#include "elephants.h"
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef long double ld;
typedef double db;
typedef string str;
typedef pair<int,int> pi;
typedef pair<ll,ll> pl;
typedef pair<db,db> pd;
typedef vector<int> vi;
typedef vector<bool> vb;
typedef vector<ll> vl;
typedef vector<db> vd;
typedef vector<str> vs;
typedef vector<pi> vpi;
typedef vector<pl> vpl;
typedef vector<pd> vpd;
#define mp make_pair
#define f first
#define s second
#define sz(x) (int)(x).size()
#define all(x) begin(x), end(x)
#define rall(x) (x).rbegin(), (x).rend()
#define sor(x) sort(all(x))
#define rsz resize
#define ins insert
#define ft front()
#define bk back()
#define pf push_front
#define pb push_back
#define eb emplace_back
#define lb lower_bound
#define ub upper_bound
#define FOR(i,a,b) for (int i = (a); i < (b); ++i)
#define F0R(i,a) FOR(i,0,a)
#define ROF(i,a,b) for (int i = (b)-1; i >= (a); --i)
#define R0F(i,a) ROF(i,0,a)
#define trav(a,x) for (auto& a: x)
const int MOD = 1e9+7; // 998244353;
const int MX = 2e5+5;
const ll INF = 1e18;
const ld PI = acos((ld)-1);
const int xd[4] = {1,0,-1,0}, yd[4] = {0,1,0,-1};
mt19937 rng((uint32_t)chrono::steady_clock::now().time_since_epoch().count());
template<class T> bool ckmin(T& a, const T& b) {
return b < a ? a = b, 1 : 0; }
template<class T> bool ckmax(T& a, const T& b) {
return a < b ? a = b, 1 : 0; }
constexpr int pct(int x) { return __builtin_popcount(x); }
constexpr int bits(int x) { return 31-__builtin_clz(x); } // floor(log2(x))
ll cdiv(ll a, ll b) { return a/b+((a^b)>0&&a%b); } // divide a by b rounded up
ll fdiv(ll a, ll b) { return a/b-((a^b)<0&&a%b); } // divide a by b rounded down
ll half(ll x) { return fdiv(x,2); }
template<class T, class U> T fstTrue(T lo, T hi, U f) {
// note: if (lo+hi)/2 is used instead of half(lo+hi) then this will loop infinitely when lo=hi
hi ++; assert(lo <= hi); // assuming f is increasing
while (lo < hi) { // find first index such that f is true
T mid = half(lo+hi);
f(mid) ? hi = mid : lo = mid+1;
}
return lo;
}
template<class T, class U> T lstTrue(T lo, T hi, U f) {
lo --; assert(lo <= hi); // assuming f is decreasing
while (lo < hi) { // find first index such that f is true
T mid = half(lo+hi+1);
f(mid) ? lo = mid : hi = mid-1;
}
return lo;
}
template<class T> void remDup(vector<T>& v) {
sort(all(v)); v.erase(unique(all(v)),end(v)); }
// INPUT
template<class A> void re(complex<A>& c);
template<class A, class B> void re(pair<A,B>& p);
template<class A> void re(vector<A>& v);
template<class A, size_t SZ> void re(array<A,SZ>& a);
template<class T> void re(T& x) { cin >> x; }
void re(db& d) { str t; re(t); d = stod(t); }
void re(ld& d) { str t; re(t); d = stold(t); }
template<class H, class... T> void re(H& h, T&... t) { re(h); re(t...); }
template<class A> void re(complex<A>& c) { A a,b; re(a,b); c = {a,b}; }
template<class A, class B> void re(pair<A,B>& p) { re(p.f,p.s); }
template<class A> void re(vector<A>& x) { trav(a,x) re(a); }
template<class A, size_t SZ> void re(array<A,SZ>& x) { trav(a,x) re(a); }
// TO_STRING
#define ts to_string
str ts(char c) { return str(1,c); }
str ts(const char* s) { return (str)s; }
str ts(str s) { return s; }
str ts(bool b) {
#ifdef LOCAL
return b ? "true" : "false";
#else
return ts((int)b);
#endif
}
template<class A> str ts(complex<A> c) {
stringstream ss; ss << c; return ss.str(); }
str ts(vector<bool> v) {
str res = "{"; F0R(i,sz(v)) res += char('0'+v[i]);
res += "}"; return res; }
template<size_t SZ> str ts(bitset<SZ> b) {
str res = ""; F0R(i,SZ) res += char('0'+b[i]);
return res; }
template<class A, class B> str ts(pair<A,B> p);
template<class T> str ts(T v) { // containers with begin(), end()
#ifdef LOCAL
bool fst = 1; str res = "{";
for (const auto& x: v) {
if (!fst) res += ", ";
fst = 0; res += ts(x);
}
res += "}"; return res;
#else
bool fst = 1; str res = "";
for (const auto& x: v) {
if (!fst) res += " ";
fst = 0; res += ts(x);
}
return res;
#endif
}
template<class A, class B> str ts(pair<A,B> p) {
#ifdef LOCAL
return "("+ts(p.f)+", "+ts(p.s)+")";
#else
return ts(p.f)+" "+ts(p.s);
#endif
}
// OUTPUT
template<class A> void pr(A x) { cout << ts(x); }
template<class H, class... T> void pr(const H& h, const T&... t) {
pr(h); pr(t...); }
void ps() { pr("\n"); } // print w/ spaces
template<class H, class... T> void ps(const H& h, const T&... t) {
pr(h); if (sizeof...(t)) pr(" "); ps(t...); }
// DEBUG
void DBG() { cerr << "]" << endl; }
template<class H, class... T> void DBG(H h, T... t) {
cerr << ts(h); if (sizeof...(t)) cerr << ", ";
DBG(t...); }
#ifdef LOCAL // compile with -DLOCAL, chk -> fake assert
#define dbg(...) cerr << "Line(" << __LINE__ << ") -> [" << #__VA_ARGS__ << "]: [", DBG(__VA_ARGS__)
#define chk(...) if (!(__VA_ARGS__)) cerr << "Line(" << __LINE__ << ") -> function(" \
<< __FUNCTION__ << ") -> CHK FAILED: (" << #__VA_ARGS__ << ")" << "\n", exit(0);
#else
#define dbg(...) 0
#define chk(...) 0
#endif
// FILE I/O
void setIn(str s) { freopen(s.c_str(),"r",stdin); }
void setOut(str s) { freopen(s.c_str(),"w",stdout); }
void unsyncIO() { cin.tie(0)->sync_with_stdio(0); }
void setIO(str s = "") {
unsyncIO();
// cin.exceptions(cin.failbit);
// throws exception when do smth illegal
// ex. try to read letter into int
if (sz(s)) { setIn(s+".in"), setOut(s+".out"); } // for USACO
}
/**
* Description: Link-Cut Tree. Given a function $f(1\ldots N)\to 1\ldots N,$
* evaluates $f^b(a)$ for any $a,b.$ \texttt{sz} is for path queries;
* \texttt{sub}, \texttt{vsub} are for subtree queries. \texttt{x->access()}
* brings \texttt{x} to the top and propagates it; its left subtree will be
* the path from \texttt{x} to the root and its right subtree will be empty.
* Then \texttt{sub} will be the number of nodes in the connected component
* of \texttt{x} and \texttt{vsub} will be the number of nodes under \texttt{x}.
* Use \texttt{makeRoot} for arbitrary path queries.
* Time: O(\log N)
* Usage: FOR(i,1,N+1)LCT[i]=new snode(i); link(LCT[1],LCT[2],1);
* Source: Dhruv Rohatgi, Eric Zhang
* https://sites.google.com/site/kc97ble/container/splay-tree/splaytree-cpp-3
* https://codeforces.com/blog/entry/67637
* Verification: (see README for links)
* ekzhang Balanced Tokens
* Dynamic Tree Test (Easy)
* https://probgate.org/viewproblem.php?pid=578 (The Applicant)
*/
typedef struct snode* sn;
struct snode { //////// VARIABLES
sn p, c[2]; // parent, children
bool flip = 0; // subtree flipped or not
int val, sz; // value in node, # nodes in current splay tree
int sub, vsub = 0; // vsub stores sum of virtual children
int vsum;
snode(int _val) : val(_val) {
p = c[0] = c[1]; calc(); }
friend int getSz(sn x) { return x?x->sz:0; }
friend int getSub(sn x) { return x?x->sub:0; }
friend int getVsum(sn x) { return x?x->vsum:0; }
void prop() { // lazy prop
if (!flip) return;
swap(c[0],c[1]); flip = 0;
F0R(i,2) if (c[i]) c[i]->flip ^= 1;
}
void calc() { // recalc vals
F0R(i,2) if (c[i]) c[i]->prop();
sz = 1+getSz(c[0])+getSz(c[1]);
sub = 1+getSub(c[0])+getSub(c[1])+vsub;
vsum = val+getVsum(c[0])+getVsum(c[1]);
}
//////// SPLAY TREE OPERATIONS
int dir() {
if (!p) return -2;
F0R(i,2) if (p->c[i] == this) return i;
return -1; // p is path-parent pointer
} // -> not in current splay tree
// test if root of current splay tree
bool isRoot() { return dir() < 0; }
friend void setLink(sn x, sn y, int d) {
if (y) y->p = x;
if (d >= 0) x->c[d] = y; }
void rot() { // assume p and p->p propagated
assert(!isRoot()); int x = dir(); sn pa = p;
setLink(pa->p, this, pa->dir());
setLink(pa, c[x^1], x); setLink(this, pa, x^1);
pa->calc(); calc();
}
void splay() {
while (!isRoot() && !p->isRoot()) {
p->p->prop(), p->prop(), prop();
dir() == p->dir() ? p->rot() : rot();
rot();
}
if (!isRoot()) p->prop(), prop(), rot();
prop();
}
sn fbo(int b) { // find by order
prop(); int z = getSz(c[0]); // of splay tree
if (b == z) { splay(); return this; }
return b < z ? c[0]->fbo(b) : c[1] -> fbo(b-z-1);
}
//////// BASE OPERATIONS
void access() { // bring this to top of tree, propagate
for (sn v = this, pre = NULL; v; v = v->p) {
v->splay(); // now switch virtual children
if (pre) v->vsub -= pre->sub;
if (v->c[1]) v->vsub += v->c[1]->sub;
v->c[1] = pre; v->calc(); pre = v;
}
splay(); assert(!c[1]); // right subtree is empty
}
void makeRoot() {
access(); flip ^= 1; access(); assert(!c[0] && !c[1]); }
//////// QUERIES
friend sn lca(sn x, sn y) {
if (x == y) return x;
x->access(), y->access(); if (!x->p) return NULL;
x->splay(); return x->p?:x; // y was below x in latter case
} // access at y did not affect x -> not connected
friend bool connected(sn x, sn y) { return lca(x,y); }
// # nodes above
int distRoot() { access(); return getSz(c[0]); }
sn getRoot() { // get root of LCT component
access(); sn a = this;
while (a->c[0]) a = a->c[0], a->prop();
a->access(); return a;
}
sn getPar(int b) { // get b-th parent on path to root
access(); b = getSz(c[0])-b; assert(b >= 0);
return fbo(b);
} // can also get min, max on path to root, etc
//////// MODIFICATIONS
void set(int v) { access(); val = v; calc(); }
friend void link(sn x, sn y, bool force = 0) {
assert(!connected(x,y));
if (force) y->makeRoot(); // make x par of y
else { y->access(); assert(!y->c[0]); }
x->access(); setLink(y,x,0); y->calc();
}
friend void cut(sn y) { // cut y from its parent
y->access(); assert(y->c[0]);
y->c[0]->p = NULL; y->c[0] = NULL; y->calc(); }
friend void cut(sn x, sn y) { // if x, y adj in tree
x->makeRoot(); y->access();
assert(y->c[0] == x && !x->c[0] && !x->c[1]); cut(y); }
};
sn LCT[400005];
const int mx = 150005;
int N, L;
int X[mx];
int par[mx];
set<pi> m;
int M;
void init(int _N, int _L, int _X[])
{
N = _N;
L = _L;
M = N+5;
for(int i = 0; i < N; i++){
X[i] = _X[i];
m.ins(mp(X[i], i));
}
X[N] = 2*MOD;
m.ins(mp(2*MOD, N));
LCT[N] = new snode(0);
for(int i = 0; i < N; i++){
LCT[i] = new snode(0);
LCT[i+M] = new snode(0);
link(LCT[i+M], LCT[i], 1);
//dbg(i+M, i);
}
//dbg(N, M);
for(int i = 0; i < N; i++){
int to = (m.lb(mp(X[i]+L+1, 0)))->s;
int nex = (next(m.find(mp(X[i], i))))->s;
//dbg(i, to, nex);
if(nex != N && (m.lb(mp(X[nex]+L+1, 0)))->s == to){
LCT[i+M]->set(0);
link(LCT[nex], LCT[i+M], 1);
par[i] = nex;
//dbg(nex, i+M);
}
else{
LCT[i+M]->set(1);
link(LCT[to], LCT[i+M], 1);
par[i] = to;
//dbg(to, i+M);
}
}
}
int spec;
void upd(int i){
if(i == -1) return;
//dbg(i);
// for(int i = 0; i < N; i++){
// dbg(i, par[i]);
// }
// dbg(i+M, par[i]);
// dbg(connected(LCT[i+M], LCT[par[i]]));
// dbg(i+M, par[i]);
if(spec != i) cut(LCT[i+M], LCT[par[i]]);
//dbg("cut", i+M, par[i]);
int to = (m.lb(mp(X[i]+L+1, 0)))->s;
int nex = (next(m.find(mp(X[i], i))))->s;
//dbg(to, nex);
if(nex != N && (m.lb(mp(X[nex]+L+1, 0)))->s == to){
LCT[i+M]->set(0);
//dbg(nex, i+M);
link(LCT[nex], LCT[i+M], 1);
par[i] = nex;
}
else{
LCT[i+M]->set(1);
//dbg(to, i+M);
link(LCT[to], LCT[i+M], 1);
par[i] = to;
}
}
int update(int i, int y)
{
spec = i;
//dbg(i, y);
//erase from lct
int prv1 = -1, prv2 = -1;
if(m.find(mp(X[i], i)) != m.begin()){
prv1 = (prev(m.find(mp(X[i], i))))->s;
}
auto it = m.lb(mp(X[i]-L, 0));
if(it != m.begin()){
prv2 = prev(it)->s;
}
//dbg(prv1, prv2);
m.erase(mp(X[i], i));
cut(LCT[i+M], LCT[par[i]]);
upd(prv1);
upd(prv2);
X[i] = y;
//add to lct
m.ins(mp(X[i], i));
prv1 = -1, prv2 = -1;
if(m.find(mp(X[i], i)) != m.begin()){
prv1 = (prev(m.find(mp(X[i], i))))->s;
}
it = m.lb(mp(X[i]-L, 0));
if(it != m.begin()){
prv2 = prev(it)->s;
}
//dbg(prv1, prv2, i);
upd(prv1);
upd(prv2);
upd(i);
LCT[N]->makeRoot();
int beg = (m.begin())->s;
LCT[beg]->access();
int ans = getVsum(LCT[beg]);
return ans;
}
