#include <bits/stdc++.h>

using namespace std;

typedef long long ll;

const int sz = 755;

const int N = 2e5 + 1;
const int LOG = 19;
vector <int> g[N];
int w[N];
int mx[N];
int tin[N];
int tout[N];
int rt[N];
int up[N][LOG];
int we[N][LOG];
int he[N];

int tt = 0;

void dfs(int v, int pr, int root, int cur, int sum)
{
    he[v] = sum;
    tin[v] = tt++;
    rt[v] = root;
    cur = max(cur, w[v]);
    mx[v] = cur;
    up[v][0] = pr;
    we[v][0] = w[v];
    for (int i = 1; i < LOG; i++)
    {
        up[v][i] = up[up[v][i - 1]][i - 1];
        we[v][i] = max(we[up[v][i - 1]][i - 1], we[v][i - 1]);
    }
    for (auto to : g[v])
    {
        if (to != pr)
        {
            dfs(to, v, root, cur, sum + 1);
        }
    }
    tout[v] = tt++;
}

int main()
{
#ifdef ONPC
    freopen("a.in", "r", stdin);
    //freopen("a.out", "w", stdout);
#else
    //freopen("a.in", "r", stdin);
    //freopen("a.out", "w", stdout);
#endif
    int n, m;
    scanf("%d%d", &n, &m);
    vector <int> pr(n);
    vector <bool> change(n);
    for (int i = 1; i < n; i++)
    {
        scanf("%d", &pr[i]);
    }
    for (int i = 1; i < n; i++)
    {
        scanf("%d", &w[i]);
    }
    vector <pair <int, int> > e;
    for (int i = 0; i < m; i++)
    {
        int u, v;
        scanf("%d%d", &u, &v);
        e.push_back({u, v});
    }
    for (int i = 0; i < m; i += sz)
    {
        vector <bool> glav(n);
        glav[0] = 1;
        int en = min(m, i + sz);
        for (int j = i; j < en; j++)
        {
            glav[e[j].first] = 1;
            glav[e[j].second] = 1;
        }
        for (int i = 0; i < n; i++)
        {
            g[i].clear();
        }
        for (int i = 0; i < n; i++)
        {
            if (!glav[i])
            {
                g[pr[i]].push_back(i);
            }
        }
        tt = 0;
        for (int i = 0; i < n; i++)
        {
            if (glav[i])
            {
                dfs(i, -1, i, 0, 0);
            }
        }
        for (int j = i; j < en; j++)
        {
            int u = e[j].first, v = e[j].second;
            if (u == v)
            {
                puts("-1");
                continue;
            }
            int cur = v;
            bool vis = 0;
            int mix = 0;
            if (rt[cur] == rt[u])
            {
                vis = true;
            }
            else
            {
                while (cur != 0)
                {
                    mix = max(mix, mx[cur]);
                    cur = pr[rt[cur]];
                    if (rt[cur] == rt[u])
                    {
                        vis = true;
                        break;
                    }
                }
            }
            if (!vis)
            {
                pr[u] = v;
            }
            else
            {
                v = cur;
                if (tin[u] <= tin[v] && tout[u] >= tout[v])
                {
                    int h = he[v] - he[u];
                    int cur = v;
                    for (int i = LOG - 1; i >= 0; i--)
                    {
                        if (h >= (1 << i))
                        {
                            h -= (1 << i);
                            mix = max(mix, we[cur][i]);
                            cur = up[cur][i];
                        }
                    }
                    printf("%d\n", mix);
                }
                else
                {
                    pr[u] = v;
                }
            }
        }
    }
}