee.push(i); // Push the current index onto the stack.
}
// Compute R (NLEE) using a stack, iterating backwards from N to 1.
stack<int> st_nlee;
for (int i = N; i >= 1; --i) {
// Pop elements from stack while the top element's value is strictly greater than A[i].
while (!st_nlee.empty() && A[st_nlee.top()] > A[i]) {
st_nlee.pop();
}
// The NLEE is the index at the top of the stack, or N+1 if the stack is empty.
R[i] = st_nlee.empty() ? N + 1 : st_nlee.top();
// Handle equal values similarly to PLEE, keeping the smallest index among equal values.
if (!st_nlee.empty() && A[st_nlee.top()] == A[i]) {
st_nlee.pop();
}
st_nlee.push(i); // Push the current index onto the stack.
}
long long total_sum_len = 0; // Use long long to store the total sum of lengths, which can be large.
// Iterate through each element A[p] considering it as the potential unique minimum of a subarray B.
for (int p = 1; p <= N; ++p) {
// Define the valid range [i_start, i_end] for the left endpoint `i` and [j_start, j_end] for the right endpoint `j`.
// For A[p] to be the unique minimum of A[i...j], we must have L[p] < i <= p <= j < R[p].
int i_start = L[p] + 1;
int i_end = p;
int j_start = p;
int j_end = R[p] - 1;
// If either the range for i or the range for j is invalid (e.g., start > end), skip this p.
if (i_start > i_end || j_start > j_end) continue;
// Calculate the lengths of the left range (for i) and right range (for j).
int len_L = i_end - i_start + 1;
int len_R = j_end - j_start + 1;
// Apply the "smaller-to-larger" optimization heuristic:
// Iterate through the smaller range and perform queries on the larger range.
// This optimization helps improve the overall time complexity.
if (len_L <= len_R) {
// Iterate through i in the left range [L[p]+1, p].
for (int i = i_start; i <= i_end; ++i) {
long long target_val = K - A[i]; // Calculate the required value A[j] such that A[i] + A[j] = K.
// Check if this target value exists in the array using map::find for efficiency.
auto it = val_indices.find(target_val);
if (it != val_indices.end()) {
// Query for indices j in the right range [p, R[p]-1] that have the target value.
pair<long long, long long> query_res = it->second.query(j_start, j_end);
long long count = query_res.first; // Number of such j's found.
long long sum_j = query_res.second; // Sum of these j indices.
if (count > 0) {
// Add the total length contributed by pairs starting at this `i`.
// The length of subarray A[i...j] is j - i + 1.
// Summing over all found j's: Sum(j - i + 1) = Sum(j) - Sum(i - 1) = Sum(j) - (i - 1) * count.
total_sum_len += sum_j - (long long)(i - 1) * count;
}
}
}
} else { // len_L > len_R
// Iterate through j in the right range [p, R[p]-1].
for (int j = j_start; j <= j_end; ++j) {
long long target_val = K - A[j]; // Calculate the required value A[i] such that A[i] + A[j] = K.
// Check if this target value exists.
auto it = val_indices.find(target_val);
if (it != val_indices.end()) {
// Query for indices i in the left range [L[p]+1, p] that have the target value.
pair<long long, long long> query_res = it->second.query(i_start, i_end);
long long count = query_res.first; // Number of such i's found.
long long sum_i = query_res.second; // Sum of these i indices.
if (count > 0) {
// Add the total length contributed by pairs ending at this `j`.
// Sum(j - i + 1) = Sum(j + 1) - Sum(i) = (j + 1) * count - Sum(i).
total_sum_len += (long long)(j + 1) * count - sum_i;
}
}
}
}
}
// Output the final computed total sum of lengths.
cout << total_sum_len << endl;
return 0;
}