#include <bits/stdc++.h>
using namespace std;
template <typename A, typename B>
string to_string(pair<A, B> p);
template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p);
template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p);
string to_string(const string& s) {
return '"' + s + '"';
}
string to_string(const char* s) {
return to_string((string) s);
}
string to_string(bool b) {
return (b ? "true" : "false");
}
string to_string(vector<bool> v) {
bool first = true;
string res = "{";
for (int i = 0; i < static_cast<int>(v.size()); i++) {
if (!first) {
res += ", ";
}
first = false;
res += to_string(v[i]);
}
res += "}";
return res;
}
template <size_t N>
string to_string(bitset<N> v) {
string res = "";
for (size_t i = 0; i < N; i++) {
res += static_cast<char>('0' + v[i]);
}
return res;
}
template <typename A>
string to_string(A v) {
bool first = true;
string res = "{";
for (const auto &x : v) {
if (!first) {
res += ", ";
}
first = false;
res += to_string(x);
}
res += "}";
return res;
}
template <typename A, typename B>
string to_string(pair<A, B> p) {
return "(" + to_string(p.first) + ", " + to_string(p.second) + ")";
}
template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p) {
return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " + to_string(get<2>(p)) + ")";
}
template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p) {
return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " + to_string(get<2>(p)) + ", " + to_string(get<3>(p)) + ")";
}
void debug_out() { cerr << endl; }
template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
cerr << " " << to_string(H);
debug_out(T...);
}
#define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__);
class segtree {
public:
struct node {
long long ad = 0;
int mu = 1;
long long sum = 0;
void apply(int l, int r, int v) {
if (v >= 0) {
ad += v;
sum += 1LL * v * (r - l + 1);
} else {
v = ~v;
ad *= v;
mu *= v;
sum *= v;
}
}
};
node unite(const node &a, const node &b) const {
node res;
res.sum = a.sum + b.sum;
return res;
}
inline void push(int x, int l, int r) {
int y = (l + r) >> 1;
int z = x + ((y - l + 1) << 1);
if (tree[x].mu != 1) {
tree[x + 1].apply(l, y, ~tree[x].mu);
tree[z].apply(y + 1, r, ~tree[x].mu);
tree[x].mu = 1;
}
if (tree[x].ad != 0) {
tree[x + 1].apply(l, y, tree[x].ad);
tree[z].apply(y + 1, r, tree[x].ad);
tree[x].ad = 0;
}
}
inline void pull(int x, int z) {
tree[x] = unite(tree[x + 1], tree[z]);
}
int n;
vector<node> tree;
void build(int x, int l, int r) {
if (l == r) {
return;
}
int y = (l + r) >> 1;
int z = x + ((y - l + 1) << 1);
build(x + 1, l, y);
build(z, y + 1, r);
pull(x, z);
}
template <typename M, typename... T>
void build(int x, int l, int r, const vector<M> &v, const T&... t) {
if (l == r) {
tree[x].apply(l, r, v[l], t...);
return;
}
int y = (l + r) >> 1;
int z = x + ((y - l + 1) << 1);
build(x + 1, l, y, v, t...);
build(z, y + 1, r, v, t...);
pull(x, z);
}
template <typename M, typename... T>
segtree(const vector<M> &v, const T&... t) {
n = v.size();
assert(n > 0);
tree.resize(2 * n - 1);
build(0, 0, n - 1, v, t...);
}
segtree(int _n) : n(_n) {
assert(n > 0);
tree.resize(2 * n - 1);
build(0, 0, n - 1);
}
segtree(){};
node get(int x, int l, int r, int ll, int rr) {
if (ll <= l && r <= rr) {
return tree[x];
}
int y = (l + r) >> 1;
int z = x + ((y - l + 1) << 1);
push(x, l, r);
node res{};
if (rr <= y) {
res = get(x + 1, l, y, ll, rr);
} else {
if (ll > y) {
res = get(z, y + 1, r, ll, rr);
} else {
res = unite(get(x + 1, l, y, ll, rr), get(z, y + 1, r, ll, rr));
}
}
pull(x, z);
return res;
}
node get(int ll, int rr) {
assert(0 <= ll && ll <= rr && rr <= n - 1);
return get(0, 0, n - 1, ll, rr);
}
node get(int p) {
assert(0 <= p && p <= n - 1);
return get(0, 0, n - 1, p, p);
}
template <typename... M>
void modify(int x, int l, int r, int ll, int rr, const M&... v) {
if (ll <= l && r <= rr) {
tree[x].apply(l, r, v...);
return;
}
int y = (l + r) >> 1;
int z = x + ((y - l + 1) << 1);
push(x, l, r);
if (ll <= y) {
modify(x + 1, l, y, ll, rr, v...);
}
if (rr > y) {
modify(z, y + 1, r, ll, rr, v...);
}
pull(x, z);
}
template <typename... M>
void modify(int ll, int rr, const M&... v) {
assert(0 <= ll && ll <= rr && rr <= n - 1);
modify(0, 0, n - 1, ll, rr, v...);
}
// find_first and find_last call all FALSE elements
// to the left (right) of the sought position exactly once
int find_first_knowingly(int x, int l, int r, const function<bool(const node&)> &f) {
if (l == r) {
return l;
}
push(x, l, r);
int y = (l + r) >> 1;
int z = x + ((y - l + 1) << 1);
int res;
if (f(tree[x + 1])) {
res = find_first_knowingly(x + 1, l, y, f);
} else {
res = find_first_knowingly(z, y + 1, r, f);
}
pull(x, z);
return res;
}
int find_first(int x, int l, int r, int ll, int rr, const function<bool(const node&)> &f) {
if (ll <= l && r <= rr) {
if (!f(tree[x])) {
return -1;
}
return find_first_knowingly(x, l, r, f);
}
push(x, l, r);
int y = (l + r) >> 1;
int z = x + ((y - l + 1) << 1);
int res = -1;
if (ll <= y) {
res = find_first(x + 1, l, y, ll, rr, f);
}
if (rr > y && res == -1) {
res = find_first(z, y + 1, r, ll, rr, f);
}
pull(x, z);
return res;
}
int find_first(int ll, int rr, const function<bool(const node&)> &f) {
assert(0 <= ll && ll <= rr && rr <= n - 1);
return find_first(0, 0, n - 1, ll, rr, f);
}
int find_last_knowingly(int x, int l, int r, const function<bool(const node&)> &f) {
if (l == r) {
return l;
}
push(x, l, r);
int y = (l + r) >> 1;
int z = x + ((y - l + 1) << 1);
int res;
if (f(tree[z])) {
res = find_last_knowingly(z, y + 1, r, f);
} else {
res = find_last_knowingly(x + 1, l, y, f);
}
pull(x, z);
return res;
}
int find_last(int x, int l, int r, int ll, int rr, const function<bool(const node&)> &f) {
if (ll <= l && r <= rr) {
if (!f(tree[x])) {
return -1;
}
return find_last_knowingly(x, l, r, f);
}
push(x, l, r);
int y = (l + r) >> 1;
int z = x + ((y - l + 1) << 1);
int res = -1;
if (rr > y) {
res = find_last(z, y + 1, r, ll, rr, f);
}
if (ll <= y && res == -1) {
res = find_last(x + 1, l, y, ll, rr, f);
}
pull(x, z);
return res;
}
int find_last(int ll, int rr, const function<bool(const node&)> &f) {
assert(0 <= ll && ll <= rr && rr <= n - 1);
return find_last(0, 0, n - 1, ll, rr, f);
}
};
struct edge {
int t;
int dis;
edge(int _t, int _dis) : t(_t), dis(_dis){};
};
vector<long long> calc(int n, int s, int t, vector<vector<edge>> g) {
vector<long long> ret(n, LLONG_MAX);
ret[s] = 0;
set<pair<long long, int>> que;
vector<int> was(n);
que.insert({ret[s], s});
while(que.size()) {
int v = que.begin()->second;
que.erase(que.begin());
if(was[v]) continue;
was[v] = 1;
for(edge e : g[v]) {
int u = e.t;
if(ret[u] > ret[v] + e.dis) {
ret[u] = ret[v] + e.dis;
que.insert(make_pair(ret[u], u));
}
}
}
return ret;
}
template <typename T, class F = function<T(const T&, const T&)>>
class SparseTable {
public:
int n;
vector<vector<T>> mat;
F func;
SparseTable(const vector<T>& a, const F& f) : func(f) {
n = static_cast<int>(a.size());
int max_log = 32 - __builtin_clz(n);
mat.resize(max_log);
mat[0] = a;
for (int j = 1; j < max_log; j++) {
mat[j].resize(n - (1 << j) + 1);
for (int i = 0; i <= n - (1 << j); i++) {
mat[j][i] = func(mat[j - 1][i], mat[j - 1][i + (1 << (j - 1))]);
}
}
}
T get(int from, int to) const {
assert(0 <= from && from <= to && to <= n - 1);
int lg = 32 - __builtin_clz(to - from + 1) - 1;
return func(mat[lg][from], mat[lg][to - (1 << lg) + 1]);
}
};
long long min_distance(vector<int> px, vector<int> h, vector<int> l, vector<int> r, vector<int> py, int s, int t) {
int n = px.size();
int m = py.size();
set<pair<int, int>> p;
auto add = [&](int x, int y) {
p.insert(make_pair(x, y));
};
map<int, int> bound;
for(int i = 0; i < n; i++) bound[px[i]] = i;
add(px[s], 0);
add(px[t], 0);
for(int i = 0; i < m; i++) {
add(px[l[i]], py[i]);
add(px[r[i]], py[i]);
}
//
SparseTable<int> d(h, [&](int i, int j){return max(i, j);});
auto find_left = [&](int l, int r, int y) {
if(d.get(l, r) < y) return -1;
while(r - l >= 1) {
int mid = (r + l) >> 1;
if(d.get(l, mid) < y) l = mid + 1;
else r = mid;
}
return r;
};
auto find_right = [&](int l, int r, int y) {
if(d.get(l, r) < y) return -1;
while(r - l >= 1) {
int mid = (r + l) >> 1;
if(d.get(mid + 1, r) < y) r = mid;
else l = mid + 1;
}
return r;
};
//near pole
for(int i = 0; i < m; i++) {
int y = py[i];
for(int rot = 0; rot < 2; rot++) {
if(l[i] < s && s < r[i]) {
int ll = find_right(l[i], s, y);
int rr = find_left(s, r[i], y);
if(ll != -1) add(px[ll], y);
if(rr != -1) add(px[rr], y);
}
swap(s, t);
}
}
vector<pair<int, int>> ad;
// near ver
{
for(auto pp : p) {
int x = pp.first, y = pp.second;
int id = lower_bound(px.begin(), px.end(), x) - px.begin();
if(id < n - 1) {
int ll = find_left(id + 1, n - 1, y);
if(ll != -1) ad.emplace_back(px[ll], y);
}
if(id > 0) {
int rr = find_right(0, id - 1, y);
if(rr != -1) ad.emplace_back(px[rr], y);
}
}
}
//near hor
{
vector<pair<int, int>> pos;
for(auto i : p) pos.push_back(i);
sort(pos.begin(), pos.end(), [&](pair<int, int> i, pair<int, int> j){return i.second < j.second;});
vector<int> order(m);
iota(order.begin(), order.end(), 0);
sort(order.begin(), order.end(), [&](int i, int j){return py[i] < py[j];});
segtree st(n);
for(int rot = 0; rot < 2; rot++) {
int cur = 0;
for(auto pp : pos) {
int x = pp.first, y = pp.second;
if(y == 0) continue;
while(py[order[cur]] != y) {
int id = order[cur];
st.modify(l[id], r[id], ~0);
st.modify(l[id], r[id], py[id]);
cur++;
}
int id = bound[x];
int yy = st.get(id, id).sum;
if(0 < yy && yy <= h[id]) ad.emplace_back(x, yy);
}
reverse(pos.begin(), pos.end());
reverse(order.begin(), order.end());
st.modify(0, n - 1, ~0);
}
}
for(auto i : ad) add(i.first, i.second);
//numerate
int sz = p.size();
map<pair<int, int>, int> id;
for(auto i : p) {
int fp = id.size();
id[i] = fp;
}
vector<vector<edge>> g(sz);
// add ver edged
int cnt = 0;
for(auto it = p.begin(); next(it) != p.end(); it++) {
if(it->first == next(it)->first) {
int u = cnt;
int v = cnt + 1;
int dis = abs(it->second - next(it)->second);
g[u].emplace_back(v, dis);
g[v].emplace_back(u, dis);
}
cnt++;
}
// add hor edge
set<pair<int, int>> rp;
for(auto i : p) rp.insert(make_pair(i.second, i.first));
for(int i = 0; i < m; i++) {
int L = px[l[i]], R = px[r[i]], y = py[i];
for(auto it = rp.lower_bound(make_pair(y, L)); it != rp.end() && next(it) != rp.end() && *next(it) <= make_pair(y, R); it++) {
int x = it->second, y = it->first, xx = next(it)->second, yy = next(it)->first;
int u = id[make_pair(x, y)], v = id[make_pair(xx, yy)];
int dis = abs(x - xx) + abs(y - yy);
g[u].emplace_back(v, dis);
g[v].emplace_back(u, dis);
}
}
{
if(p.size() >= 100000) return -1;
}
//
int ss = id[make_pair(px[s], 0)];
int tt = id[make_pair(px[t], 0)];
vector<long long> ret = calc(sz, ss, tt, g);
if(ret[tt] == LLONG_MAX) return -1;
return ret[tt];
}
/*int main() {
cout << min_distance({0, 3, 5, 7, 10, 12, 14},
{8, 7, 9, 7, 6, 6, 9},
{0, 0, 0, 2, 2, 3, 4},
{1, 2, 6, 3, 6, 4, 6},
{1, 6, 8, 1, 7, 2, 5},
1, 5);
return 0;
}
*/
