最长路径异或和

题目大意

给定一棵边权树, 要求一条两个点之间的最短路径, 使得路径上所有边的权值的异或最大.

题解

考虑异或的性质:

• 交换律\(a \oplus b = b \oplus a\)
• 结合律\((a \oplus b) \oplus c = a \oplus (b \oplus c)\)
• 消去率\(a \oplus b = a \oplus c \oplus b \oplus c\)

所以我们发现, 假设\(anc\)是\(u\)和\(v\)的祖先, \(u\)和\(v\)分别是路径的两端, 则\(u \to anc \to v = (root \to anc \to u) \oplus (root \to anc \to v)\)

DFS一次, 到达每个点的时候都在trie树上找最大异或值即可.

#include 
#include 
#include 
#include 

const int N = (int)1e5;

namespace Zeonfai
{
    inline int getInt()
    {
        int a = 0, sgn = 1;
        char c;
        while(! isdigit(c = getchar()))
            if(c == '-')
                sgn *= -1;
        while(isdigit(c))
            a = a *10 + c - '0', c = getchar();
        return a * sgn;
    }
}

struct trieTree
{
    struct node
    {
        node *suc[2];

        inline node()
        {
            suc[0] = suc[1] = NULL;
        }
    };
    
    node *rt, *lst;

    inline void initialize()
    {
        rt = new node;
    }

    inline void _insert(int k)
    {
        node *u = lst->suc[k];
        if(lst->suc[k] == NULL)
            lst->suc[k] = new node;
        lst = lst->suc[k];
    }
    
    inline void insert(int w)
    {
        lst = rt;
        for(int i = 30; ~ i; -- i)
            _insert(w >> i & 1);
    }

    inline int query(int w)
    {
        int res = 0;
        node *u = rt;
        for(int i = 30; ~ i; -- i)
        {
            int k = w >> i & 1;
            if(u->suc[k ^ 1] != NULL)
                u = u->suc[k ^ 1], res += 1 << i;
            else
                u = u->suc[k];
        }
        return res;
    }
}trie;

int ans = 0;

struct tree
{   
    struct edge
    {
        int v, w, nxt;
    }edg[N << 1];
    
    int tp;
    int hd[N + 1];
    
    inline void initialize()
    {
        tp = 0;
        memset(hd, -1, sizeof(hd));
    }
    
    inline void addEdge(int u, int v, int w)
    {
        edg[tp].v = v, edg[tp].w = w, edg[tp].nxt = hd[u];
        hd[u] = tp ++;
    }
    
    inline void DFS(int u, int pre, int w)
    {
        trie.insert(w);
        ans = std::max(ans, trie.query(w));
        for(int p = hd[u]; ~ p; p = edg[p].nxt)
            if(edg[p].v ^ pre)
                DFS(edg[p].v, u, w ^ edg[p].w);
    }
}org;

int main()
{
    #ifndef ONLINE_JUDGE
    freopen("XSY2340.in", "r", stdin);
    #endif
    using namespace Zeonfai;
    int n = getInt();
    org.initialize();
    for(int i = 1; i < n; ++ i)
    {
        int u = getInt(), v = getInt(), w = getInt();
        org.addEdge(u, v, w), org.addEdge(v, u, w);
    }
    trie.initialize();
    org.DFS(1, -1, 0);
    printf("%d\n", ans);
}

转载于:https://www.cnblogs.com/ZeonfaiHo/p/7114834.html