Link to the original text

The question

Given a tree with n Tree of nodes .

Every point has a weight .

For each of these $i$ Given three parameters $a_i,b_i,c_i$ , From $i$ Starting from one point, the next point you can reach x You need to meet one of three requirements :

1. x stay $a_i$ To $b_i$ On the simple path of .

2. x stay $a_i$ To $c_i$ On the simple path of .

3. x stay $c_i$ To $b_i$ On the simple path of .

Starting from any point , The weight of the passing node and the front k What's the length of the small route . Note that you can go through the nodes repeatedly .

Space restriction 100MB, The time limit 3s

$n,k\leq 5\times 10^5$

Make sure the answer is less than $10^8$.

Answer key

because $k,n$ Isomorphism , So the later complexity analysis doesn't distinguish $n$ and $k$ .

First of all, the nodes that can be reached by each point are divided into no more than two chains .

Partition with tree chain + Line segment tree + Preprocessing heavy chain prefixes min , To achieve $O(\log n)$ Find the minimum weight point on a chain .

If we come violent , So obviously :

Initially, all the points are thrown directly into a pile , The key words are the sum of node weights , Take out the smallest elements in turn , Every time I take out one , Just add a solution to the heap that you can get from this point , So just take it out k The next time .

Obviously, the complexity of time and space has withered .

So we consider splitting the tree chain 、 Mark these lines on the tree , The keywords for these tags are “ From the current state , Then walk to the point with the smallest weight in the interval to get the path length ”, When it's taken out of the heap, it splits the markers , You get a time complexity $O(n\log^2 n)$ Spatial complexity $O(n\log n)$ How to do it , It's still not enough to pass the question .

Consider changing the form of the tag to $(a,b)$ , Two endpoints in a chain . So when you take out this chain , Just update the heap state in two parts :

1. Split the chain from the minimum weight point of the chain , Into two shorter chains , Add to the heap .

2. Starting from the minimum weight point of this chain , Up to two chains , Add to the heap .

So we get a spatial complexity $O(n)$ , Time complexity $O(n\log n)$ How to do it .

And then because the blogger is so big , Still stuck in space .

So I forced vector The array is written as an array simulation list , Write the system heap as a handwriting heap ……

Finally, I suddenly found out that he promised that the answer was less than 1e8 , It's not that the edge weight is less than 1e8 I was stunned when I went to school .

So many constants stuck before , In fact, you only need to change the data type of the storage path length from longlong Change to int Just fine ……


#include <bits/stdc++.h>
using namespace std;
typedef long long LL;
LL read(){
LL x=0,f=0;
char ch=getchar();
while (!isdigit(ch))
while (isdigit(ch))
return f?-x:x;
const int N=500005;
int n,k;
struct Info{
int a,b,c,d;
struct Gragh{
static const int M=N*2;
int cnt,y[M],nxt[M],fst[N];
void clear(){
memset(fst,0,sizeof fst);
void add(int a,int b){
int w[N];
int fa[N],depth[N],size[N],son[N],top[N],I[N],O[N],aI[N],Min[N],Time=0;
void dfs1(int x,int pre,int d){
for (int i=g.fst[x];i;i=g.nxt[i]){
int y=g.y[i];
if (y!=pre){
if (!son[x]||size[y]>size[son[x]])
void ckwMin(int &x,int y){
if (w[x]>w[y])
void dfs2(int x,int Top){
if (son[x])
for (int i=g.fst[x];i;i=g.nxt[i]){
int y=g.y[i];
if (y!=fa[x]&&y!=son[x])
int LCA(int x,int y){
int fx=top[x],fy=top[y];
while (fx!=fy){
if (depth[fx]<depth[fy])
return depth[x]<depth[y]?x:y;
int isanc(int x,int y){
return I[x]<=I[y]&&I[y]<=O[x];
int go_up(int x,int y){
if (isanc(aI[I[y]+1],x))
return aI[I[y]+1];
int fx=top[x];
while (depth[fx]>depth[y]){
if (depth[fa[x]]>depth[y])
return x;
namespace Seg{
int v[N<<2];
void build(int rt,int L,int R){
if (L==R)
return (void)(v[rt]=aI[L]);
int mid=(L+R)>>1,ls=rt<<1,rs=ls|1;
int query(int rt,int L,int R,int xL,int xR){
if (xL<=L&&R<=xR)
return v[rt];
int mid=(L+R)>>1,ls=rt<<1,rs=ls|1;
if (xR<=mid)
return query(ls,L,mid,xL,xR);
if (xL>mid)
return query(rs,mid+1,R,xL,xR);
int res=query(ls,L,mid,xL,xR);
return res;
int query(int x,int y){
int ans=x,fx=top[x],fy=top[y];
while (fx!=fy){
if (depth[fx]<depth[fy])
if (depth[x]>depth[y])
return ans;
const int Size=N*5;
struct Node{
int a,b,len;
Node(int _a,int _b,int pre){
len=pre+w[Seg :: query(a,b)];
int stt=0;
bool cmp(int a,int b){
return s[a].len>s[b].len;
struct priority_Queue{
int v[Size],s;
bool empty(){
return s==0;
void up(int x){
int y=x>>1;
while (y){
if (cmp(v[y],v[x]))
void down(int x){
int y=x<<1;
while (y<=s){
if (y<s&&cmp(v[y],v[y+1]))
if (cmp(v[x],v[y]))
void pop(){
int top(){
return v[1];
void push(int x){
void push(Node x){
void Split(Node x,int Mi){
int a=x.a,b=x.b,m=Mi;
if (!isanc(m,a))
if (m!=a){
int aa=go_up(a,m);
if (m!=b){
int bb;
if (isanc(m,b))
void solve(){
while (!Q.empty())
for (int i=1;i<=n;i++)
while (k--){
Node now=s[];
int x=Seg :: query(now.a,now.b);
if (~v[x].a)
if (~v[x].c)
int main(){
for (int i=1;i<=n;i++)
for (int i=2;i<=n;i++){
int a=read();
for (int i=1;i<=n;i++)
for (int i=1;i<=n;i++){
int a=v[i].a,b=v[i].b,c=v[i].c;
if (depth[c]<depth[b])
if (depth[b]<depth[a])
if (depth[c]<depth[b])
if (a==b||b==c){
int ab=LCA(a,b),ac=LCA(a,c),bc=LCA(b,c);
if (ab==ac&&ab==bc){
int g=ab;
if (depth[a]>depth[g]){
if (ab==bc)
else if (ac==bc)
if (depth[c]<depth[b])
if (isanc(b,c)){
int d=go_up(c,bc);
Seg :: build(1,1,n);
return 0;

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