# Heavy Light White Falcon

### Problem Statement :

```Our lazy white falcon finally decided to learn heavy-light decomposition. Her teacher gave an assignment for her to practice this new technique. Please help her by solving this problem.

You are given a tree with N nodes and each node's value is initially 0. The problem asks you to operate the following two types of queries:

"1 u x" assign x to the value of the node .
"2 u v" print the maximum value of the nodes on the unique path between u and v.

Input Format

First line consists of two integers seperated by a space: N and Q.
Following N - 1  lines consisting of two integers denotes the undirectional edges of the tree.
Following Q lines consist of the queries you are asked to operate.

Constraints

1  <=   N , Q , x  <=  50000

It is guaranteed that input denotes a connected tree with N nodes. Nodes are enumerated with 0-based indexing.

Output Format

For each second type of query print single integer in a single line, denoting the asked maximum value.```

### Solution :

```                            ```Solution in C :

In   C++   :

#include <cmath>
#include <cstdio>
#include <vector>
#include <iostream>
#include <algorithm>
using namespace std;

const int N = 50010;

class node {

public :
int l, r, mx;
node *left, *right;

void update(int idx, int val) {
if(l >= r) {
mx = val;
return;
}
int mid = (l + r) / 2;
(idx <= mid ? left : right)->update(idx, val);
mx = max(left->mx, right->mx);
}

int query(int a, int b) {
if(b < l or r < a) return 0;
if(a <= l and r <= b) return mx;
return max(left->query(a, b), right->query(a, b));
}

node(int _l, int _r) :
l(_l), r(_r), mx(0), left(NULL), right(NULL) {}
};

node* init(int l, int r) {
node *p = new node(l, r);
if(l < r) {
int mid = (l + r) / 2;
p->left = init(l, mid);
p->right = init(mid+1, r);
}
return p;
}

int n, q;

vector<int> Path[N];
int sz[N], H[N], P[N], G[N], pos[N];

void dfs_init(int u, int p, int h) {
P[u] = p;
H[u] = h;
sz[u] = 1;
if(v == p) {
continue;
}
dfs_init(v, u, h+1);
sz[u] += sz[v];
}
}
void dfs_HLD(int u) {
Path[u].push_back(u);
for(int i = 0;i < Path[u].size();i++) {
int v = Path[u][i];
G[v] = u;
pos[v] = i;
if(vv == P[v]) continue;
if(2*sz[vv] >= sz[v]) {
Path[u].push_back(vv);
}else {
dfs_HLD(vv);
}
}
}
head[u] = init(0, Path[u].size() - 1);
}
int query(int u, int v) {
int ans = 0;
while(G[u] != G[v]) {
if(H[G[u]] < H[G[v]]) {
swap(u, v);
}
u = P[G[u]];
}
if(pos[u] > pos[v]) {
swap(u, v);
}
return ans;
}
int main() {

ios::sync_with_stdio(false);
cin >> n >> q;
for(int i = 0;i < n-1;i++) {
int u, v;
cin >> u >> v;
}

dfs_init(0, 0, 0);
dfs_HLD(0);

for(int i = 0;i < q;i++) {
int type;
cin >> type;
if(type == 1) {
int u, x;
cin >> u >> x;
}else {
int u, v;
cin >> u >> v;
cout << query(u, v) << "\n";
}
}

return 0;
}

In   Java  :

import java.io.*;
import java.util.*;
import java.text.*;
import java.math.*;
import java.util.regex.*;

public class Solution {

// array to store values for each node(vertices)
static int[] v;
// array to store connections (edges)
static ArrayList<Integer>[] e;

// some node with single connection will be tree root
static int root;
// direction to reach root for each node
static int[] dirToRoot;
// distance to root
static int[] distToRoot;

// maximum node value
static int totalMax = 0;

static void markRoot(){
for(int i=0; i<v.length; i++){
if(e[i].size()==1){
root = i;
break;
}
}
}

static void markPathToRoot(int node, int dist, boolean[] visited){
distToRoot[node] = dist;
for(Integer item : e[node]){
int index = item.intValue();
if(visited[index]==false){
dirToRoot[index] = node;
visited[index] = true;
markPathToRoot(index, dist+1, visited);
}
}
}

static int findMax2(int start, int finish){
int sDistance = distToRoot[start];
int fDistance = distToRoot[finish];

int sIndex = start;
int fIndex = finish;
int max = Math.max(v[sIndex], v[fIndex]);

// decrease distance from the one that is more far from root
while(sDistance>fDistance && max<totalMax){
sIndex = dirToRoot[sIndex];
max = max >= v[sIndex] ? max : v[sIndex];
sDistance--;
}
while(fDistance>sDistance && max<totalMax){
fIndex = dirToRoot[fIndex];
max = max >= v[fIndex] ? max : v[fIndex];
fDistance--;
}

// run both of them
while(sIndex!=fIndex){
fIndex = dirToRoot[fIndex];
sIndex = dirToRoot[sIndex];
max = max >= v[fIndex] ? max : v[fIndex];
max = max >= v[sIndex] ? max : v[sIndex];
if(sIndex==root || max==totalMax)
break;
}
return Math.max(max, v[sIndex]);
}

// calculate distance to root node from each node
static void resetRoot(){
// direction to the root of tree for each node
dirToRoot = new int[v.length];
distToRoot= new int[v.length];

// mark node with only one edge as root
markRoot();
//        System.out.println("root="+root);

dirToRoot[root] = root;
boolean[] visited = new boolean[v.length];
visited[root] = true;
markPathToRoot(root, 0, visited);
}

public static void main(String[] args) {
sc.init(System.in);
int N = sc.nextInt();
int Q = sc.nextInt();

// array to store values for each node(vertices)
v = new int[N];
// array to store connections (edges)
e = new ArrayList[N];

for(int i=0; i<N; i++)
e[i] = new ArrayList<Integer>(2);

for(int i=0; i<N-1; i++){
int v1 = sc.nextInt();
int v2 = sc.nextInt();
// add to both because undirectional
}

resetRoot();

while(Q-->0){
int type = sc.nextInt();
if(type==1){
int node    = sc.nextInt();
int value   = sc.nextInt();
v[node] = value;
totalMax = value > totalMax ? value : totalMax;
}else{
int start   = sc.nextInt();
int finish  = sc.nextInt();
//search for solution
System.out.println( findMax2(start, finish) );
}
}
}

// code from internet
/** Class for buffered reading int and double values */
static StringTokenizer tokenizer;

/** call this method to initialize reader for InputStream */
static void init(InputStream input) {
tokenizer = new StringTokenizer("");
}

/** get next word */
static String next() {
while ( ! tokenizer.hasMoreTokens() ) {
//TODO add check for eof if necessary
try{
}catch(Exception e){}
}
}

static int nextInt() {
return Integer.parseInt( next() );
}

static double nextDouble() {
return Double.parseDouble( next() );
}
}
}

In   Python3  :

def segtree_init(ary):
ary = list(ary)
seg = [ary]
while len(ary) > 1:
if len(ary) & 1: ary.append(0)
ary = [max(ary[i],ary[i+1]) for i in range(0,len(ary),2)]
seg.append(ary)
return seg
def segtree_set(seg, i, x):
ary = seg
ary[i] = x
for j in range(1, len(seg)):
x = max(ary[i], ary[i^1])
ary = seg[j]
i >>= 1
ary[i] = x
def segtree_max(seg, lo, hi):
m = 0
j = 0
while lo < hi:
ary = seg[j]
if lo & 1:
x = ary[lo]
if x > m: m = x
lo += 1
if hi & 1:
hi -= 1
x = ary[hi]
if x > m: m = x
lo >>= 1
hi >>= 1
j += 1
return m
class heavy_light_node:
def __init__(self, segtree):
self.parent = None
self.pos = -1
self.segtree = segtree
def build_tree(i, edges, location):
children = []
members = [i]
ed = edges[i]
while ed:
for j in range(1,len(ed)):
child = build_tree(ed[j], edges, location)
child.pos = len(members) - 1
children.append(child)
i = ed
members.append(i)
ed = edges[i]
node = heavy_light_node(segtree_init(0 for j in members))
for child in children:
child.parent = node
for j in range(len(members)):
location[members[j]] = (node, j)
return node
edges = [[] for i in range(N)]
for i in range(N-1):
x, y = map(int, input().split())
edges[x].append(y)
edges[y].append(x)
size =  * N
active = 
while active:
i = active[-1]
if size[i] == 0:
size[i] = 1
for j in edges[i]:
edges[j].remove(i)
active.append(j)
else:
active.pop()
edges[i].sort(key=lambda j: -size[j])
size[i] = 1 + sum(size[j] for j in edges[i])
location = [None] * N
build_tree(0, edges, location)
return location
def root_path(i, location):
loc = location[i]
path = [ loc ]
loc = loc
while loc.parent != None:
path.append((loc.parent, loc.pos))
loc = loc.parent
path.reverse()
return path
def max_weight(x, y):
px = root_path(x, location)
py = root_path(y, location)
m = 1
stop = min(len(px), len(py))
while m < stop and px[m] == py[m]: m += 1
loc, a = px[m-1]
b = py[m-1]
if a > b: a, b = b, a
w = segtree_max(loc.segtree, a, b+1)
for j in range(m, len(px)):
loc, i = px[j]
x = segtree_max(loc.segtree, 0, i+1)
if x > w: w = x
for j in range(m, len(py)):
loc, i = py[j]
x = segtree_max(loc.segtree, 0, i+1)
if x > w: w = x
return w
N, Q = map(int, input().split())
for i in range(Q):
t, x, y = map(int, input().split())
if t == 1:
loc, i = location[x]
segtree_set(loc.segtree, i, y)
elif t == 2:
print(max_weight(x, y))```
```

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