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 :



title-img


                            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;

vector<int> adj[N];
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);
    for(int v : adj[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;
        adj[u].push_back(v);
        adj[v].push_back(u);
    }
    
    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.
Set<TreeNode> linkedNodes, children;
TreeNode parent;
BranchContainer branch;

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

if (!head.branch.isTrunk) {
sum -= head.branch.list.get(branchSize-1).value; 
}

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

return ret;
}

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

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);
ret.addAll(tmpPath);
}
else if (branches==0) {
addSameBranch(node1, node2, ret);
}
else {
tmp = advanceBranch(node1, branches, ret);
if (!findPathFixOne(node2, tmp, ret)) {
System.err.println("3 Cannot find path between "
 + node1.toString() + " and " + tmp.toString());
}
}
}
else if (branches==0) {
addSameBranch(node1, node2, ret);
}
else {
tmp = advanceBranch(node2, branches, ret);
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;
}
}

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

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

addSameBranch(fixed, node, path);

return true;
}

void addSameBranch(final TreeNode node1, 
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;
}

path.add(node1.branch.list.subList(leafLevel-level1,
 leafLevel-level2+1));
}

void organizeTree() {
root = null;
int maxLinks = 0;
for (int i = 0; i < nNodes; ++i) {
final TreeNode node = nodes[i];
final int links = node.linkedNodes.size(); 
if (links>maxLinks) {
maxLinks = links;
root = node;
}
}

setChildren();
enumerateBranches();

return;
}

void setChildren() {
int level = 0;
root.level = level;
Map<TreeNode, Set<TreeNode>> pcMap = 
new HashMap<>();
pcMap.put(root, root.linkedNodes);
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;
parent.children.addAll(list);
if (parent.parent!=null) {
parent.children.remove(parent.parent);
}

for (TreeNode node : parent.children) {
node.parent = parent;
newMap.put(node, node.linkedNodes);
}
}

++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;
}

node.branch.list.add(node);
node.branch.set.add(node);
TreeNode tmpNode = node.parent;
while (tmpNode!=null) {
node.branch.list.add(tmpNode);
node.branch.set.add(tmpNode);
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();

BufferedReader br = 
new BufferedReader(new InputStreamReader(System.in));
String line = br.readLine();
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) {
//Read input.
line = br.readLine();
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;
}

node1.linkedNodes.add(node2);
node2.linkedNodes.add(node1);
}

falcon.organizeTree();

for (int i = 0; i < falcon.nQueries; ++i) 
{
line = br.readLine();
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],
chain_head[100000],chain_len[100000]={0};
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{
chain_head[cn]=x->x;
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;
chain_head[0]=0;
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]]);
x=DP[0][chain_head[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
def read_tree(N):
   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())
location = read_tree(N)
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))
                        








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Delete a Node

Delete the node at a given position in a linked list and return a reference to the head node. The head is at position 0. The list may be empty after you delete the node. In that case, return a null value. Example: list=0->1->2->3 position=2 After removing the node at position 2, list'= 0->1->-3. Function Description: Complete the deleteNode function in the editor below. deleteNo

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