#include <bits/stdc++.h>
#define LL long long
#define INF_LL 0x3f3f3f3f3f3f3f3f
int N,hd[100009],to[200009],nxt[200009],val[200009],k,siz[100009],son[100009],
dfn[100009],top[100009],pa[100009],pa2[100009],kk,v[100009],dep2[100009];
LL dep[100009],xx,yy;
void dfs(int n,int f) {
    siz[n]=1;
    pa2[n]=f;
    pa[n]=f;
    dep2[n]=dep2[f]+1;
    for(int i=hd[n];i;i=nxt[i]) {
        if(to[i]==f) continue;
        dep[to[i]]=dep[n]+val[i];
        dfs(to[i],n);
        siz[n]+=siz[to[i]];
        if(siz[son[n]]<siz[to[i]]) son[n]=to[i];
    }
}
void dfs2(int n,int tp) {
    top[n]=tp;
    if(!son[n]) return;
    dfs2(son[n],tp);
    for(int i=hd[n];i;i=nxt[i]) {
        if(to[i]==pa2[n]||to[i]==son[n]) continue;
        dfs2(to[i],to[i]);
    }
}
void l(int u,int v,int w) {
    to[++k]=v;nxt[k]=hd[u];val[k]=w;hd[u]=k;
}
inline LL dis(int x,int y) {
    LL as=dep[x]+dep[y];
    while(top[x]!=top[y]) {
        if(dep2[top[x]]<dep2[top[y]]) std::swap(x,y);
        x=pa2[top[x]];
    }
    if(dep2[x]<dep2[y]) std::swap(x,y);
    return as-2*dep[y];
}
struct n_t{
    LL a;int b,i;
} f[100009];
std::vector<n_t> tp[100009];
bool cmp(n_t x,n_t y) {
    return x.b>y.b;
}
void up(int n,LL v,int sl) {
    int x=n;
    while(n) {
        LL vv=v+1ll*sl*dis(x,n);
        while(tp[n].size()>=1) {
            int sz=tp[n].size();
            if(tp[n][sz-1].b==sl&&tp[n][sz-1].a<vv) {
                return;
            }
            if(tp[n][sz-1].b==sl&&tp[n][sz-1].a>=vv) {
                tp[n].pop_back();
                continue;
            }
            if(sz<2) break;
            if((tp[n][sz-2].a-tp[n][sz-1].a)*(tp[n][sz-1].b-sl)>
                (tp[n][sz-1].a-vv)*(tp[n][sz-2].b-tp[n][sz-1].b)) {
                break;
            }
            tp[n].pop_back();
        }
        tp[n].push_back((n_t){vv,sl,0});
        n=pa[n];
    }
}
LL qr(int n) {
    int x=n;
    LL ans=xx+dep[n]*yy;
    while(n) {
        if(tp[n].size()) {
            LL va=dis(x,n);
            int l=0,r=tp[n].size()-1,md;
            while(l<r) {
                md=(l+r)/2;
                if((tp[n][md].a-tp[n][md+1].a)>-va*(tp[n][md].b-tp[n][md+1].b)) {
                    l=md+1;
                } else {
                    r=md;
                }
            }
            ans=std::min(ans,tp[n][l].a+tp[n][l].b*va);
        }
        n=pa[n];
    }
    return ans;
}
void cl(int n,int f) {
    siz[n]=1;son[n]=0;
    for(int i=hd[n];i;i=nxt[i]) {
        if(to[i]==f||v[to[i]]) continue;
        cl(to[i],n);
        siz[n]+=siz[to[i]];
        if(siz[son[n]]<siz[to[i]]) son[n]=to[i];
    }
}
void sv(int n,int ff) {
    cl(n,0);
    int x=n;
    while(son[x]&&siz[son[x]]>=(siz[n]+1)/2) {
        x=son[x];
    }
    n=x;
    pa[n]=ff;
    v[n]=1;
    for(int i=hd[n];i;i=nxt[i]) {
        if(!v[to[i]]) sv(to[i],n);
    }
}
signed main(void) {
    scanf("%d",&N);
    for(int i=1;i<=N;i++) {
        scanf("%lld",&f[i].a);
    }
    for(int i=1;i<=N;i++) {
        scanf("%d",&f[i].b);
        f[i].i=i;
    }
    xx=f[1].a;yy=f[1].b;
    for(int i=1;i<N;i++) {
        int u,v,w;
        scanf("%d %d %d",&u,&v,&w);
        l(u,v,w),l(v,u,w);
    }
    dfs(1,0);
    dfs2(1,1);
    sv(1,0);
    std::sort(f+1,f+N+1,cmp);
    for(int i=1;i<=N;i++) {
        up(f[i].i,qr(f[i].i)+f[i].a,f[i].b);
    } 
    for(int i=2;i<=N;i++) printf("%lld ",qr(i));
}