P3302 [SDOI2013] 森林 题解

题目链接：森林

有意思的树上可持久化线段树变形题，建议先看这个：P2633 Count on a tree 题解

对于本题而言，我们重新阐述树上可持久化线段树的核心思想，对于点路径/边路径上的第 \(k\) 大问题，我们使用树上前缀和问题的思想，将其转化为可差性问题：一条路径上的权值线段树可以拆分为几棵权值线段树进行贡献拿到，具体可以参照链接文章提到的树上前缀和的基本思想。

本题观察到既有连接新边操作，又有查找操作，相对来说较为复杂，我们重新关注树上可持久化线段树树需要关注什么：

1. \(u\) 与 \(v\) 处的前缀点对应的可持久化线段树。

2. \(lca\) 与 \(fa[lca]\) 处的前缀点对应的可持久化线段树。

注意到本题为森林，不一定是一棵完整的树了，我们注意如果 \(u\) 与 \(v\) 建边：

那么很显然，\(u\) 可以作为 \(v\) 的父亲，而 \(v\) 也可以作为 \(u\) 的父亲，并且对于它们各自对应的 \(root\) 上的可持久化线段树查找是正确的，可以看做局部每棵树的树上前缀和建出来的可持久化线段树在局部上的查找显然正确。其实抽象出来局部的一棵树不也是一棵完整的树吗？我们发现合并的影响仅仅体现在某一个点成为另一个点的父亲：

如图所示，如果 \(u \rightarrow v\)，即 \(v\) 成为 \(u\) 的父亲，那么影响的仅仅是 \(u\) 为根对应的整棵树的所有点，它们的倍增数组和父亲还有深度之类的包括可持久化线段树都得重建。我们考虑暴力一次更新是 \(O(n\log{n})\)。那显然如果每次都随便选一个点暴力更新，复杂度则为 \(O(n^2\log{n})\)，你不觉得这玩意不就是树合并吗？树合并优化复杂度的办法多了去了，我们考虑使用启发式合并，小树的根作为合并对象，大树的根则为父亲。

当然有人疑惑它们一开始就有父亲了，怎么写？给两个一开始我过了但并不是很符合正确的启发式合并的写法：

参照写法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'
//用于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;
}

参照写法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'
//用于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;
}

上述一个是按照 \(u\) 和 \(v\) 处对应的可持久化线段树大小比较，其实也就是 \(u\) 和 \(v\) 图上紫色父亲部分的大小启发式合并，这并不是很正确的：

因为我们关注到 \(u\) 变为 \(v\) 的父亲以后，那么相当于以 \(u\) 为根重建了原来的树，包括了父亲和儿子都要发生遍历改变。

第二种则是按照 \(u\) 和 \(v\) 的子树大小启发式合并，这个很显然只考虑到了儿子贡献，第一个则是只考虑到了父亲贡献。

其实这玩意就是连通块合并，直接带权并查集维护连通快大小就能确定合并一方是谁了。

参照写法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'
//用于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个树的局部树上可持久化线段树，查找也按照树上前缀和的方式计算贡献从而二分第 \(k\) 大，最后注意下离散化就行。

\[最坏时间复杂度为:\ O(n\log^2{n}) \]