#include <bits/stdc++.h>
using namespace std;
using ll = long long;
#define pb emplace_back
#define AI(i) begin(i), end(i)
template<class T> bool chmin(T &a, T b) { return b < a && (a = b, true); }
template<class T> bool chmax(T &a, T b) { return a < b && (a = b, true); }
#ifdef KEV
#define DE(args...) kout("[ " + string(#args) + " ] = ", args)
void kout() { cerr << endl; }
template<class T, class ...U> void kout(T a, U ...b) { cerr << a << ' ', kout(b...); }
template<class T> void debug(T l, T r) { while (l != r) cerr << *l << " \n"[next(l)==r], ++l; }
#else
#define DE(...) 0
#define debug(...) 0
#endif
// TODO Watch out for the size
const int MAX_N = 400010 * 10;
const ll inf = 1ll << 55;
#include "walk.h"
// graph is 0 based
int n, m;
vector<int> xup[MAX_N];
vector<int> x, h, pfsz;
vector<pair<int,int>> edge[MAX_N];
int get_id(int i, int j) {
assert(binary_search(AI(xup[i]), j));
return lower_bound(AI(xup[i]), j) - begin(xup[i]) + pfsz[i];
}
const int C = 3;
void sweep(vector<int> l, vector<int> r, vector<int> y) {
set<int> ally;
vector<vector<int>> in(n + 10), out(n + 10);
for (int i = 0;i < m;++i) {
if (r[i] - l[i] + 1 <= 2) continue;
DE(l[i], r[i], y[i]);
in[l[i]+1].pb(y[i]);
out[r[i]].pb(y[i]);
}
for (int i = 0;i < n;++i) {
for (int j : out[i]) ally.erase(j);
for (int j : in[i]) ally.insert(j);
DE(i);
debug(AI(ally));
auto it = begin(ally), rit = ally.upper_bound(h[i]);
for (int j = 0;j < C && it != rit;++j) {
xup[i].pb(*it++);
}
for (int j = 0;j < C && it != rit;++j)
xup[i].pb(*--rit);
}
}
void init_ver(std::vector<int> l, std::vector<int> r, std::vector<int> y, int s, int g) {
sweep(l, r, y);
// for (int i = 0;i < m;++i) {
// for (int j = l[i];j <= r[i];++j) {
// if (h[j] < y[i]) continue;
// xup[j].pb(y[i]);
// }
// }
// for (int i = 0;i < n;++i) {
// sort(AI(xup[i]));
// xup[i].erase(unique(AI(xup[i])), end(xup[i]));
// if (xup[i].size() < C * 2) continue;
// vector<int> e(end(xup[i])-C, end(xup[i]));
// xup[i].resize(C);
// xup[i].insert(end(xup[i]), AI(e));
// }
//
xup[s].pb(0), xup[g].pb(0);
for (int i = 0;i < m;++i) {
xup[ l[i] ].pb( y[i] );
xup[ r[i] ].pb( y[i] );
}
pfsz.resize(n+1);
for (int i = 0;i < n;++i) {
sort(AI(xup[i]));
xup[i].erase(unique(AI(xup[i])), end(xup[i]));
pfsz[i+1] = pfsz[i] + xup[i].size();
for (int j = 1;j < xup[i].size();++j) {
int a = pfsz[i]+j-1, b = pfsz[i] + j
//int a = get_id(i, xup[i][j-1]), b = get_id(i, xup[i][j]),
, d = xup[i][j] - xup[i][j-1];
edge[a].pb(b, d);
edge[b].pb(a, d);
}
}
}
void mbuild(vector<int> l, vector<int> r, vector<int> y) {
map<int, vector<int>> xs;
for (int i = 0;i < n;++i) {
for (int j : xup[i])
xs[j].pb(i);
}
for (int i = 0;i < m;++i) {
vector< int > loc;
auto it = lower_bound(AI( xs[y[i]] ), l[i]);
while (it != end(xs[y[i]])) {
if (*it > r[i]) break;
loc.pb(*it++);
}
for (int j = 1;j < loc.size();++j) {
int a = get_id(loc[j-1], y[i]), b = get_id(loc[j], y[i]),
d = x[ loc[j] ] - x[ loc[j-1] ];
edge[a].pb(b, d);
edge[b].pb(a, d);
}
}
}
ll sho(int s, int t) {
int V = pfsz.back() + 10;
vector<ll> dis(V, inf);
priority_queue<pair<ll,ll>, vector<pair<ll,ll>>, greater<pair<ll,ll>>> pq;
auto upd = [&](int id, ll d) {
if (chmin(dis[id], d))
pq.emplace(d, id);
};
upd(s, 0);
while (pq.size()) {
auto [d, x] = pq.top(); pq.pop();
if (d != dis[x]) continue;
if (x == t) return d;
DE(x, d);
for (auto [u, w] : edge[x])
upd(u, d + w);
}
return -1;
}
long long min_distance(std::vector<int> x, std::vector<int> h, std::vector<int> l, std::vector<int> r, std::vector<int> y, int s, int g) {
n = x.size();
m = l.size();
::x = x;
::h = h;
init_ver(l, r, y, s, g);
mbuild(l, r, y);
return sho( get_id(s, 0), get_id(g, 0) );
}
