P3302 [SDOI2013] 森林 題解

Athanasy發表於2024-03-17

題目連結：森林

有意思的樹上可持久化線段樹變形題，建議先看這個：P2633 Count on a tree 題解

對於本題而言，我們重新闡述樹上可持久化線段樹的核心思想，對於點路徑/邊路徑上的第 \(k\) 大問題，我們使用樹上字首和問題的思想，將其轉化為可差性問題：一條路徑上的權值線段樹可以拆分為幾棵權值線段樹進行貢獻拿到，具體可以參照連結文章提到的樹上字首和的基本思想。

本題觀察到既有連線新邊操作，又有查詢操作，相對來說較為複雜，我們重新關注樹上可持久化線段樹樹需要關注什麼：

  1. \(u\) 與 \(v\) 處的字首點對應的可持久化線段樹。

  2. \(lca\) 與 \(fa[lca]\) 處的字首點對應的可持久化線段樹。

注意到本題為森林，不一定是一棵完整的樹了，我們注意如果 \(u\) 與 \(v\) 建邊：

那麼很顯然，\(u\) 可以作為 \(v\) 的父親，而 \(v\) 也可以作為 \(u\) 的父親，並且對於它們各自對應的 \(root\) 上的可持久化線段樹查詢是正確的，可以看做區域性每棵樹的樹上字首和建出來的可持久化線段樹在區域性上的查詢顯然正確。其實抽象出來區域性的一棵樹不也是一棵完整的樹嗎？我們發現合併的影響僅僅體現在某一個點成為另一個點的父親：

如圖所示，如果 \(u \rightarrow v\)，即 \(v\) 成為 \(u\) 的父親，那麼影響的僅僅是 \(u\) 為根對應的整棵樹的所有點，它們的倍增陣列和父親還有深度之類的包括可持久化線段樹都得重建。我們考慮暴力一次更新是 \(O(n\log{n})\)。那顯然如果每次都隨便選一個點暴力更新，複雜度則為 \(O(n^2\log{n})\)，你不覺得這玩意不就是樹合併嗎？樹合併最佳化複雜度的辦法多了去了，我們考慮使用啟發式合併，小樹的根作為合併物件，大樹的根則為父親。

當然有人疑惑它們一開始就有父親了，怎麼寫？給兩個一開始我過了但並不是很符合正確的啟發式合併的寫法：

參照寫法1 
#include <bits/stdc++.h>

// #pragma GCC optimize(2)
// #pragma GCC optimize("Ofast,no-stack-protector,unroll-loops,fast-math")
// #pragma GCC target("sse,sse2,sse3,ssse3,sse4.1,sse4.2,avx,avx2,popcnt,tune=native")

#define isPbdsFile

#ifdef isPbdsFile

#include <bits/extc++.h>

#else

#include <ext/pb_ds/priority_queue.hpp>
#include <ext/pb_ds/hash_policy.hpp>
#include <ext/pb_ds/tree_policy.hpp>
#include <ext/pb_ds/trie_policy.hpp>
#include <ext/pb_ds/tag_and_trait.hpp>
#include <ext/pb_ds/hash_policy.hpp>
#include <ext/pb_ds/list_update_policy.hpp>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/exception.hpp>
#include <ext/rope>

#endif

using namespace std;
using namespace __gnu_cxx;
using namespace __gnu_pbds;
typedef long long ll;
typedef long double ld;
typedef pair<int, int> pii;
typedef pair<ll, ll> pll;
typedef tuple<int, int, int> tii;
typedef tuple<ll, ll, ll> tll;
typedef unsigned int ui;
typedef unsigned long long ull;
typedef __int128 i128;
#define hash1 unordered_map
#define hash2 gp_hash_table
#define hash3 cc_hash_table
#define stdHeap std::priority_queue
#define pbdsHeap __gnu_pbds::priority_queue
#define sortArr(a, n) sort(a+1,a+n+1)
#define all(v) v.begin(),v.end()
#define yes cout<<"YES"
#define no cout<<"NO"
#define Spider ios_base::sync_with_stdio(false);cin.tie(nullptr);cout.tie(nullptr);
#define MyFile freopen("..\\input.txt", "r", stdin),freopen("..\\output.txt", "w", stdout);
#define forn(i, a, b) for(int i = a; i <= b; i++)
#define forv(i, a, b) for(int i=a;i>=b;i--)
#define ls(x) (x<<1)
#define rs(x) (x<<1|1)
#define endl '\n'
//用於Miller-Rabin
[[maybe_unused]] static int Prime_Number[13] = {0, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37};

template <typename T>
int disc(T* a, int n)
{
    return unique(a + 1, a + n + 1) - (a + 1);
}

template <typename T>
T lowBit(T x)
{
    return x & -x;
}

template <typename T>
T Rand(T l, T r)
{
    static mt19937 Rand(time(nullptr));
    uniform_int_distribution<T> dis(l, r);
    return dis(Rand);
}

template <typename T1, typename T2>
T1 modt(T1 a, T2 b)
{
    return (a % b + b) % b;
}

template <typename T1, typename T2, typename T3>
T1 qPow(T1 a, T2 b, T3 c)
{
    a %= c;
    T1 ans = 1;
    for (; b; b >>= 1, (a *= a) %= c)if (b & 1)(ans *= a) %= c;
    return modt(ans, c);
}

template <typename T>
void read(T& x)
{
    x = 0;
    T sign = 1;
    char ch = getchar();
    while (!isdigit(ch))
    {
        if (ch == '-')sign = -1;
        ch = getchar();
    }
    while (isdigit(ch))
    {
        x = (x << 3) + (x << 1) + (ch ^ 48);
        ch = getchar();
    }
    x *= sign;
}

template <typename T, typename... U>
void read(T& x, U&... y)
{
    read(x);
    read(y...);
}

template <typename T>
void write(T x)
{
    if (typeid(x) == typeid(char))return;
    if (x < 0)x = -x, putchar('-');
    if (x > 9)write(x / 10);
    putchar(x % 10 ^ 48);
}

template <typename C, typename T, typename... U>
void write(C c, T x, U... y)
{
    write(x), putchar(c);
    write(c, y...);
}


template <typename T11, typename T22, typename T33>
struct T3
{
    T11 one;
    T22 tow;
    T33 three;

    bool operator<(const T3 other) const
    {
        if (one == other.one)
        {
            if (tow == other.tow)return three < other.three;
            return tow < other.tow;
        }
        return one < other.one;
    }

    T3() { one = tow = three = 0; }

    T3(T11 one, T22 tow, T33 three) : one(one), tow(tow), three(three)
    {
    }
};

template <typename T1, typename T2>
void uMax(T1& x, T2 y)
{
    if (x < y)x = y;
}

template <typename T1, typename T2>
void uMin(T1& x, T2 y)
{
    if (x > y)x = y;
}

struct Hash
{
    static uint64_t splitmix64(uint64_t x)
    {
        x += 0x9e3779b97f4a7c15;
        x = (x ^ x >> 30) * 0xbf58476d1ce4e5b9;
        x = (x ^ x >> 27) * 0x94d049bb133111eb;
        return x ^ x >> 31;
    }

    static size_t get(const uint64_t x)
    {
        static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();
        return splitmix64(x + FIXED_RANDOM);
    }

    template <typename T>
    size_t operator()(T x) const
    {
        return get(std::hash<T>()(x));
    }

    template <typename F, typename S>
    size_t operator()(pair<F, S> p) const
    {
        return get(std::hash<F>()(p.first)) ^ std::hash<S>()(p.second);
    }
};

constexpr int N = 1e5 + 10;

struct Node
{
    int left, right, cnt;
} node[N << 7];

#define left(x) node[x].left
#define right(x) node[x].right
#define cnt(x) node[x].cnt
int n, m, q, mx, cnt;
int a[N], ord[N];
hash2<int, int, Hash> mp;

inline void add(const int pre, int& curr, const int val, const int l = 1, const int r = mx)
{
    node[curr = ++cnt] = node[pre];
    cnt(curr)++;
    const int mid = l + r >> 1;
    if (l == r)return;
    if (val <= mid)add(left(pre),left(curr), val, l, mid);
    else add(right(pre),right(curr), val, mid + 1, r);
}

inline int query(const int u, const int v, const int lca, const int lca_fa, const int k, const int l = 1,
                 const int r = mx)
{
    if (l == r)return ord[l];
    const int mid = l + r >> 1;
    const int leftSize = cnt(left(u)) + cnt(left(v)) - cnt(left(lca)) - cnt(left(lca_fa));
    if (leftSize >= k)return query(left(u),left(v),left(lca),left(lca_fa), k, l, mid);
    return query(right(u),right(v),right(lca),right(lca_fa), k - leftSize, mid + 1, r);
}

constexpr int T = 20;
int root[N], fa[N][T + 1], deep[N];
vector<int> child[N];

inline void dfs(const int curr, const int pa)
{
    deep[curr] = deep[fa[curr][0] = pa] + 1;
    add(root[pa], root[curr], a[curr]);
    forn(i, 1, T)fa[curr][i] = fa[fa[curr][i - 1]][i - 1];
    for (const auto nxt : child[curr])if (nxt != pa)dfs(nxt, curr);
}

inline int LCA(int x, int y)
{
    if (deep[x] < deep[y])swap(x, y);
    forv(i, T, 0)if (deep[fa[x][i]] >= deep[y])x = fa[x][i];
    if (x == y)return x;
    forv(i, T, 0)if (fa[x][i] != fa[y][i])x = fa[x][i], y = fa[y][i];
    return fa[x][0];
}

int last;

inline void solve()
{
    cin >> n >> m >> q;
    forn(i, 1, n)cin >> a[i], ord[i] = a[i];
    sortArr(ord, n), mx = disc(ord, n);
    forn(i, 1, mx)mp[ord[i]] = i;
    forn(i, 1, n)a[i] = mp[a[i]];
    forn(i, 1, m)
    {
        int u, v;
        cin >> u >> v;
        child[u].push_back(v);
        child[v].push_back(u);
    }
    forn(i, 1, n)if (!deep[i])dfs(i, 0);
    while (q--)
    {
        char op;
        cin >> op;
        if (op == 'L')
        {
            int u, v;
            cin >> u >> v, u ^= last, v ^= last;
            child[u].push_back(v), child[v].push_back(u);
            if (cnt(root[u]) < cnt(root[v]))swap(u, v);
            dfs(v, u);
        }
        else
        {
            int u, v, k;
            cin >> u >> v >> k, u ^= last, v ^= last, k ^= last;
            const int lca = LCA(u, v);
            const int lca_fa = fa[lca][0];
            cout << (last = query(root[u], root[v], root[lca], root[lca_fa], k)) << endl;
        }
    }
}

signed int main()
{
    // MyFile
    Spider
    //------------------------------------------------------
    // clock_t start = clock();
    int test = 1;
    //    read(test);
    cin >> test;
    test = 1;
    forn(i, 1, test)solve();
    //    while (cin >> n, n)solve();
    //    while (cin >> test)solve();
    // clock_t end = clock();
    // cerr << "time = " << double(end - start) / CLOCKS_PER_SEC << "s" << endl;
}
參照寫法2 
#include <bits/stdc++.h>

// #pragma GCC optimize(2)
// #pragma GCC optimize("Ofast,no-stack-protector,unroll-loops,fast-math")
// #pragma GCC target("sse,sse2,sse3,ssse3,sse4.1,sse4.2,avx,avx2,popcnt,tune=native")

#define isPbdsFile

#ifdef isPbdsFile

#include <bits/extc++.h>

#else

#include <ext/pb_ds/priority_queue.hpp>
#include <ext/pb_ds/hash_policy.hpp>
#include <ext/pb_ds/tree_policy.hpp>
#include <ext/pb_ds/trie_policy.hpp>
#include <ext/pb_ds/tag_and_trait.hpp>
#include <ext/pb_ds/hash_policy.hpp>
#include <ext/pb_ds/list_update_policy.hpp>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/exception.hpp>
#include <ext/rope>

#endif

using namespace std;
using namespace __gnu_cxx;
using namespace __gnu_pbds;
typedef long long ll;
typedef long double ld;
typedef pair<int, int> pii;
typedef pair<ll, ll> pll;
typedef tuple<int, int, int> tii;
typedef tuple<ll, ll, ll> tll;
typedef unsigned int ui;
typedef unsigned long long ull;
typedef __int128 i128;
#define hash1 unordered_map
#define hash2 gp_hash_table
#define hash3 cc_hash_table
#define stdHeap std::priority_queue
#define pbdsHeap __gnu_pbds::priority_queue
#define sortArr(a, n) sort(a+1,a+n+1)
#define all(v) v.begin(),v.end()
#define yes cout<<"YES"
#define no cout<<"NO"
#define Spider ios_base::sync_with_stdio(false);cin.tie(nullptr);cout.tie(nullptr);
#define MyFile freopen("..\\input.txt", "r", stdin),freopen("..\\output.txt", "w", stdout);
#define forn(i, a, b) for(int i = a; i <= b; i++)
#define forv(i, a, b) for(int i=a;i>=b;i--)
#define ls(x) (x<<1)
#define rs(x) (x<<1|1)
#define endl '\n'
//用於Miller-Rabin
[[maybe_unused]] static int Prime_Number[13] = {0, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37};

template <typename T>
int disc(T* a, int n)
{
    return unique(a + 1, a + n + 1) - (a + 1);
}

template <typename T>
T lowBit(T x)
{
    return x & -x;
}

template <typename T>
T Rand(T l, T r)
{
    static mt19937 Rand(time(nullptr));
    uniform_int_distribution<T> dis(l, r);
    return dis(Rand);
}

template <typename T1, typename T2>
T1 modt(T1 a, T2 b)
{
    return (a % b + b) % b;
}

template <typename T1, typename T2, typename T3>
T1 qPow(T1 a, T2 b, T3 c)
{
    a %= c;
    T1 ans = 1;
    for (; b; b >>= 1, (a *= a) %= c)if (b & 1)(ans *= a) %= c;
    return modt(ans, c);
}

template <typename T>
void read(T& x)
{
    x = 0;
    T sign = 1;
    char ch = getchar();
    while (!isdigit(ch))
    {
        if (ch == '-')sign = -1;
        ch = getchar();
    }
    while (isdigit(ch))
    {
        x = (x << 3) + (x << 1) + (ch ^ 48);
        ch = getchar();
    }
    x *= sign;
}

template <typename T, typename... U>
void read(T& x, U&... y)
{
    read(x);
    read(y...);
}

template <typename T>
void write(T x)
{
    if (typeid(x) == typeid(char))return;
    if (x < 0)x = -x, putchar('-');
    if (x > 9)write(x / 10);
    putchar(x % 10 ^ 48);
}

template <typename C, typename T, typename... U>
void write(C c, T x, U... y)
{
    write(x), putchar(c);
    write(c, y...);
}


template <typename T11, typename T22, typename T33>
struct T3
{
    T11 one;
    T22 tow;
    T33 three;

    bool operator<(const T3 other) const
    {
        if (one == other.one)
        {
            if (tow == other.tow)return three < other.three;
            return tow < other.tow;
        }
        return one < other.one;
    }

    T3() { one = tow = three = 0; }

    T3(T11 one, T22 tow, T33 three) : one(one), tow(tow), three(three)
    {
    }
};

template <typename T1, typename T2>
void uMax(T1& x, T2 y)
{
    if (x < y)x = y;
}

template <typename T1, typename T2>
void uMin(T1& x, T2 y)
{
    if (x > y)x = y;
}

struct Hash
{
    static uint64_t splitmix64(uint64_t x)
    {
        x += 0x9e3779b97f4a7c15;
        x = (x ^ x >> 30) * 0xbf58476d1ce4e5b9;
        x = (x ^ x >> 27) * 0x94d049bb133111eb;
        return x ^ x >> 31;
    }

    static size_t get(const uint64_t x)
    {
        static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();
        return splitmix64(x + FIXED_RANDOM);
    }

    template <typename T>
    size_t operator()(T x) const
    {
        return get(std::hash<T>()(x));
    }

    template <typename F, typename S>
    size_t operator()(pair<F, S> p) const
    {
        return get(std::hash<F>()(p.first)) ^ std::hash<S>()(p.second);
    }
};

constexpr int N = 1e5 + 10;

struct Node
{
    int left, right, cnt;
} node[N << 8];

#define left(x) node[x].left
#define right(x) node[x].right
#define cnt(x) node[x].cnt
int n, m, q, mx, cnt;
int a[N], ord[N], siz[N];
hash2<int, int, Hash> mp;

inline void add(const int pre, int& curr, const int val, const int l = 1, const int r = mx)
{
    node[curr = ++cnt] = node[pre];
    cnt(curr)++;
    const int mid = l + r >> 1;
    if (l == r)return;
    if (val <= mid)add(left(pre),left(curr), val, l, mid);
    else add(right(pre),right(curr), val, mid + 1, r);
}

inline int query(const int u, const int v, const int lca, const int lca_fa, const int k, const int l = 1,
                 const int r = mx)
{
    if (l == r)return ord[l];
    const int mid = l + r >> 1;
    const int leftSize = cnt(left(u)) + cnt(left(v)) - cnt(left(lca)) - cnt(left(lca_fa));
    if (leftSize >= k)return query(left(u),left(v),left(lca),left(lca_fa), k, l, mid);
    return query(right(u),right(v),right(lca),right(lca_fa), k - leftSize, mid + 1, r);
}

constexpr int T = 20;
int root[N], fa[N][T + 1], deep[N];
vector<int> child[N];

inline void dfs(const int curr, const int pa)
{
    deep[curr] = deep[fa[curr][0] = pa] + 1;
    add(root[pa], root[curr], a[curr]);
    forn(i, 1, T)fa[curr][i] = fa[fa[curr][i - 1]][i - 1];
    siz[curr] = 1;
    for (const auto nxt : child[curr])
    {
        if (nxt == pa)continue;
        dfs(nxt, curr);
        siz[curr] += siz[nxt];
    }
}

inline int LCA(int x, int y)
{
    if (deep[x] < deep[y])swap(x, y);
    forv(i, T, 0)if (deep[fa[x][i]] >= deep[y])x = fa[x][i];
    if (x == y)return x;
    forv(i, T, 0)if (fa[x][i] != fa[y][i])x = fa[x][i], y = fa[y][i];
    return fa[x][0];
}

int last;

inline void solve()
{
    cin >> n >> m >> q;
    forn(i, 1, n)cin >> a[i], ord[i] = a[i];
    sortArr(ord, n), mx = disc(ord, n);
    forn(i, 1, mx)mp[ord[i]] = i;
    forn(i, 1, n)a[i] = mp[a[i]];
    forn(i, 1, m)
    {
        int u, v;
        cin >> u >> v;
        child[u].push_back(v);
        child[v].push_back(u);
    }
    forn(i, 1, n)if (!deep[i])dfs(i, 0);
    while (q--)
    {
        char op;
        cin >> op;
        if (op == 'L')
        {
            int u, v;
            cin >> u >> v, u ^= last, v ^= last;
            child[u].push_back(v), child[v].push_back(u);
            if (siz[u] < siz[v])swap(u, v);
            dfs(v, u);
        }
        else
        {
            int u, v, k;
            cin >> u >> v >> k, u ^= last, v ^= last, k ^= last;
            const int lca = LCA(u, v);
            const int lca_fa = fa[lca][0];
            cout << (last = query(root[u], root[v], root[lca], root[lca_fa], k)) << endl;
        }
    }
}

signed int main()
{
    // MyFile
    Spider
    //------------------------------------------------------
    // clock_t start = clock();
    int test = 1;
    //    read(test);
    cin >> test;
    test = 1;
    forn(i, 1, test)solve();
    //    while (cin >> n, n)solve();
    //    while (cin >> test)solve();
    // clock_t end = clock();
    // cerr << "time = " << double(end - start) / CLOCKS_PER_SEC << "s" << endl;
}

上述一個是按照 \(u\) 和 \(v\) 處對應的可持久化線段樹大小比較，其實也就是 \(u\) 和 \(v\) 圖上紫色父親部分的大小啟發式合併，這並不是很正確的：

因為我們關注到 \(u\) 變為 \(v\) 的父親以後，那麼相當於以 \(u\) 為根重建了原來的樹，包括了父親和兒子都要發生遍歷改變。

第二種則是按照 \(u\) 和 \(v\) 的子樹大小啟發式合併，這個很顯然只考慮到了兒子貢獻，第一個則是隻考慮到了父親貢獻。

其實這玩意就是連通塊合併，直接帶權並查集維護連通快大小就能確定合併一方是誰了。

參照寫法3 
#include <bits/stdc++.h>

// #pragma GCC optimize(2)
// #pragma GCC optimize("Ofast,no-stack-protector,unroll-loops,fast-math")
// #pragma GCC target("sse,sse2,sse3,ssse3,sse4.1,sse4.2,avx,avx2,popcnt,tune=native")

#define isPbdsFile

#ifdef isPbdsFile

#include <bits/extc++.h>

#else

#include <ext/pb_ds/priority_queue.hpp>
#include <ext/pb_ds/hash_policy.hpp>
#include <ext/pb_ds/tree_policy.hpp>
#include <ext/pb_ds/trie_policy.hpp>
#include <ext/pb_ds/tag_and_trait.hpp>
#include <ext/pb_ds/hash_policy.hpp>
#include <ext/pb_ds/list_update_policy.hpp>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/exception.hpp>
#include <ext/rope>

#endif

using namespace std;
using namespace __gnu_cxx;
using namespace __gnu_pbds;
typedef long long ll;
typedef long double ld;
typedef pair<int, int> pii;
typedef pair<ll, ll> pll;
typedef tuple<int, int, int> tii;
typedef tuple<ll, ll, ll> tll;
typedef unsigned int ui;
typedef unsigned long long ull;
typedef __int128 i128;
#define hash1 unordered_map
#define hash2 gp_hash_table
#define hash3 cc_hash_table
#define stdHeap std::priority_queue
#define pbdsHeap __gnu_pbds::priority_queue
#define sortArr(a, n) sort(a+1,a+n+1)
#define all(v) v.begin(),v.end()
#define yes cout<<"YES"
#define no cout<<"NO"
#define Spider ios_base::sync_with_stdio(false);cin.tie(nullptr);cout.tie(nullptr);
#define MyFile freopen("..\\input.txt", "r", stdin),freopen("..\\output.txt", "w", stdout);
#define forn(i, a, b) for(int i = a; i <= b; i++)
#define forv(i, a, b) for(int i=a;i>=b;i--)
#define ls(x) (x<<1)
#define rs(x) (x<<1|1)
#define endl '\n'
//用於Miller-Rabin
[[maybe_unused]] static int Prime_Number[13] = {0, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37};

template <typename T>
int disc(T* a, int n)
{
    return unique(a + 1, a + n + 1) - (a + 1);
}

template <typename T>
T lowBit(T x)
{
    return x & -x;
}

template <typename T>
T Rand(T l, T r)
{
    static mt19937 Rand(time(nullptr));
    uniform_int_distribution<T> dis(l, r);
    return dis(Rand);
}

template <typename T1, typename T2>
T1 modt(T1 a, T2 b)
{
    return (a % b + b) % b;
}

template <typename T1, typename T2, typename T3>
T1 qPow(T1 a, T2 b, T3 c)
{
    a %= c;
    T1 ans = 1;
    for (; b; b >>= 1, (a *= a) %= c)if (b & 1)(ans *= a) %= c;
    return modt(ans, c);
}

template <typename T>
void read(T& x)
{
    x = 0;
    T sign = 1;
    char ch = getchar();
    while (!isdigit(ch))
    {
        if (ch == '-')sign = -1;
        ch = getchar();
    }
    while (isdigit(ch))
    {
        x = (x << 3) + (x << 1) + (ch ^ 48);
        ch = getchar();
    }
    x *= sign;
}

template <typename T, typename... U>
void read(T& x, U&... y)
{
    read(x);
    read(y...);
}

template <typename T>
void write(T x)
{
    if (typeid(x) == typeid(char))return;
    if (x < 0)x = -x, putchar('-');
    if (x > 9)write(x / 10);
    putchar(x % 10 ^ 48);
}

template <typename C, typename T, typename... U>
void write(C c, T x, U... y)
{
    write(x), putchar(c);
    write(c, y...);
}


template <typename T11, typename T22, typename T33>
struct T3
{
    T11 one;
    T22 tow;
    T33 three;

    bool operator<(const T3 other) const
    {
        if (one == other.one)
        {
            if (tow == other.tow)return three < other.three;
            return tow < other.tow;
        }
        return one < other.one;
    }

    T3() { one = tow = three = 0; }

    T3(T11 one, T22 tow, T33 three) : one(one), tow(tow), three(three)
    {
    }
};

template <typename T1, typename T2>
void uMax(T1& x, T2 y)
{
    if (x < y)x = y;
}

template <typename T1, typename T2>
void uMin(T1& x, T2 y)
{
    if (x > y)x = y;
}

struct Hash
{
    static uint64_t splitmix64(uint64_t x)
    {
        x += 0x9e3779b97f4a7c15;
        x = (x ^ x >> 30) * 0xbf58476d1ce4e5b9;
        x = (x ^ x >> 27) * 0x94d049bb133111eb;
        return x ^ x >> 31;
    }

    static size_t get(const uint64_t x)
    {
        static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();
        return splitmix64(x + FIXED_RANDOM);
    }

    template <typename T>
    size_t operator()(T x) const
    {
        return get(std::hash<T>()(x));
    }

    template <typename F, typename S>
    size_t operator()(pair<F, S> p) const
    {
        return get(std::hash<F>()(p.first)) ^ std::hash<S>()(p.second);
    }
};

constexpr int N = 1e5 + 10;

struct Node
{
    int left, right, cnt;
} node[N << 7];

#define left(x) node[x].left
#define right(x) node[x].right
#define cnt(x) node[x].cnt
int n, m, q, mx, cnt;
int a[N], ord[N];
hash2<int, int, Hash> mp;

struct DSU
{
    int fa[N], siz[N];

    void init(const int n)
    {
        forn(i, 1, n)fa[i] = i, siz[i] = 1;
    }

    int find(const int x)
    {
        return x == fa[x] ? x : fa[x] = find(fa[x]);
    }

    void merge(int x, int y)
    {
        x = find(x), y = find(y);
        siz[y] += siz[x], fa[x] = y;
    }
} dsu;

inline void add(const int pre, int& curr, const int val, const int l = 1, const int r = mx)
{
    node[curr = ++cnt] = node[pre];
    cnt(curr)++;
    const int mid = l + r >> 1;
    if (l == r)return;
    if (val <= mid)add(left(pre),left(curr), val, l, mid);
    else add(right(pre),right(curr), val, mid + 1, r);
}

inline int query(const int u, const int v, const int lca, const int lca_fa, const int k, const int l = 1,
                 const int r = mx)
{
    if (l == r)return ord[l];
    const int mid = l + r >> 1;
    const int leftSize = cnt(left(u)) + cnt(left(v)) - cnt(left(lca)) - cnt(left(lca_fa));
    if (leftSize >= k)return query(left(u),left(v),left(lca),left(lca_fa), k, l, mid);
    return query(right(u),right(v),right(lca),right(lca_fa), k - leftSize, mid + 1, r);
}

constexpr int T = 20;
int root[N], fa[N][T + 1], deep[N];
vector<int> child[N];

inline void dfs(const int curr, const int pa)
{
    deep[curr] = deep[fa[curr][0] = pa] + 1;
    add(root[pa], root[curr], a[curr]);
    forn(i, 1, T)fa[curr][i] = fa[fa[curr][i - 1]][i - 1];
    for (const auto nxt : child[curr])if (nxt != pa)dfs(nxt, curr);
}

inline int LCA(int x, int y)
{
    if (deep[x] < deep[y])swap(x, y);
    forv(i, T, 0)if (deep[fa[x][i]] >= deep[y])x = fa[x][i];
    if (x == y)return x;
    forv(i, T, 0)if (fa[x][i] != fa[y][i])x = fa[x][i], y = fa[y][i];
    return fa[x][0];
}

int last;

inline void solve()
{
    cin >> n >> m >> q;
    dsu.init(n);
    forn(i, 1, n)cin >> a[i], ord[i] = a[i];
    sortArr(ord, n), mx = disc(ord, n);
    forn(i, 1, mx)mp[ord[i]] = i;
    forn(i, 1, n)a[i] = mp[a[i]];
    forn(i, 1, m)
    {
        int u, v;
        cin >> u >> v;
        child[u].push_back(v);
        child[v].push_back(u);
        dsu.merge(u, v);
    }
    forn(i, 1, n)if (!deep[i])dfs(i, 0);
    while (q--)
    {
        char op;
        cin >> op;
        if (op == 'L')
        {
            int u, v;
            cin >> u >> v, u ^= last, v ^= last;
            child[u].push_back(v), child[v].push_back(u);
            if (dsu.siz[dsu.find(u)] < dsu.siz[dsu.find(v)])swap(u, v);
            dfs(v, u), dsu.merge(u, v);
        }
        else
        {
            int u, v, k;
            cin >> u >> v >> k, u ^= last, v ^= last, k ^= last;
            const int lca = LCA(u, v);
            const int lca_fa = fa[lca][0];
            cout << (last = query(root[u], root[v], root[lca], root[lca_fa], k)) << endl;
        }
    }
}

signed int main()
{
    // MyFile
    Spider
    //------------------------------------------------------
    // clock_t start = clock();
    int test = 1;
    //    read(test);
    cin >> test;
    test = 1;
    forn(i, 1, test)solve();
    //    while (cin >> n, n)solve();
    //    while (cin >> test)solve();
    // clock_t end = clock();
    // cerr << "time = " << double(end - start) / CLOCKS_PER_SEC << "s" << endl;
}

初始化就正常初始化各個樹的區域性樹上可持久化線段樹，查詢也按照樹上字首和的方式計算貢獻從而二分第 \(k\) 大，最後注意下離散化就行。

\[最壞時間複雜度為:\ O(n\log^2{n}) \]

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