Please check previous sets of Proto Van Emde Boas Tree article first. It is highly recommended.
Procedure for finding minimum:
- Base Case: If the size of Proto-VEB is 2 then we will return smallest key present in the cluster if no keys present then we will return -1 as the symbol that no keys are present.
-
Recursion:
- We will start recursion over summary until we reach the first true value(In the code, first not nullptr in the summary cluster) which shows that there is a key present in that cluster.
- Now We will find the position of the key in that cluster using again recursively call over a cluster to find the first true value (In the code, first not nullptr in the cluster) in a cluster as we have done above.
- Finally, we will return the index of that key according to cluster number we get from procedure over summary and position we get from the procedure over the cluster in the last step.
See the image below for the basic understanding of the operation:
Observe the light green circles from top to bottom:
See the image below for real Proto – VEB minimum operation:
Follow the instructions in order of numberings.
You can easily get the idea of Maximum from the minimum procedure. See the image below:
Observe the light green circles from top to bottom:
Implementation of the above algorithm:
// C++ implementation of the approach #include <bits/stdc++.h> using namespace std;
class Proto_Van_Emde_Boas {
public :
// Total number of keys
int universe_size;
// Summary
Proto_Van_Emde_Boas* summary;
// Clusters array of Proto-VEB pointers
vector<Proto_Van_Emde_Boas*> clusters;
int root( int u)
{
return ( int ) sqrt (u);
}
// Function to return cluster numbers
// in which key is present
int high( int x)
{
return x / root(universe_size);
}
// Function to return position of x in cluster
int low( int x)
{
return x % root(universe_size);
}
// Function to return the index from
// cluster number and position
int generate_index( int cluster, int position)
{
return cluster * root(universe_size) + position;
}
// Constructor
Proto_Van_Emde_Boas( int size)
{
universe_size = size;
// Base case
if (size <= 2) {
// Set summary to nullptr as there is no
// more summary for size 2
summary = nullptr;
// Vector of two pointers
// nullptr in starting
clusters = vector<Proto_Van_Emde_Boas*>(size, nullptr);
}
else {
// Assigning Proto-VEB(sqrt(u)) to summary
summary = new Proto_Van_Emde_Boas(root(size));
// Creating array of Proto-VEB Tree pointers of size sqrt(u)
// first all nullptrs are going to assign
clusters = vector<Proto_Van_Emde_Boas*>(root(size), nullptr);
// Assigning Proto-VEB(sqrt(u)) to all its clusters
for ( int i = 0; i < root(size); i++) {
clusters[i] = new Proto_Van_Emde_Boas(root(size));
}
}
}
}; // Function that returns true if the // key is present in the tree bool isMember(Proto_Van_Emde_Boas* helper, int key)
{ // If key is greater then universe_size then
// returns false
if (key >= helper->universe_size)
return false ;
// If we reach at base case
// the just return whether
// pointer is nullptr then false
// else return true
if (helper->universe_size == 2) {
return helper->clusters[key];
}
else {
// Recursively go deep into the
// level of Proto-VEB tree using its
// cluster index and its position
return isMember(helper->clusters[helper->high(key)],
helper->low(key));
}
} // Function to insert a key in the tree void insert(Proto_Van_Emde_Boas*& helper, int key)
{ // If we reach at base case
// then assign Proto-VEB(1) in place
// of nullptr
if (helper->universe_size == 2) {
helper->clusters[key] = new Proto_Van_Emde_Boas(1);
}
else {
// Recursively using index of cluster and its
// position in cluster
insert(helper->clusters[helper->high(key)],
helper->low(key));
// Also do the same recursion in summary VEB
insert(helper->summary, helper->high(key));
}
} // Function to return the minimum key from the tree int minimum(Proto_Van_Emde_Boas* helper)
{ // Base case choses the least key
// present in the cluster
if (helper->universe_size == 2) {
if (helper->clusters[0]) {
return 0;
}
else if (helper->clusters[1]) {
return 1;
}
// No keys present then return -1
return -1;
}
else {
// Recursively find in summary for
// first 1 present in Proto-VEB
int minimum_cluster = minimum(helper->summary);
int offset;
// If no key is present in
// the cluster then return -1
if (minimum_cluster == -1) {
return -1;
}
else {
// Recursively find the position of the key
// in the minimum_cluster
offset = minimum(helper->clusters[minimum_cluster]);
// Returns overall index of minimum key
return helper->generate_index(minimum_cluster, offset);
}
}
} // Function to return the maximum key from the tree int maximum(Proto_Van_Emde_Boas* helper)
{ // Return the maximum key present in
// the cluster
if (helper->universe_size == 2) {
if (helper->clusters[1]) {
return 1;
}
else if (helper->clusters[0]) {
return 0;
}
// Return -1 if no keys present in the
// cluster
return -1;
}
else {
// Recursively find the last 1 present
// in the summary
int maximum_cluster = maximum(helper->summary);
int offset;
// If no key is present in
// the cluster then return -1
if (maximum_cluster == -1) {
return -1;
}
else {
// Recursively find the position of the key
// in the maximum_cluster
offset = maximum(helper->clusters[maximum_cluster]);
return helper->generate_index(maximum_cluster, offset);
}
}
} // Function to delete a key from the tree void pveb_delete(Proto_Van_Emde_Boas*& helper, int key)
{ // Base case: If the key is present
// then make it nullptr
if (helper->universe_size == 2) {
if (helper->clusters[key]) {
delete helper->clusters[key];
helper->clusters[key] = nullptr;
}
}
else {
// Recursive delete to reach at the base case
pveb_delete(helper->clusters[helper->high(key)], helper->low(key));
bool isanyinCluster = false ;
// Iterate over the cluster of keys to check whether
// any other key is present within that cluster
// If yes then we should not update summary to 0
// else update summary to 0
for ( int i = helper->high(key) * helper->root(helper->universe_size);
i < (helper->high(key) + 1) * helper->root(helper->universe_size);
i++) {
// If member is present then break the loop
if (isMember(helper->clusters[helper->high(key)], i)) {
isanyinCluster = true ;
break ;
}
}
// If no member is present then
// update summary to zero
if (isanyinCluster == false ) {
pveb_delete(helper->summary, helper->high(key));
}
}
} // Driver code int main()
{ Proto_Van_Emde_Boas* hello = new Proto_Van_Emde_Boas(4);
cout << boolalpha;
insert(hello, 2);
insert(hello, 3);
cout << minimum(hello) << endl;
cout << maximum(hello) << endl;
} |
// Java implementation import java.util.ArrayList;
class Proto_Van_Emde_Boas {
public int universe_size;
public Proto_Van_Emde_Boas summary;
// Clusters array of Proto-VEB pointers
public ArrayList<Proto_Van_Emde_Boas> clusters;
public Proto_Van_Emde_Boas( int size)
{
universe_size = size;
if (size <= 2 ) {
summary = null ;
clusters
= new ArrayList<Proto_Van_Emde_Boas>(size);
for ( int i = 0 ; i < size; i++) {
clusters.add( null );
}
}
else {
summary = new Proto_Van_Emde_Boas(root(size));
clusters = new ArrayList<Proto_Van_Emde_Boas>(
root(size));
for ( int i = 0 ; i < root(size); i++) {
clusters.add(
new Proto_Van_Emde_Boas(root(size)));
}
}
}
public int root( int u) { return ( int )Math.sqrt(u); }
// Function to return cluster numbers
// in which key is present
public int high( int x)
{
return x / root(universe_size);
}
// Function to return position of x in cluster
public int low( int x)
{
return x % root(universe_size);
}
// Function to return the index from
// cluster number and position
public int generate_index( int cluster, int position)
{
return cluster * root(universe_size) + position;
}
} class Main {
// Function to insert a key in the tree
public static void insert(Proto_Van_Emde_Boas helper,
int key)
{
// If we reach at base case
// then assign Proto-VEB(1) in place
// of nullptr
if (helper.universe_size == 2 ) {
helper.clusters.set(key,
new Proto_Van_Emde_Boas( 1 ));
}
else {
// Recursively using index of cluster and its
// position in cluster
insert(helper.clusters.get(helper.high(key)),
helper.low(key));
insert(helper.summary, helper.high(key));
}
}
// Function to return the minimum key from the tree
public static int minimum(Proto_Van_Emde_Boas helper)
{
// Base case choses the least key
// present in the cluster
if (helper.universe_size == 2 ) {
if (helper.clusters.get( 0 ) != null ) {
return 0 ;
}
else if (helper.clusters.get( 1 ) != null ) {
return 1 ;
}
return - 1 ;
}
else {
// Recursively find in summary for
// first 1 present in Proto-VEB
int minimum_cluster = minimum(helper.summary);
int offset;
if (minimum_cluster
== - 1 ) { // If no key then return -1
return - 1 ;
}
else {
// Recursively find the position of the key
// in the minimum_cluster
offset = minimum(
helper.clusters.get(minimum_cluster));
return helper.generate_index(
minimum_cluster, offset);
}
}
}
// Function to return the maximum key from the tree
public static int maximum(Proto_Van_Emde_Boas helper)
{
if (helper.universe_size == 2 ) {
if (helper.clusters.get( 1 ) != null ) {
return 1 ;
}
else if (helper.clusters.get( 0 ) != null ) {
return 0 ;
}
return - 1 ; // If no key then return -1
}
else {
// Recursively find in summary for
// first 1 present in Proto-VEB
int maximum_cluster = maximum(helper.summary);
int offset;
if (maximum_cluster == - 1 ) {
return - 1 ; // If no key then return -1
}
else {
// Recursively find the position of the key
// in the minimum_cluster
offset = maximum(
helper.clusters.get(maximum_cluster));
return helper.generate_index(
maximum_cluster, offset);
}
}
}
// Driver code
public static void main(String[] args)
{
Proto_Van_Emde_Boas hello
= new Proto_Van_Emde_Boas( 4 );
// inserting
insert(hello, 2 );
insert(hello, 3 );
// finding minimun by minimum func.
System.out.println(minimum(hello));
// fiding maximum by maximum func.
System.out.println(maximum(hello));
}
} |
# Python implementation of the program import math
class Proto_Van_Emde_Boas:
def __init__( self , size):
self .universe_size = size
if size < = 2 :
self .summary = None
self .clusters = [ None ] * size
else :
self .summary = Proto_Van_Emde_Boas( int (math.sqrt(size)))
self .clusters = [Proto_Van_Emde_Boas(
int (math.sqrt(size))) for _ in range ( int (math.sqrt(size)))]
def root( self , u):
return int (math.sqrt(u))
# Function to return cluster numbers
# in which key is present
def high( self , x):
return x / / self .root( self .universe_size)
# Function to return position of x in cluster
def low( self , x):
return x % self .root( self .universe_size)
# Function to return the index from # cluster number and position def generate_index( self , cluster, position):
return cluster * self .root( self .universe_size) + position
# Function to insert a key in the tree def insert(helper, key):
if helper.universe_size = = 2 :
helper.clusters[key] = Proto_Van_Emde_Boas( 1 )
else :
insert(helper.clusters[helper.high(key)], helper.low(key))
insert(helper.summary, helper.high(key))
# Function to find minimum def minimum(helper):
if helper.universe_size = = 2 :
if helper.clusters[ 0 ] is not None :
return 0
elif helper.clusters[ 1 ] is not None :
return 1
return - 1
else :
minimum_cluster = minimum(helper.summary)
if minimum_cluster = = - 1 :
return - 1
else :
offset = minimum(helper.clusters[minimum_cluster])
return helper.generate_index(minimum_cluster, offset)
# Function to find maximum def maximum(helper):
if helper.universe_size = = 2 :
if helper.clusters[ 1 ] is not None :
return 1
elif helper.clusters[ 0 ] is not None :
return 0
return - 1
else :
maximum_cluster = maximum(helper.summary)
if maximum_cluster = = - 1 :
return - 1
else :
offset = maximum(helper.clusters[maximum_cluster])
return helper.generate_index(maximum_cluster, offset)
# Driver code hello = Proto_Van_Emde_Boas( 4 )
# inserting insert(hello, 2 );
insert(hello, 3 );
print (minimum(hello))
print (maximum(hello))
|
// JavaScript implementation of the program class Proto_Van_Emde_Boas { constructor(size) {
this .universe_size = size;
if (size <= 2) {
this .summary = null ;
this .clusters = new Array(size).fill( null );
} else {
this .summary = new Proto_Van_Emde_Boas(Math.sqrt(size));
this .clusters = Array(Math.sqrt(size)).fill().map(() => {
return new Proto_Van_Emde_Boas(Math.sqrt(size));
});
}
}
root(u) {
return Math.sqrt(u);
}
// Function to return cluster numbers
// in which key is present
high(x) {
return Math.floor(x / this .root( this .universe_size));
}
// Function to return position of x in cluster
low(x) {
return x % this .root( this .universe_size);
}
// Function to return the index from
// cluster number and position
generate_index(cluster, position) {
return cluster * this .root( this .universe_size) + position;
}
} // Function to insert a key in the tree function insert(helper, key) {
if (helper.universe_size == 2) {
helper.clusters[key] = new Proto_Van_Emde_Boas(1);
} else {
insert(helper.clusters[helper.high(key)], helper.low(key));
insert(helper.summary, helper.high(key));
}
} // Function to find minimum function minimum(helper) {
if (helper.universe_size == 2) {
if (helper.clusters[0] !== null ) {
return 0;
} else if (helper.clusters[1] !== null ) {
return 1;
}
return -1;
} else {
let minimum_cluster = minimum(helper.summary);
if (minimum_cluster == -1) {
return -1;
} else {
let offset = minimum(helper.clusters[minimum_cluster]);
return helper.generate_index(minimum_cluster, offset);
}
}
} // Function to find maximum function maximum(helper) {
if (helper.universe_size == 2) {
if (helper.clusters[1] !== null ) {
return 1;
} else if (helper.clusters[0] !== null ) {
return 0;
}
return -1;
} else {
let maximum_cluster = maximum(helper.summary);
if (maximum_cluster == -1) {
return -1;
} else {
let offset = maximum(helper.clusters[maximum_cluster]);
return helper.generate_index(maximum_cluster, offset);
}
}
} // Driver code let hello = new Proto_Van_Emde_Boas(4);
// inserting insert(hello, 2); insert(hello, 3); console.log(minimum(hello)); console.log(maximum(hello)); |
using System;
using System.Collections.Generic;
public class Proto_Van_Emde_Boas {
public int universe_size;
public Proto_Van_Emde_Boas summary;
public List<Proto_Van_Emde_Boas> clusters;
public Proto_Van_Emde_Boas( int size)
{
universe_size = size;
if (size <= 2) {
summary = null ;
clusters = new List<Proto_Van_Emde_Boas>(size);
for ( int i = 0; i < size; i++) {
clusters.Add( null );
}
}
else {
summary = new Proto_Van_Emde_Boas(root(size));
clusters
= new List<Proto_Van_Emde_Boas>(root(size));
for ( int i = 0; i < root(size); i++) {
clusters.Add(
new Proto_Van_Emde_Boas(root(size)));
}
}
}
public int root( int u) { return ( int )Math.Sqrt(u); }
public int high( int x)
{
return x / root(universe_size);
}
public int low( int x)
{
return x % root(universe_size);
}
public int generate_index( int cluster, int position)
{
return cluster * root(universe_size) + position;
}
} public class GFG {
public static void insert(Proto_Van_Emde_Boas helper,
int key)
{
if (helper.universe_size == 2) {
helper.clusters[key]
= new Proto_Van_Emde_Boas(1);
}
else {
insert(helper.clusters[helper.high(key)],
helper.low(key));
insert(helper.summary, helper.high(key));
}
}
public static int minimum(Proto_Van_Emde_Boas helper)
{
if (helper.universe_size == 2) {
if (helper.clusters[0] != null ) {
return 0;
}
else if (helper.clusters[1] != null ) {
return 1;
}
return -1;
}
else {
int minimum_cluster = minimum(helper.summary);
int offset;
if (minimum_cluster == -1) {
return -1;
}
else {
offset = minimum(
helper.clusters[minimum_cluster]);
return helper.generate_index(
minimum_cluster, offset);
}
}
}
public static int maximum(Proto_Van_Emde_Boas helper)
{
if (helper.universe_size == 2) {
if (helper.clusters[1] != null ) {
return 1;
}
else if (helper.clusters[0] != null ) {
return 0;
}
return -1;
}
else {
int maximum_cluster = maximum(helper.summary);
int offset;
if (maximum_cluster == -1) {
return -1;
}
else {
offset = maximum(
helper.clusters[maximum_cluster]);
return helper.generate_index(
maximum_cluster, offset);
}
}
}
static void Main()
{
Proto_Van_Emde_Boas hello
= new Proto_Van_Emde_Boas(4);
insert(hello, 2);
insert(hello, 3);
Console.WriteLine(minimum(hello));
// fiding maximum by maximum func.
Console.WriteLine(maximum(hello));
}
} // The code is contributed by Nidhi goel. |
Output
2 3
Both Minimum and Maximum query runs in O(log2(u)) time complexity.
Recurrence Relation:
T(u) = 2T() + O(1)