# Lazy White Falcon

### Problem Statement :

```White Falcon just solved the data structure problem below using heavy-light decomposition. Can you help her find a new solution that doesn't require implementing any fancy techniques?

There are 2 types of query operations that can be performed on a tree:

1 u x: Assign x as the value of node u.
2 u v: Print the sum of the node values in the unique path from node u to node v.
Given a tree with N nodes where each node's value is initially 0, execute Q queries.

Input Format

The first line contains 2 space-separated integers, N and Q, respectively.
The N-1 subsequent lines each contain 2 space-separated integers describing an undirected edge in the tree.
Each of the Q subsequent lines contains a query you must execute.

Constraints

1  <=  N, Q  <=  10^5
1  <=   x  <=   1000

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

Output Format

For each type-2 query, print its integer result on a new line.```

### Solution :

```                            ```Solution in C :

In    C++  :

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

const int N = 100010;
const int LG_N = 20;

int n, q;

int tree[2*N];
vector<int> euler;
int first[N], last[N];

int H[N], P[N][LG_N];
int val[N];

void dfs(int u, int p, int h) {
H[u] = h;
P[u][0] = p;
for(int i = 1;i < LG_N;i++) {
P[u][i] = P[P[u][i-1]][i-1];
}
first[u] = euler.size();
euler.push_back(u);
if(v == p) {
continue;
}
dfs(v, u, h+1);
}
last[u] = euler.size();
euler.push_back(u);
}
int lca(int u, int v) {
if(H[u] < H[v]) swap(u, v);
for(int i = LG_N-1;i >= 0;i--) {
if(H[P[u][i]] >= H[v]) {
u = P[u][i];
}
}
if(u == v) {
return u;
}
for(int i = LG_N-1;i >= 0;i--) {
if(P[u][i] != P[v][i]) {
u = P[u][i];
v = P[v][i];
}
}
return P[u][0];
}
void update(int idx, int val) {
while(idx < euler.size()) {
tree[idx] += val;
idx += idx & (-idx);
}
}
int query(int idx) {
int ans = 0;
while(idx > 0) {
ans += tree[idx];
idx -= idx & (-idx);
}
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;
}

euler.resize(1, 0);
dfs(0, 0, 0);

for(int i = 0;i < q;i++) {
int type;
cin >> type;
if(type == 1) {
int u, x;
cin >> u >> x;
update(first[u], x - val[u]);
update(last[u],  val[u] - x);
val[u] = x;
}else {
int u, v;
cin >> u >> v;
int l = lca(u, v);
int ans = query(first[u]) + query(first[v]);
ans = ans - 2 * query(first[l]) + val[l];
cout << ans << "\n";
}
}
return 0;
}

In   Java :

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

class TreeNode implements Comparable<TreeNode> {
int index;
int value;
int level = -1;    //0 is root.
TreeNode parent;
BranchContainer branch;

TreeNode(int i) {
index = i;
children = new HashSet<TreeNode>();
branch = new BranchContainer();
}

void updateValue(int v) {
int diff = v - value;
value = v;
branch.sum += diff;
}

@Override
public String toString() {
return "i=" + index + " L=" + level;
}

@Override
public int hashCode() {
final int prime = 31;
int result = 1;
result = prime * result + index;
return result;
}

@Override
public boolean equals(Object obj) {
if (this == obj)
return true;
if (obj == null)
return false;
if (getClass() != obj.getClass())
return false;
TreeNode other = (TreeNode) obj;
return index == other.index;
}

@Override
public int compareTo(TreeNode o) {
return index - o.index;
}
}

class BranchContainer {
ArrayList<TreeNode> list = new ArrayList<>();
HashSet<TreeNode> set = new HashSet<>();
int sum = 0;
boolean isTrunk = false;
}

public class Solution {
TreeNode[] nodes;
int nNodes, nQueries, treeHeight;
TreeNode root;

int getSum(final int index1, final int index2) {
final List<List<TreeNode>> path =
findPath(nodes[index1], nodes[index2]);
int ret = 0;
for (List<TreeNode> list : path) {
if (list.isEmpty()) {
continue;
}

final int segSize = list.size();
if (branchSize>2*segSize) {
for (TreeNode node : list) {
ret += node.value;
}
}
else {
tail = list.get(segSize-1);
final List<TreeNode> list1 =
final List<TreeNode> list2 =
leaf.branch.list.subList(leaf.level-tail.level+1,
branchSize);
int sum = 0;
for (TreeNode node : list1) {
sum += node.value;
}
for (TreeNode node : list2) {
sum += node.value;
}

}

ret += leaf.branch.sum - sum;
}
}

return ret;
}

List<List<TreeNode>> findPath(final TreeNode node1,
final TreeNode node2) {
List<List<TreeNode>> ret =

if (node1.branch.isTrunk ||
node1.branch.list.get(0).level==0) {
if (!findPathFixOne(node1, node2, ret)) {
System.err.println("1 Cannot find path between "
+ node1.toString() + " and " + node2.toString());
}
return ret;
}
else if (node2.branch.isTrunk ||
node2.branch.list.get(0).level==0) {
if (!findPathFixOne(node2, node1, ret)) {
System.err.println("2 Cannot find path between "
+ node2.toString() + " and " + node1.toString());
}
return ret;
}

int branches = countBrancheDist(node1, node2);
TreeNode tmp = null;
if (branches<0) {
branches = countBrancheDist(node2, node1);
if (branches<0) {
TreeNode n1 = advanceBranch(node1, 1, ret);
TreeNode n2 = advanceBranch(node2, 1, ret);
List<List<TreeNode>> tmpPath = findPath(n1, n2);
}
else if (branches==0) {
}
else {
if (!findPathFixOne(node2, tmp, ret)) {
System.err.println("3 Cannot find path between "
+ node1.toString() + " and " + tmp.toString());
}
}
}
else if (branches==0) {
}
else {
if (!findPathFixOne(node1, tmp, ret)) {
System.err.println("4 Cannot find path between "
+ node2.toString() + " and " + tmp.toString());
}
}

return ret;
}

int countBrancheDist(final TreeNode fixed,
final TreeNode node) {
int ret = 0;
boolean found = fixed.branch.set.contains(node);
if (found) {
return ret;
}

TreeNode end =
node.branch.list.get(node.branch.list.size()-1);
while (end.level>0) {
++ret;
if (fixed.branch.set.contains(end)) {
return ret;
}
end = end.branch.list.get(end.branch.list.size()-1);
}

if (fixed.branch.set.contains(end)) {
return ++ret;
}
else {
return -1;
}
}

final int n, List<List<TreeNode>> path) {
TreeNode ret = node;
for (int i = 0; i < n; ++i) {
int size = ret.branch.list.size()-1;
level-ret.level, size));
ret = ret.branch.list.get(size);
}

return ret;
}

boolean findPathFixOne(final TreeNode fixed,
TreeNode node,
List<List<TreeNode>> path) {
while (node.level>0 &&
!fixed.branch.set.contains(node)) {
final int end = node.branch.list.size() - 1;
list.get(0).level - node.level, end));
node = node.branch.list.get(end);
}

if(!fixed.branch.set.contains(node)) {
return false;
}

return true;
}

final TreeNode node2, List<List<TreeNode>> path) {
int leafLevel = node1.branch.list.get(0).level;
int level1 = node1.level,
level2 = node2.level;
if (level1<level2) {
int tmpI = level1;
level1 = level2;
level2 = tmpI;
}

leafLevel-level2+1));
}

void organizeTree() {
root = null;
for (int i = 0; i < nNodes; ++i) {
final TreeNode node = nodes[i];
root = node;
}
}

setChildren();
enumerateBranches();

return;
}

void setChildren() {
int level = 0;
root.level = level;
Map<TreeNode, Set<TreeNode>> pcMap =
new HashMap<>();
while (!pcMap.isEmpty()) {
Map<TreeNode, Set<TreeNode>> newMap =
new HashMap<>();
for (Map.Entry<TreeNode,
Set<TreeNode>> entry : pcMap.entrySet()) {
final TreeNode parent = entry.getKey();
final Set<TreeNode> list = entry.getValue();
parent.level = level;
if (parent.parent!=null) {
parent.children.remove(parent.parent);
}

for (TreeNode node : parent.children) {
node.parent = parent;
}
}

++level;
pcMap = newMap;
}

treeHeight = level;
}

void enumerateBranches() {
boolean foundTrunk = false;
for (int i = 0; i < nNodes; ++i) {
final TreeNode node = nodes[i];
if (!node.children.isEmpty()) {
continue;
}

TreeNode tmpNode = node.parent;
while (tmpNode!=null) {
if (tmpNode.branch.list.isEmpty()) {
tmpNode.branch = node.branch;
tmpNode = tmpNode.parent;
}
else {
break;
}
}

if (!foundTrunk && tmpNode==null) {
foundTrunk = true;
node.branch.isTrunk = true;
}
}

return;
}

public static void main(String[] args) {
try {
long t1 = System.currentTimeMillis();

Solution falcon = new Solution();

int index1 = 0,
index2 = line.indexOf(' ', index1);
falcon.nNodes =
Integer.parseInt(line.substring(index1, index2));
index1 = index2+1;
index2 = line.length();
falcon.nQueries =
Integer.parseInt(line.substring(index1, index2));

falcon.nodes = new TreeNode[falcon.nNodes];
PrintWriter out =
new PrintWriter(new BufferedWriter(
new OutputStreamWriter(new FileOutputStream(
java.io.FileDescriptor.out), "UTF-8"), 512));

for (int i = 0; i < falcon.nNodes-1; ++i) {
index1 = 0;
index2 = line.indexOf(' ', index1);
final int n1 = Integer.parseInt(
line.substring(index1, index2));
index1 = index2+1;
index2 = line.length();
final int n2 = Integer.parseInt(
line.substring(index1, index2));

TreeNode node1, node2;

if (falcon.nodes[n1]!=null) {
node1 = falcon.nodes[n1];
}
else {
node1 = new TreeNode(n1);
falcon.nodes[n1] = node1;
}

if (falcon.nodes[n2]!=null) {
node2 = falcon.nodes[n2];
}
else {
node2 = new TreeNode(n2);
falcon.nodes[n2] = node2;
}

}

falcon.organizeTree();

for (int i = 0; i < falcon.nQueries; ++i)
{
index1 = 0;
index2 = line.indexOf(' ', index1);
final int q = Integer.parseInt(
line.substring(index1, index2));
index1 = index2+1;
index2 = line.indexOf(' ', index1);
final Integer u = new Integer(
line.substring(index1, index2));
index1 = index2+1;
index2 = line.length();
final Integer v = new Integer(
line.substring(index1, index2));

switch(q) {
case 1: falcon.nodes[u].updateValue(v);
break;
case 2:
out.println(falcon.getSum(u, v));
break;
default:    System.err.println("Invalid query " + q);
}
}
out.flush();

}
catch (Exception e) {
e.printStackTrace( System.err );
}
}
}

In   C   :

#include <stdio.h>
#include <stdlib.h>
typedef struct _lnode{
int x;
int w;
struct _lnode *next;
} lnode;
typedef struct _tree{
int sum;
} tree;
void insert_edge(int x,int y,int w);
void dfs0(int u);
void dfs1(int u,int c);
void preprocess();
int lca(int a,int b);
int sum(int v,int tl,
int tr,int l,int r,tree *t);
void update(int v,int tl,
int tr,int pos,int new_val,tree *t);
int min(int x,int y);
int max(int x,int y);
int solve(int x,int ancestor);
int N,cn,level[100000],DP[18][100000],
subtree_size[100000],special[100000],
node_chain[100000],node_idx[100000],
lnode *table[100000]={0};
tree *chain[100000];

int main(){
int Q,x,y,i;
scanf("%d%d",&N,&Q);
for(i=0;i<N-1;i++){
scanf("%d%d",&x,&y);
insert_edge(x,y,1);
}
preprocess();
while(Q--){
scanf("%d",&x);
switch(x){
case 1:
scanf("%d%d",&x,&y);
update(1,0,chain_len[node_chain[x]]
-1,node_idx[x],y,chain[node_chain[x]]);
break;
default:
scanf("%d%d",&x,&y);
i=lca(x,y);
printf("%d\n",
solve(x,i)+solve(y,i)-
sum(1,0,chain_len[node_chain[i]]
-1,node_idx[i],node_idx[i],chain[node_chain[i]]));
}
}
return 0;
}
void insert_edge(int x,int y,int w){
lnode *t=malloc(sizeof(lnode));
t->x=y;
t->w=w;
t->next=table[x];
table[x]=t;
t=malloc(sizeof(lnode));
t->x=x;
t->w=w;
t->next=table[y];
table[y]=t;
return;
}
void dfs0(int u){
lnode *x;
subtree_size[u]=1;
special[u]=-1;
for(x=table[u];x;x=x->next)
if(x->x!=DP[0][u]){
DP[0][x->x]=u;
level[x->x]=level[u]+1;
dfs0(x->x);
subtree_size[u]+=subtree_size[x->x];
if(special[u]==-1 ||
subtree_size[x->x]>subtree_size[special[u]])
special[u]=x->x;
}
return;
}
void dfs1(int u,int c){
lnode *x;
node_chain[u]=c;
node_idx[u]=chain_len[c]++;
for(x=table[u];x;x=x->next)
if(x->x!=DP[0][u])
if(x->x==special[u])
dfs1(x->x,c);
else{
dfs1(x->x,cn++);
}
return;
}
void preprocess(){
int i,j;
level[0]=0;
DP[0][0]=0;
dfs0(0);
for(i=1;i<18;i++)
for(j=0;j<N;j++)
DP[i][j] = DP[i-1][DP[i-1][j]];
cn=1;
dfs1(0,0);
for(i=0;i<cn;i++)
chain[i]=(tree*)malloc(
4*chain_len[i]*sizeof(tree));
for(i=0;i<N;i++)
update(1,0,chain_len[node_chain[i]]-1,
node_idx[i],0,chain[node_chain[i]]);
return;
}
int lca(int a,int b){
int i;
if(level[a]>level[b]){
i=a;
a=b;
b=i;
}
int d = level[b]-level[a];
for(i=0;i<18;i++)
if(d&(1<<i))
b=DP[i][b];
if(a==b)return a;
for(i=17;i>=0;i--)
if(DP[i][a]!=DP[i][b])
a=DP[i][a],b=DP[i][b];
return DP[0][a];
}
int sum(int v,int tl,int tr,int l,
int r,tree *t){
if(l>r)
return 0;
if(l==tl && r==tr)
return t[v].sum;
int tm=(tl+tr)/2;
return sum(v*2,tl,tm,l,min(r,tm),t)+
sum(v*2+1,tm+1,tr,max(l,tm+1),r,t);
}
void update(int v,int tl,int tr,
int pos,int new_val,tree *t){
if(tl==tr)
t[v].sum=new_val;
else{
int tm=(tl+tr)/2;
if(pos<=tm)
update(v*2,tl,tm,pos,new_val,t);
else
update(v*2+1,tm+1,tr,pos,new_val,t);
t[v].sum=t[v*2].sum+t[v*2+1].sum;
}
}
int min(int x,int y){
return (x<y)?x:y;
}
int max(int x,int y){
return (x>y)?x:y;
}
int solve(int x,int ancestor){
int ans=0;
while(node_chain[x]!=node_chain[ancestor]){
ans+=sum(1,0,chain_len[node_chain[x]]-1,
0,node_idx[x],chain[node_chain[x]]);
}
ans+=sum(1,0,chain_len[node_chain[x]]-1,
node_idx[ancestor],node_idx[x],
chain[node_chain[x]]);
return ans;
}

In   Python3  :

class heavy_light_node:
def __init__(self, size):
self.parent = None
self.pos = -1
self.weight = [0] * size
self.fenwick = [0] * size
def set_weight(self, i, x):
d = x - self.weight[i]
self.weight[i] = x
N = len(self.weight)
while i < N:
self.fenwick[i] += d
i |= i + 1
def sum_weight(self, i):
if i < 0: return 0
x = self.fenwick[i]
i &= i + 1
while i:
x += self.fenwick[i-1]
i &= i - 1
return x
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[0]
members.append(i)
ed = edges[i]
node = heavy_light_node(len(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 = [0] * N
active = [0]
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[0]
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][0] == py[m][0]: m += 1
loc, a = px[m-1]
b = py[m-1][1]
if a > b: a, b = b, a
w = loc.sum_weight(b) - loc.sum_weight(a-1)
for j in range(m, len(px)):
loc, i = px[j]
w += loc.sum_weight(i)
for j in range(m, len(py)):
loc, i = py[j]
w += loc.sum_weight(i)
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]
loc.set_weight(i, y)
elif t == 2:
print(max_weight(x, y))```
```

## Array Manipulation

Starting with a 1-indexed array of zeros and a list of operations, for each operation add a value to each of the array element between two given indices, inclusive. Once all operations have been performed, return the maximum value in the array. Example: n=10 queries=[[1,5,3], [4,8,7], [6,9,1]] Queries are interpreted as follows: a b k 1 5 3 4 8 7 6 9 1 Add the valu

## Print the Elements of a Linked List

This is an to practice traversing a linked list. Given a pointer to the head node of a linked list, print each node's data element, one per line. If the head pointer is null (indicating the list is empty), there is nothing to print. Function Description: Complete the printLinkedList function in the editor below. printLinkedList has the following parameter(s): 1.SinglyLinkedListNode

## Insert a Node at the Tail of a Linked List

You are given the pointer to the head node of a linked list and an integer to add to the list. Create a new node with the given integer. Insert this node at the tail of the linked list and return the head node of the linked list formed after inserting this new node. The given head pointer may be null, meaning that the initial list is empty. Input Format: You have to complete the SinglyLink