BST maintenance
Problem Statement :
Consider a binary search tree T which is initially empty. Also, consider the first N positive integers {1, 2, 3, 4, 5, ....., N} and its permutation P {a1, a2, ..., aN}.
If we start adding these numbers to the binary search tree T, starting from a1, continuing with a2, ... (and so on) ..., ending with aN. After every addition we ask you to output the sum of distances between every pair of T's nodes.
Input Format
The first line of the input consists of the single integer N, the size of the list.
The second line of the input contains N single space separated numbers the permutation a1, a2, ..., aN itself.
Constraints
1 ≤ N ≤ 250000
Output Format
Output N lines.
On the ith line output the sum of distances between every pair of nodes after adding the first i numbers from the permutation to the binary search tree T
Solution :
Solution in C :
In C++ :
#include <bits/stdc++.h>
using namespace std;
const int
MAXV = 250005;
int V;
int p[MAXV];
int pos[MAXV];
set< int > S;
int L[MAXV];
int R[MAXV];
int P[MAXV];
bool seen[MAXV];
vector< int > childs;
int depth[MAXV];
int size[MAXV];
vector< int > G[MAXV];
int root;
long long sum[MAXV];
long long nodeCount[3];
long long distancesSum[3];
int branch[MAXV];
int curBranch;
void computeSizes(int u)
{
seen[u] = 1;
size[u] = 1;
for (int v : G[u]) if (!seen[v])
{
computeSizes(v);
size[u] += size[v];
}
seen[u] = 0;
}
void dfs(int u)
{
branch[u] = curBranch;
seen[u] = 1;
childs.push_back(u);
for (int v : G[u]) if (!seen[v])
{
depth[v] = depth[u] + 1;
dfs(v);
}
seen[u] = 0;
}
int findCenter(int u)
{
bool isCenter =
((size[root] - size[u]) * 2 <= size[root]);
for (int v : G[u]) if (!seen[v])
isCenter &= (size[v] * 2 <= size[root]);
if (isCenter)
return u;
seen[u] = 1;
for (int v : G[u]) if (!seen[v])
{
int c = findCenter(v);
if (c != -1)
{
seen[u] = 0;
return c;
}
}
seen[u] = 0;
return -1;
}
bool posCmp(int a, int b)
{
return pos[a] < pos[b];
}
void solve(int u)
{
computeSizes(u);
if (size[u] == 1) // single leaf
return ;
root = u;
u = findCenter(u);
seen[u] = 1;
childs.clear();
curBranch = 0;
for (int v : G[u]) if (!seen[v]) {
depth[v] = 1;
dfs(v);
// fill 0
nodeCount[curBranch] = 0;
distancesSum[curBranch] = 0;
curBranch++;
}
assert(curBranch <= 3);
childs.push_back(u); // root
// sort can be avoided
sort(childs.begin(), childs.end(), posCmp);
bool isRootAdded = false;
for (int c : childs)
{
if (c == u) // add root
{
for (int i = 0; i < curBranch; ++i)
sum[ pos[c] ] += distancesSum[i];
isRootAdded = true;
}
else {
int b = branch[c];
if (isRootAdded)
{
for (int i = 0; i < curBranch; ++i) if (i != b)
sum[ pos[c] ] +=
distancesSum[i] + nodeCount[i] * depth[c];
// add root
sum[ pos[c] ] += depth[c];
}
distancesSum[b] += depth[c];
nodeCount[b]++;
}
}
for (int v : G[u]) if (!seen[v])
solve(v);
seen[u] = 0;
}
int main()
{
ios_base::sync_with_stdio(0);
cin.tie(0);
cin >> V;
memset(L, -1, sizeof(L));
memset(R, -1, sizeof(R));
for (int i = 0; i < V; ++i)
{
cin >> p[i];
--p[i];
pos[ p[i] ] = i;
auto lb = S.lower_bound(p[i]);
if (lb != S.end())
{
if (L[*lb] < 0) {
L[*lb] = p[i];
G[*lb].push_back(p[i]);
G[p[i]].push_back(*lb);
}
}
if (lb != S.begin()) {
--lb;
if (R[*lb] < 0) {
R[*lb] = p[i];
G[*lb].push_back(p[i]);
G[p[i]].push_back(*lb);
}
}
S.insert(p[i]);
}
solve(p[0]);
for (int i = 1; i < V; ++i)
sum[i] += sum[i - 1];
for (int i = 0; i < V; ++i)
cout << sum[i] << '\n';
return 0;
}
In Java :
import java.io.*;
import java.util.*;
public class Solution {
BufferedReader br;
PrintWriter out;
StringTokenizer st;
boolean eof;
static int getSize(Node v) {
return v == null ? 0 : v.size;
}
static class Node implements Comparable<Node>
{
Node left, right;
int val;
int depth;
int size;
int pathNum, inPathPos;
@Override
public int compareTo(Node o) {
return Integer.compare(val, o.val);
}
public Node(int val) {
this.val = val;
}
@Override
public String toString() {
StringBuilder sb = new StringBuilder();
if (left != null) {
sb.append('[');
sb.append(left.toString());
sb.append("] ");
}
sb.append(val);
if (right != null) {
sb.append(" [");
sb.append(right.toString());
sb.append(']');
}
return sb.toString();
}
}
int nPaths = 0;
List<Node> nextNode;
int[] pathSize;
FenwickTree[] fen;
void addUptoRoot(Node v) {
do {
int pathNum = v.pathNum;
int inPathPos = v.inPathPos;
fen[pathNum].add(0, inPathPos);
v = nextNode.get(pathNum);
} while (v != null);
}
long getUptoRoot(Node v) {
long ret = 0;
do {
int pathNum = v.pathNum;
int inPathPos = v.inPathPos;
ret += fen[pathNum].get(0, inPathPos);
v = nextNode.get(pathNum);
} while (v != null);
return ret;
}
void solve() throws IOException {
int n = nextInt();
TreeSet<Node> set = new TreeSet<>();
Node root = null;
Node[] order = new Node[n];
for (int i = 0; i < n; i++) {
int x = nextInt();
Node v = new Node(x);
order[i] = v;
set.add(v);
if (set.size() == 1) {
root = v;
continue;
}
Node before = set.lower(v);
Node after = set.higher(v);
if (before != null
&& before.right == null
&& (after == null ||
after.left != null || before.depth < after.depth))
{
before.right = v;
v.depth = before.depth + 1;
} else {
after.left = v;
v.depth = after.depth + 1;
}
}
dfs1(root);
nextNode = new ArrayList<Node>(n);
pathSize = new int[n];
dfs2(root, 0, 0, null);
fen = new FenwickTree[nPaths];
for (int i = 0; i < nPaths; i++) {
fen[i] = new FenwickTree(pathSize[i]);
}
long outp = 0;
long sumDistRoot = 0;
for (int i = 0; i < n; i++) {
Node v = order[i];
sumDistRoot += v.depth;
addUptoRoot(v);
long sum = getUptoRoot(v) - (i + 1);
long delta = (long) (i + 1) * v.depth - 2 * sum;
outp += sumDistRoot + delta;
out.println(outp);
}
}
void dfs1(Node v) {
if (v == null) {
return;
}
dfs1(v.left);
dfs1(v.right);
v.size = getSize(v.left) + getSize(v.right) + 1;
}
void dfs2(Node v, int pathNum, int inPathPos, Node par) {
if (v == null) {
return;
}
if (inPathPos == 0) {
pathNum = nPaths++;
nextNode.add(par);
}
v.pathNum = pathNum;
v.inPathPos = inPathPos;
if (v.left == null && v.right == null) {
pathSize[pathNum] = inPathPos + 1;
return;
}
if (getSize(v.left) > getSize(v.right)) {
dfs2(v.left, pathNum, inPathPos + 1, v);
dfs2(v.right, -1, 0, v);
} else {
dfs2(v.right, pathNum, inPathPos + 1, v);
dfs2(v.left, -1, 0, v);
}
}
static class IntList {
/**
* Never shrinks
*/
private int[] data;
private int size;
public IntList(int cap) {
data = new int[cap];
size = 0;
}
void add(int x) {
data[size++] = x;
}
void clear() {
size = 0;
}
int get(int idx) {
if (idx < 0 || idx >= size) {
throw new IndexOutOfBoundsException();
}
return data[idx];
}
}
static class FenwickTree {
private int n;
private long[] c0;
private long[] c1;
public FenwickTree(int n) {
this.n = n;
this.c0 = new long[n];
this.c1 = new long[n];
}
void add(int low, int high) {
/**
* [low, high]
*/
internalUpdate(low, -(low - 1), 1);
internalUpdate(high, high, -1);
}
private void internalUpdate(int x, int d0, int d1) {
for (int i = x; i < n; i |= i + 1) {
c0[i] += d0;
c1[i] += d1;
}
}
long get(int low, int high) {
/**
* [low, high]
*/
return get(high) - get(low - 1);
}
long get(int x) {
/**
* [0, x]
*/
long a1 = 0;
long a0 = 0;
for (int i = x; i >= 0; i = (i & (i + 1)) - 1) {
a1 += c1[i];
a0 += c0[i];
}
return a1 * x + a0;
}
}
Solution() throws IOException {
br = new BufferedReader(new InputStreamReader(System.in));
out = new PrintWriter(System.out);
solve();
out.close();
}
public static void main(String[] args) throws IOException {
new Solution();
}
String nextToken() {
while (st == null || !st.hasMoreTokens()) {
try {
st = new StringTokenizer(br.readLine());
} catch (Exception e) {
eof = true;
return null;
}
}
return st.nextToken();
}
String nextString() {
try {
return br.readLine();
} catch (IOException e) {
eof = true;
return null;
}
}
int nextInt() throws IOException {
return Integer.parseInt(nextToken());
}
long nextLong() throws IOException {
return Long.parseLong(nextToken());
}
double nextDouble() throws IOException {
return Double.parseDouble(nextToken());
}
}
In C :
#include <stdio.h>
#include <stdlib.h>
typedef struct _node{
int x;
struct _node *next;
} lnode;
void init( int n ,int *tree);
void range_increment( int i, int j,
int val ,int *tree);
int query( int i ,int *tree);
void insert_edge(int x,int y);
void dfs0(int u);
void preprocess();
int lca(int a,int b);
int dist(int u,int v);
void dfs1(int u,int p);
int dfs2(int u,int p);
void decompose(int root,int p);
int a[250000],cut[250000]={0},parent[250000],
DP[18][250000],mid[750000],left[750000],
right[750000],level[250000],
sub[250000],N,NN,nn;
long long count[250000]={0},
sum[250000]={0},con[250000]={0};
lnode *table[250000]={0};
int main(){
int x,y,z,leftd,rightd,i;
long long ans,aa=0;
scanf("%d",&NN);
for(i=0;i<NN;i++)
scanf("%d",a+i);
init(NN,mid);
init(NN,left);
init(NN,right);
for(i=0;i<NN;i++){
leftd=x=query(a[i]-1,left);
if(!x)
leftd=1;
rightd=y=query(a[i]-1,right);
if(!y)
rightd=NN;
z=query(a[i]-1,mid);
if(z)
insert_edge(z-1,a[i]-1);
range_increment(leftd-1,rightd-1,a[i]-z,mid);
range_increment(a[i]-1,rightd-1,a[i]-x,left);
range_increment(leftd-1,a[i]-1,a[i]-y,right);
}
preprocess();
decompose(a[NN/2]-1,-1);
for(i=0;i<NN;i++){
for(ans=sum[a[i]-1],x=a[i]-1;1;x=parent[x]){
if(parent[x]==-1)
break;
ans+=sum[parent[x]]-con[x]+dist(a[i]-1,
parent[x])*(count[parent[x]]-count[x]);
}
for(x=a[i]-1;x!=-1;x=parent[x]){
sum[x]+=dist(a[i]-1,x);
count[x]++;
if(parent[x]!=-1)
con[x]+=dist(a[i]-1,parent[x]);
}
printf("%lld\n",aa+=ans);
}
return 0;
}
void init( int n ,int *tree){
N = 1;
while( N < n ) N *= 2;
int i;
for( i = 1; i < N + n; i++ ) tree[i] = 0;
}
void range_increment( int i, int j,
int val ,int *tree){
for( i += N, j += N; i <= j;
i = ( i + 1 ) / 2, j = ( j - 1 ) / 2 )
{
if( i % 2 == 1 ) tree[i] += val;
if( j % 2 == 0 ) tree[j] += val;
}
}
int query( int i ,int *tree){
int ans = 0,j;
for( j = i + N; j; j /= 2 ) ans += tree[j];
return ans;
}
void insert_edge(int x,int y){
lnode *t=malloc(sizeof(lnode));
t->x=y;
t->next=table[x];
table[x]=t;
t=malloc(sizeof(lnode));
t->x=x;
t->next=table[y];
table[y]=t;
return;
}
void dfs0(int u){
lnode *x;
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);
}
return;
}
void preprocess(){
int i,j;
level[a[0]-1]=0;
DP[0][a[0]-1]=a[0]-1;
dfs0(a[0]-1);
for(i=1;i<18;i++)
for(j=0;j<NN;j++)
DP[i][j] = DP[i-1][DP[i-1][j]];
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 dist(int u,int v){
return level[u] + level[v] - 2*level[lca(u,v)];
}
void dfs1(int u,int p){
sub[u]=1;
nn++;
lnode *x;
for(x=table[u];x;x=x->next)
if(x->x!=p && !cut[x->x]){
dfs1(x->x,u);
sub[u]+=sub[x->x];
}
return;
}
int dfs2(int u,int p){
lnode *x;
for(x=table[u];x;x=x->next)
if(x->x!=p && sub[x->x]>nn/2 && !cut[x->x])
return dfs2(x->x,u);
return u;
}
void decompose(int root,int p){
nn=0;
dfs1(root,root);
int centroid = dfs2(root,root);
parent[centroid]=p;
cut[centroid]=1;
lnode *x;
for(x=table[centroid];x;x=x->next)
if(!cut[x->x])
decompose(x->x,centroid);
return;
}
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