2846. Minimum Edge Weight Equilibrium Queries in a Tree

There is an undirected tree with n nodes labeled from 0 to n - 1. You are given the integer n and a 2D integer array edges of length n - 1, where edges[i] = [ui, vi, wi] indicates that there is an edge between nodes ui and vi with weight wi in the tree.

You are also given a 2D integer array queries of length m, where queries[i] = [ai, bi]. For each query, find the minimum number of operations required to make the weight of every edge on the path from ai to bi equal. In one operation, you can choose any edge of the tree and change its weight to any value.

Note that:

• Queries are independent of each other, meaning that the tree returns to its initial state on each new query.
• The path from ai to bi is a sequence of distinct nodes starting with node ai and ending with node bi such that every two adjacent nodes in the sequence share an edge in the tree.

Return an array answer of length m where answer[i] is the answer to the ith query.

c++

#include <bits/stdc++.h>

#pragma optimization_level 3
#pragma GCC optimize("Ofast,no-stack-protector,unroll-loops,fast-math,O3")
#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx")
#pragma GCC optimize("Ofast")//Comment optimisations for interactive problems (use endl)
#pragma GCC target("avx,avx2,fma")
#pragma GCC optimization ("unroll-loops")

using namespace std;

struct PairHash {inline std::size_t operator()(const std::pair<int, int> &v) const { return v.first * 31 + v.second; }};

// speed
#define Code ios_base::sync_with_stdio(false);
#define By ios::sync_with_stdio(0);
#define Sumfi cout.tie(NULL);

// alias
using ll = long long;
using ld = long double;
using ull = unsigned long long;

// constants
const ld PI = acosl(-1.0);  /* pi */
const ll INF = 1e18;
const ld EPS = 1e-9;
const ll MAX_N = 10101;
const ll mod = 1e9 + 7;

// typedef
typedef pair<int,int> pii;
typedef pair<ll,ll> pll;
typedef vector<pll> vpll;
typedef array<int,3> ai3;
typedef array<ll,3> all3;
typedef array<ll,4> all4;
typedef array<ll,5> all5;
typedef vector<all3> vall3;
typedef vector<all4> vall4;
typedef vector<all5> vall5;
typedef pair<ld, ld> pld;
typedef vector<pld> vpld;
typedef vector<ld> vld;
typedef vector<ll> vll;
typedef vector<ull> vull;
typedef vector<vll> vvll;
typedef vector<int> vi;
typedef vector<bool> vb;
typedef deque<ll> dqll;
typedef deque<pll> dqpll;
typedef pair<string, string> pss;
typedef vector<pss> vpss;
typedef vector<string> vs;
typedef vector<vs> vvs;
typedef unordered_set<ll> usll;
typedef unordered_set<pll, PairHash> uspll;
typedef unordered_map<ll, ll> umll;
typedef unordered_map<pll, ll, PairHash> umpll;

// macros
#define precision(x) cout<<fixed;cout.precision(x);
#define rep(i,m,n) for(ll i=m;i<n;i++)
#define rrep(i,m,n) for(ll i=n;i>=m;i--)
#define all(a) begin(a), end(a)
#define rall(a) rbegin(a), rend(a)
#define ZERO(a) memset(a,0,sizeof(a))
#define MINUS(a) memset(a,0xff,sizeof(a))
#define INF(a) memset(a,0x3f3f3f3f3f3f3f3fLL,sizeof(a))
#define ASCEND(a) iota(all(a),0)
#define sz(x) ll((x).size())
#define BIT(a,i) ((a>>i)&1)
#define BITSHIFT(a,i,n) (((a<<i) & ((1ll<<n) - 1)) | (a>>(n-i)))
#define MAXBIT(a) (64ll - __builtin_clzll(a) - 1ll)
#define pyes cout<<"YES\n";
#define pno cout<<"NO\n";
#define endl "\n"
#define pneg1 cout<<"-1\n";
#define ppossible cout<<"Possible\n";
#define pimpossible cout<<"Impossible\n";
#define TC(x) cout<<"Case #"<<x<<": ";
#define X first
#define Y second

// debug
void __print(int x) { cerr << x; }
void __print(long x) { cerr << x; }
void __print(long long x) { cerr << x; }
void __print(unsigned x) { cerr << x; }
void __print(unsigned long x) { cerr << x; }
void __print(unsigned long long x) { cerr << x; }
void __print(float x) { cerr << x; }
void __print(double x) { cerr << x; }
void __print(long double x) { cerr << x; }
void __print(char x) { cerr << '\'' << x << '\''; }
void __print(const char *x) { cerr << '\"' << x << '\"'; }
void __print(const string &x) { cerr << '\"' << x << '\"'; }
void __print(bool x) { cerr << (x ? "true" : "false"); }
template <typename A>
void __print(const A &x);
template <typename A, typename B>
void __print(const pair<A, B> &p);
template <typename... A>
void __print(const tuple<A...> &t);
template <typename T>
void __print(stack<T> s);
template <typename T>
void __print(queue<T> q);
template <typename T, typename... U>
void __print(priority_queue<T, U...> q);
template <typename A>
void __print(const A &x) {
    bool first = true;
    cerr << '{';
    for (const auto &i : x) {
        cerr << (first ? "" : ","), __print(i);
        first = false;
    }
    cerr << '}';
}
template <typename A, typename B>
void __print(const pair<A, B> &p) {
    cerr << '(';
    __print(p.first);
    cerr << ',';
    __print(p.second);
    cerr << ')';
}
template <typename... A>
void __print(const tuple<A...> &t) {
    bool first = true;
    cerr << '(';
    apply([&first](const auto &...args) { ((cerr << (first ? "" : ","), __print(args), first = false), ...); }, t);
    cerr << ')';
}
template <typename T>
void __print(stack<T> s) {
    vector<T> debugVector;
    while (!s.empty()) {
        T t = s.top();
        debugVector.push_back(t);
        s.pop();
    }
    reverse(debugVector.begin(), debugVector.end());
    __print(debugVector);
}
template <typename T>
void __print(queue<T> q) {
    vector<T> debugVector;
    while (!q.empty()) {
        T t = q.front();
        debugVector.push_back(t);
        q.pop();
    }
    __print(debugVector);
}
template <typename T, typename... U>
void __print(priority_queue<T, U...> q) {
    vector<T> debugVector;
    while (!q.empty()) {
        T t = q.top();
        debugVector.push_back(t);
        q.pop();
    }
    __print(debugVector);
}
void _print() { cerr << "]\n"; }
template <typename Head, typename... Tail>
void _print(const Head &H, const Tail &...T) {
    __print(H);
    if (sizeof...(T))
        cerr << ", ";
    _print(T...);
}
#ifndef ONLINE_JUDGE
#define debug(...) cerr << "Line:" << __LINE__ << " [" << #__VA_ARGS__ << "] = ["; _print(__VA_ARGS__);
#else
#define debug(...)
#endif

// utility functions
template <typename T>
void print(T &&t)  { cout << t << "\n"; }
template<typename T>
void printv(vector<T>v){ll n=v.size();rep(i,0,n){cout<<v[i];if(i+1!=n)cout<<' ';}cout<<endl;}
template<typename T>
void printvv(vector<vector<T>>v){ll n=v.size();rep(i,0,n)printv(v[i]);}
template<typename T>
void printvln(vector<T>v){ll n=v.size();rep(i,0,n)cout<<v[i]<<endl;}
void fileIO(string in = "input.txt", string out = "output.txt") {freopen(in.c_str(),"r",stdin); freopen(out.c_str(),"w",stdout);}
void readf() {freopen("", "rt", stdin);}
template <typename... T>
void in(T &...a) { ((cin >> a), ...); }
template<typename T>
void readv(vector<T>& v){rep(i,0,sz(v)) in(v[i]);}
template<typename T, typename U>
void readp(pair<T,U>& A) {cin>>A.first>>A.second;}
template<typename T, typename U>
void readvp(vector<pair<T,U>>& A) {rep(i,0,sz(A)) readp(A[i]); }
template<typename T>
void readvall(vector<T>& v) {rep(i,0,sz(v)) rep(j,0,sz(v[i])) in(v[i][j]);}
template<typename T>
void readvv(vector<vector<T>>& A) {rep(i,0,sz(A)) readv(A[i]);}

struct Combination {
    vll fac, inv;
    ll n, MOD;

    ll modpow(ll n, ll x, ll MOD = mod) { if(!x) return 1; ll res = modpow(n,x>>1,MOD); res = (res * res) % MOD; if(x&1) res = (res * n) % MOD; return res; }

    Combination(ll _n, ll MOD = mod): n(_n + 1), MOD(MOD) {
        inv = fac = vll(n,1);
        rep(i,1,n) fac[i] = fac[i-1] * i % MOD;
        inv[n - 1] = modpow(fac[n - 1], MOD - 2, MOD);
        rrep(i,1,n - 2) inv[i] = inv[i + 1] * (i + 1) % MOD;
    }

    ll fact(ll n) {return fac[n];}
    ll nCr(ll n, ll r) {
        if(n < r or n < 0 or r < 0) return 0;
        return fac[n] * inv[r] % MOD * inv[n-r] % MOD;
    }
};

struct Matrix {
    ll r,c,MOD;
    vvll matrix;
    Matrix(ll r, ll c, ll v = 0, ll MOD = mod): r(r), c(c), matrix(vvll(r,vll(c,v))), MOD(MOD) {}
    Matrix(vvll m, ll MOD = mod) : r(sz(m)), c(sz(m[0])), matrix(m), MOD(MOD) {}

    vector<ll>& operator[](ll pos) {return matrix[pos];}
    Matrix operator*(const Matrix& B) const {
        Matrix res(r, B.c, 0,MOD);
        rep(i,0,r) rep(j,0,B.c) rep(k,0,B.r) {
                    res[i][j] = (res[i][j] + matrix[i][k] * B.matrix[k][j] % MOD) % MOD;
                }
        return res;
    }

    Matrix copy() {
        Matrix copy(r,c,0,MOD);
        copy.matrix = matrix;
        return copy;
    }


    Matrix pow(ll n) {
        assert(r == c);
        Matrix res(r,r, 0,MOD);
        Matrix now = copy();
        rep(i,0,r) res[i][i] = 1;
        while(n) {
            if(n & 1) res = res * now;
            now = now * now;
            n /= 2;
        }
        return res;
    }

    ll det() {
        if(r == 1) return matrix[0][0];
        if(r == 2) return matrix[0][0] * matrix[1][1] - matrix[0][1] * matrix[1][0];
        ll res = 0;
        rep(p,0,c) {
            Matrix mat(c-1,c-1);
            rep(i,1,r) rep(j,0,c) {
                    if(j == p) continue;
                    mat[i][j - (j >= p)] = matrix[i][j];
                }
            res += matrix[0][p] * (p & 1 ? -1 : 1) * mat.det();
        }
        return res;
    }
};

// geometry data structures
template <typename T>
struct Point {
    T y,x;
    Point(T y, T x) : y(y), x(x) {}
    Point(pair<T,T> p) : y(p.first), x(p.second) {}
    Point() {}
    void input() {cin>>y>>x;}
    friend ostream& operator<<(ostream& os, const Point<T>& p) { os<<p.y<<' '<<p.x<<'\n'; return os;}
    Point<T> operator+(Point<T>& p) {return Point<T>(y + p.y, x + p.x);}
    Point<T> operator-(Point<T>& p) {return Point<T>(y - p.y, x - p.x);}
    Point<T> operator*(ll n) {return Point<T>(y*n,x*n); }
    Point<T> operator/(ll n) {return Point<T>(y/n,x/n); }
    bool operator==(const Point<T> &a) {return x == a.x && y == a.y;}
    bool operator<(const Point &other) const {if (x == other.x) return y < other.y;return x < other.x;}
    Point<T> rotate(Point<T> center, ld angle) {
        ld si = sin(angle * PI / 180.), co = cos(angle * PI / 180.);
        ld y = this->y - center.y;
        ld x = this->x - center.x;

        return Point<T>(y * co - x * si + center.y, y * si + x * co + center.x);
    }
    ld distance(Point<T> other) {
        T dy = abs(this->y - other.y);
        T dx = abs(this->x - other.x);
        return sqrt(dy * dy + dx * dx);
    }

    T norm() { return x * x + y * y; }
};

template<typename T>
struct Line {
    Point<T> A, B;
    Line(Point<T> A, Point<T> B) : A(A), B(B) {}
    Line() {}

    void input() {
        A = Point<T>();
        B = Point<T>();
        A.input();
        B.input();
    }

    T ccw(Point<T> &a, Point<T> &b, Point<T> &c) {
        T res = a.x * b.y + b.x * c.y + c.x * a.y;
        res -= (a.x * c.y + b.x * a.y + c.x * b.y);
        return res;
    }

    bool on(Point<T> x) {
        return ccw(A,x,B) == 0;
    }

    bool isIntersect(Line<T> o) {
        T p1p2 = ccw(A,B,o.A) * ccw(A,B,o.B);
        T p3p4 = ccw(o.A,o.B,A) * ccw(o.A,o.B,B);
        if (p1p2 == 0 && p3p4 == 0) {
            pair<T,T> p1(A.y, A.x), p2(B.y,B.x), p3(o.A.y, o.A.x), p4(o.B.y, o.B.x);
            if (p1 > p2) swap(p2, p1);
            if (p3 > p4) swap(p3, p4);
            return p3 <= p2 && p1 <= p4;
        }
        return p1p2 <= 0 && p3p4 <= 0;
    }

    pair<bool,Point<ld>> intersection(Line<T> o) {
        if(!this->intersection(o)) return {false, {}};
        ld det = 1. * (o.B.y-o.A.y)*(B.x-A.x) - 1.*(o.B.x-o.A.x)*(B.y-A.y);
        ld t = ((o.B.x-o.A.x)*(A.y-o.A.y) - (o.B.y-o.A.y)*(A.x-o.A.x)) / det;
        return {true, {A.y + 1. * t * (B.y - A.y), B.x + 1. * t * (B.x - A.x)}};
    }

    //@formula for : y = ax + pre
    //@return {a,pre};
    pair<ld, ld> formula() {
        T y1 = A.y, y2 = B.y;
        T x1 = A.x, x2 = B.x;
        if(y1 == y2) return {0, (ld)y1};
        if(x1 == x2) return {INF, (ld)x1};
        ld a = 1. * (y2 - y1) / (x2 - x1);
        ld b = -x1 * a + y1;
        return {a, b};
    }
};

template<typename T>
struct Circle {
    Point<T> center;
    T radius;
    Circle(T y, T x, T radius) : center(Point<T>(y,x)), radius(radius) {}
    Circle(Point<T> center, T radius) : center(center), radius(radius) {}
    Circle() {}

    void input() {
        center = Point<T>();
        center.input();
        cin>>radius;
    }

    bool circumference(Point<T> p) {
        return (center.x - p.x) * (center.x - p.x) + (center.y - p.y) * (center.y - p.y) == radius * radius;
    }

    bool intersect(Circle<T> c) {
        T d = (center.x - c.center.x) * (center.x - c.center.x) + (center.y - c.center.y) * (center.y - c.center.y);
        return (radius - c.radius) * (radius - c.radius) <= d and d <= (radius + c.radius) * (radius + c.radius);
    }

    bool include(Circle<T> c) {
        T d = (center.x - c.center.x) * (center.x - c.center.x) + (center.y - c.center.y) * (center.y - c.center.y);
        return d <= radius * radius;
    }
};

ll __gcd(ll x, ll y) { return !y ? x : __gcd(y, x % y); }
all3 __exgcd(ll x, ll y) { if(!y) return {x,1,0}; auto [g,x1,y1] = __exgcd(y, x % y); return {g, y1, x1 - (x/y) * y1}; }
ll __lcm(ll x, ll y) { return x / __gcd(x,y) * y; }
ll modpow(ll n, ll x, ll MOD = mod) { if(x < 0) return modpow(modpow(n,-x,MOD), MOD-2,MOD); n%=MOD; if(!x) return 1; ll res = modpow(n,x>>1,MOD); res = (res * res) % MOD; if(x&1) res = (res * n) % MOD; return res; }

ll LCA[MAX_N][22], dp[MAX_N][22][27], level[MAX_N];
ll query(ll u, ll v) {
    if(level[u] < level[v]) swap(u, v);
    ll diff = level[u] - level[v];
    for(ll i = 0; diff; i++, diff /= 2) {
        if(diff & 1) u = LCA[u][i];
    }

    if(u != v) {
        rrep(i, 0, 21) {
            if(LCA[u][i] == LCA[v][i]) continue;
            u = LCA[u][i];
            v = LCA[v][i];
        }
        u = LCA[u][0];
    }
    return u;
}
ll distance(ll u, ll v) {
    ll lca = query(u,v);
    return level[u] + level[v] - 2 * level[lca];
}
int ask(ll u, ll k, ll w) {
    ll res = 0;
    for(ll i = 0; k and i < 22; i++, k /= 2) {
        if(k & 1) {
            res += dp[u][i][w];
            u = LCA[u][i];
        }
    }
    return res;
}
class Solution {
    vector<vpll> adj;
    void dfs1(ll u, ll par, ll dep = 0) {
        level[u] = dep;
        for(auto [v,w] : adj[u]) {
            if(v == par) continue;
            LCA[v][0] = u;
            dp[v][0][w] = 1;
            dfs1(v,u,dep+1);
        }
    }
public:
    vector<int> minOperationsQueries(int n, vector<vector<int>>& edges, vector<vector<int>>& queries) {
        MINUS(LCA);
        ZERO(dp);
        adj = vector<vpll>(n);
        for(int i = 0; i < edges.size(); i++) {
            int u = edges[i][0], v = edges[i][1], w = edges[i][2];
            adj[u].push_back({v,w});
            adj[v].push_back({u,w});
        }
        dfs1(0,-1);
        rep(j,0,22) {
            rep(i,0,n) {
                if(LCA[i][j] == -1) continue;
                LCA[i][j+1] = LCA[LCA[i][j]][j];
                rep(k,1,27) {
                    dp[i][j+1][k] = dp[LCA[i][j]][j][k] + dp[i][j][k];
                }
            }
        }
        vi res;
        rep(i,0,sz(queries)) {
            auto u = queries[i][0], v = queries[i][1];
            int lca = query(u,v), dis = distance(u,v), now = dis;
            int disu = distance(lca,u), disv = distance(lca,v);
            rep(j,1,27) {
                now = min(now, dis - ask(u,disu,j) - ask(v,disv,j));
            }
            res.push_back(now);
        }
        return res;
    }
};