abc351F Double Sum
给定数组A[N]，对所有1<=i<j<=N，计算max(A[j]-A[i],0)之和。
2<=N<=4E5; 0<=A[i]<=1E8
分析：从左到后依次处理，用平衡树维护左侧A[i]，对于A[j]，只需要统计值小于A[j]的那些A[i]即可，可以合并求和过程转化为前缀和。
#include <bits/stdc++.h>
using i64 = long long;
template <typename TYPE>
struct SumTreap {
struct Node {
TYPE data, sum;
int rnd, siz, dup, son[2];
Node() { data = sum = rnd = siz = dup = son[0] = son[1] = 0; }
Node(const TYPE &d, int cnt=1) { init(d, cnt); }
void init(const TYPE & d, int cnt) {
data = d; dup = siz = cnt; sum = d * cnt; rnd = rand(); son[0] = son[1] = 0;
}
};
SumTreap(int multi=1):multiple(multi) { reset(); }
void setmulti(int multi) { multiple = multi; }
int newnode(const TYPE & d, int cnt) {
if (!reuse.empty()) {
int r = reuse.front();
reuse.pop_front();
node[r].init(d, cnt);
return r;
}
node.push_back({d, cnt});
return node.size() - 1;
}
void reset() { root = 0; node.resize(1); reuse.clear(); }
void maintain(int x) {
node[x].siz = node[x].dup;
node[x].sum = node[x].data * node[x].dup;
if (node[x].son[0]) {
node[x].siz += node[node[x].son[0]].siz;
node[x].sum += node[node[x].son[0]].sum;
}
if (node[x].son[1]) {
node[x].siz += node[node[x].son[1]].siz;
node[x].sum += node[node[x].son[1]].sum;
}
}
void rotate(int d, int &r) {
int k = node[r].son[d^1];
node[r].son[d^1] = node[k].son[d];
node[k].son[d] = r;
maintain(r);
maintain(k);
r = k;
}
void insert(const TYPE &data, int cnt, int &r) {
if (r) {
if (!(data < node[r].data) && !(node[r].data < data)) {
if (multiple) {
node[r].dup += cnt;
maintain(r);
}
} else {
int d = data < node[r].data ? 0 : 1;
int u = node[r].son[d];
insert(data, cnt, u);
node[r].son[d] = u;
if (node[node[r].son[d]].rnd > node[r].rnd) {
rotate(d^1, r);
} else {
maintain(r);
}
}
} else {
r = newnode(data, cnt);
}
}
TYPE kth(int k) {
assert(1 <= k && k <= size());
for (int r = root; r; ) {
int x = node[r].son[0] ? node[node[r].son[0]].siz : 0;
int y = node[r].dup;
if (k <= x) {
r = node[r].son[0];
} else if (k <= x + y) {
return node[r].data;
} else {
k -= x + y;
r = node[r].son[1];
}
}
assert(k == 0);
return 0;
}
TYPE Kth(int k) {
assert(1 <= k && k <= size());
for (int r = root; r; ) {
int x = node[r].son[1] ? node[node[r].son[1]].siz : 0;
int y = node[r].dup;
if (k <= x) {
r = node[r].son[1];
} else if (k <= x + y) {
return node[r].data;
} else {
k -= x + y;
r = node[r].son[0];
}
}
assert(k == 0);
return 0;
}
TYPE ksum(int k) {
assert(0 <= k && k <= node[root].siz);
TYPE ans = 0;
for (int r = root; r && k; ) {
int x = node[r].son[0] ? node[node[r].son[0]].siz : 0;
int y = node[r].dup;
if (k <= x) {
r = node[r].son[0];
} else if (k <= x + y) {
ans += node[node[r].son[0]].sum + node[r].data * (k - x);
k = 0;
} else {
ans += node[node[r].son[0]].sum + node[r].data * y;
k -= x + y;
r = node[r].son[1];
}
}
return ans;
}
TYPE kSum(int k) {
assert(0 <= k && k <= node[root].siz);
TYPE ans = 0;
for (int r = root; r && k; ) {
int x = node[r].son[1] ? node[node[r].son[1]].siz : 0;
int y = node[r].dup;
if (k <= x) {
r = node[r].son[1];
} else if (k <= x + y) {
ans += node[node[r].son[1]].sum + node[r].data * (k - x);
k = 0;
} else {
ans += node[node[r].son[1]].sum + node[r].data * y;
k -= x + y;
r = node[r].son[0];
}
}
return ans;
}
void erase(const TYPE& data, int cnt, int & r) {
if (r == 0) return;
int d = -1;
if (data < node[r].data) {
d = 0;
} else if (node[r].data < data) {
d = 1;
}
if (d == -1) {
node[r].dup -= cnt;
if (node[r].dup > 0) {
maintain(r);
} else {
if (node[r].son[0] == 0) {
reuse.push_back(r);
r = node[r].son[1];
} else if (node[r].son[1] == 0) {
reuse.push_back(r);
r = node[r].son[0];
} else {
int dd = node[node[r].son[0]].rnd > node[node[r].son[1]].rnd ? 1 : 0;
rotate(dd, r);
erase(data, cnt, node[r].son[dd]);
}
}
} else {
erase(data, cnt, node[r].son[d]);
}
if (r) maintain(r);
}
int ltcnt(const TYPE &data) {
int ans = 0;
for (int r = root; r; ) {
int x = node[r].son[0] ? node[node[r].son[0]].siz : 0;
if (node[r].data < data) {
ans += node[r].dup + x;
r = node[r].son[1];
} else if (data < node[r].data) {
r = node[r].son[0];
} else {
ans += x;
break;
}
}
return ans;
}
int gtcnt(const TYPE &data) {
int ans = 0;
for (int r = root; r; ) {
int x = node[r].son[1] ? node[node[r].son[1]].siz : 0;
if (node[r].data < data) {
r = node[r].son[1];
} else if (data < node[r].data) {
ans += node[r].dup + x;
r = node[r].son[0];
} else {
ans += x;
break;
}
}
return ans;
}
int count(const TYPE &data) {
for (int r = root; r; ) {
if (data < node[r].data) {
r = node[r].son[0];
} else if (node[r].data < data) {
r = node[r].son[1];
} else {
return node[r].dup;
}
}
return 0;
}
std::pair<bool,TYPE> prev(const TYPE &data) {
std::pair<bool,TYPE> ans = {false, 0};
for (int r = root; r; ) {
if (node[r].data < data) {
if (ans.first) {
ans.second = std::max(ans.second, node[r].data);
} else {
ans = {true, node[r].data};
}
r = node[r].son[1];
} else {
r = node[r].son[0];
}
}
return ans;
}
std::pair<bool,TYPE> next(const TYPE &data) {
std::pair<bool,TYPE> ans = {false, 0};
for (int r = root; r; ) {
if (data < node[r].data) {
if (ans.first) {
ans.second = std::min(ans.second, node[r].data);
} else {
ans = {true, node[r].data};
}
r = node[r].son[0];
} else {
r = node[r].son[1];
}
}
return ans;
}
std::vector<Node> node;
std::deque<int> reuse;
int root, multiple;
void insert(const TYPE& data, int cnt=1) { insert(data, cnt, root); }
void erase(const TYPE& data, int cnt=1) { erase(data, cnt, root); }
int size() const { return root ? node[root].siz : 0; }
int lecnt(const TYPE& data) { return size() - gtcnt(data, root); }
int gecnt(const TYPE& data) { return size() - ltcnt(data, root); }
};
void solve() {
int N;
std::cin >> N;
SumTreap<i64> tr;
std::vector<int> A(N);
for (int i = 0; i < N; i++) {
std::cin >> A[i];
}
i64 ans = 0;
for (int i = 0; i < N; i++) {
i64 cnt1 = tr.ltcnt(A[i]);
ans += cnt1 * A[i] - tr.ksum(cnt1);
tr.insert(A[i]);
}
std::cout << ans << "\n";
}
int main() {
std::cin.tie(0)->sync_with_stdio(0);
std::cout << std::fixed << std::setprecision(10);
int t = 1;
// std::cin >> t;
while (t--) solve();
return 0;
}