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# Proto Van Emde Boas Trees | Set 4 | Deletion

Please check previous sets of Proto Van Emde Boas Tree article first. It is highly recommended.
Procedure for delete:

1. Base Case: If we reach at Proto VEB with size 2 then we will check for whether the key is present or not if yes then we assign the pointer to nullptr which will set false to it presence.
2. Recursion:
• We recursively call delete function over the cluster of the keys i.e. high(key) and its position low(key).
• After we delete the key from the cluster (after we reach to the base case) we check whether there are any other keys are present in the cluster. If there is any key present then we can not set the summary to nullptr otherwise we will set the summary to nullptr by calling delete over summary.

Lets understand 1 delete on Proto-VEB of size 4:
First it will recursively call delete(cluster[0], 1).
So now the base case is satisfied so it will go at position 1 in the cluster[0] Proto-VEB and will set it to nullptr if it is present.
Now we will check if any more keys are present in cluster[0] (see the for loop in delete), 0 is present so delete(summary, 0) call is not going to execute and summary will remain same.
See the image below to understand it:
Follow the instructions written near the boxes from top to bottom.

Below is the implementation:

## CPP

 // C++ implementation of the approach#include 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 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(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(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 treebool 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 treevoid 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 delete a key from the treevoid 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 codeint main(){    Proto_Van_Emde_Boas* hello = new Proto_Van_Emde_Boas(4);     cout << isMember(hello, 2);     insert(hello, 2);     insert(hello, 3);     cout << isMember(hello, 2);     pveb_delete(hello, 2);     cout << isMember(hello, 2);}

## Java

 // Java implementation of the approachimport java.util.*; class Proto_Van_Emde_Boas {   // Total number of keys  int universe_size;  Proto_Van_Emde_Boas summary;   // Clusters array of Proto-VEB pointers  ArrayList clusters;   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 = null;      clusters = new ArrayList(size);      for (int i = 0; i < size; i++) {        clusters.add(null);      }    } else {      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 = new ArrayList(root(size));       // Assigning Proto-VEB(sqrt(u)) to all its clusters      for (int i = 0; i < root(size); i++) {        clusters.add(new Proto_Van_Emde_Boas(root(size)));      }    }  }   int root(int u) {    return (int) Math.sqrt(u);  }  // Function to return cluster numbers  // in which key is present  int high(int x) {    return x / root(universe_size);  }   int low(int x) {    return x % root(universe_size);  }  // return the index from cluster number and position  int generate_index(int cluster, int position) {    return cluster * root(universe_size) + position;  } } class Solution {   //returns true if the key is present in the tree  public static boolean 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;    }     // pointer is nullptr then false    // else return true    if (helper.universe_size == 2) {      return helper.clusters.get(key) != null;    } else {       // Recursively go deep into the      // level of Proto-VEB tree using its      // cluster index and its position      return isMember(helper.clusters.get(helper.high(key)), helper.low(key));    }  }   // Function to insert a key in the tree  public static void insert(Proto_Van_Emde_Boas helper, int key) {     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 delete a key from the tree  public static void pveb_delete(Proto_Van_Emde_Boas helper, int key) {     // If the key is present then make it nullptr    if (helper.universe_size == 2) {      if (helper.clusters.get(key) != null) {        helper.clusters.set(key, null);      }    } else {      pveb_delete(helper.clusters.get(helper.high(key)), helper.low(key));       boolean 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 (isMember(helper.clusters.get(helper.high(key)), i)) {          isanyinCluster = true;          break;        }      }      // If no member is present then      // update summary to zero      if (!isanyinCluster) {        pveb_delete(helper.summary, helper.high(key));      }    }  }    // Driver code  public static void main(String[] args) {     Proto_Van_Emde_Boas hello = new Proto_Van_Emde_Boas(4);     // checking it is member or not by isMember    System.out.println(isMember(hello, 2));     // inserting    insert(hello, 2);    insert(hello, 3);     // again checking it is member or not by isMember    System.out.println(isMember(hello, 2));     // deleting    pveb_delete(hello, 2);     // checking it is member or not by isMember    System.out.println(isMember(hello, 2));  }}

## Python3

 # Python implementationimport math class ProtoVanEmdeBoas:  def __init__(self, size):    self.universeSize = size        # Set summary to null as there is no    # more summary for size 2    if size <= 2:      self.summary = None      self.clusters = [None] * size    else:      self.summary = ProtoVanEmdeBoas(self.Root(size))       self.clusters = [ProtoVanEmdeBoas(self.Root(size)) for i in range(self.Root(size))]   def Root(self, u):    return int(math.sqrt(u))   def High(self, x):    return int(x / self.Root(self.universeSize))   def Low(self, x):    return x % self.Root(self.universeSize)   def GenerateIndex(self, cluster, position):    return cluster * self.Root(self.universeSize) + position      #  Function that returns true if the key is present in    #  the treedef IsMember(helper, key):         #  If key is greater than or equal to universeSize    #  then return false  if key >= helper.universeSize:    return False      #  If we reach at base case    #  the just return whether    #  pointer is nullptr then false    #  else return true  if helper.universeSize == 2:    return helper.clusters[key] != None  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 treedef Insert(helper, key):         # If we reach at base case    # then assign Proto-VEB(1) in place    # of nullptr  if helper.universeSize == 2:    helper.clusters[key] = ProtoVanEmdeBoas(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 delete a key from the treedef PvebDelete(helper, key):         # Base case: If the key is present    # then make it nullptr  if helper.universeSize == 2:    if helper.clusters[key] != None:      helper.clusters[key] = None  else:    # Recursive delete to reach at the base case    PvebDelete(helper.clusters[helper.High(key)], helper.Low(key))     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 i in range(helper.High(key) * helper.Root(helper.universeSize), (helper.High(key) + 1) * helper.Root(helper.universeSize)):               # 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 not isAnyInCluster:      PvebDelete(helper.summary, helper.High(key)) # Driver Codehello = ProtoVanEmdeBoas(4) # Checking the membershipprint(IsMember(hello, 2)) # InsertingInsert(hello, 2)Insert(hello, 3) # Checking the membershipprint(IsMember(hello, 2)) # deletingPvebDelete(hello, 2)print(IsMember(hello, 2))

## Javascript

 class ProtoVanEmdeBoas {  constructor(size) {    this.universeSize = size;     if (size <= 2) {            // Set summary to null as there is no            // more summary for size 2      this.summary = null;      this.clusters = new Array(size).fill(null);    } else {      this.summary = new ProtoVanEmdeBoas(this.Root(size));                 // Creating array of Proto-VEB Tree pointers of size sqrt(u)        // first all nulls are going to assign      this.clusters = new Array(this.Root(size)).fill(null).map(() => new ProtoVanEmdeBoas(this.Root(size)));    }  }   Root(u) {    return Math.floor(Math.sqrt(u));  }     // Function to return cluster numbers    // in which key is present  High(x) {    return Math.floor(x / this.Root(this.universeSize));  }   // Function to return position of x in cluster  Low(x) {    return x % this.Root(this.universeSize);  } // return the index from cluster number and position  GenerateIndex(cluster, position) {    return cluster * this.Root(this.universeSize) + position;  }}     // Function that returns true if the key is present in    // the treefunction IsMember(helper, key) {         // If key is greater than or equal to universeSize        // then return false  if (key >= helper.universeSize) {    return false;  }     // If we reach at base case    // the just return whether    // pointer is nullptr then false    // else return true  if (helper.universeSize === 2) {    return helper.clusters[key] !== null;  } 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 treefunction Insert(helper, key) {         // If we reach at base case    // then assign Proto-VEB(1) in place    // of nullptr  if (helper.universeSize === 2) {    helper.clusters[key] = new ProtoVanEmdeBoas(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 delete a key from the treefunction PvebDelete(helper, key) {         // Base case: If the key is present    // then make it nullptr  if (helper.universeSize === 2) {    if (helper.clusters[key] !== null) {      helper.clusters[key] = null;    }  } else {             // Recursive delete to reach at the base case    PvebDelete(helper.clusters[helper.High(key)], helper.Low(key));     let 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 (let i = helper.High(key) * helper.Root(helper.universeSize); i < (helper.High(key) + 1) * helper.Root(helper.universeSize); 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) {      PvebDelete(helper.summary, helper.High(key));    }  }} // Driver codeconst hello = new ProtoVanEmdeBoas(4); console.log(IsMember(hello, 2));Insert(hello, 2);Insert(hello, 3);console.log(IsMember(hello, 2));PvebDelete(hello, 2);console.log(IsMember(hello, 2));

## C#

 using System;using System.Collections.Generic; public class  ProtoVanEmdeBoas{    // Total number of keys   public int universeSize;   public  ProtoVanEmdeBoas summary;     // Clusters array of Proto-VEB pointers    public List< ProtoVanEmdeBoas> clusters;     public  ProtoVanEmdeBoas(int size)    {        universeSize = size;         // Base case        if (size <= 2)        {            // Set summary to null as there is no            // more summary for size 2            summary = null;            clusters = new List(size);            for (int i = 0; i < size; i++)            {                clusters.Add(null);            }        }        else        {            summary = new  ProtoVanEmdeBoas(Root(size));             // Creating array of Proto-VEB Tree pointers of size sqrt(u)            // first all nulls are going to assign            clusters = new List< ProtoVanEmdeBoas>(Root(size));             // Assigning Proto-VEB(sqrt(u)) to all its clusters            for (int i = 0; i < Root(size); i++)            {                clusters.Add(new ProtoVanEmdeBoas(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(universeSize);    }    public int Low(int x)    {        return x % Root(universeSize);    }     // return the index from cluster number and position    int GenerateIndex(int cluster, int position)    {        return cluster * Root(universeSize) + position;    }} public class Solution{        // Function that returns true if the key is present in    // the tree    static bool IsMember(ProtoVanEmdeBoas helper, int key)    {         // If key is greater than or equal to universeSize        // then return false        if (key >= helper.universeSize)        {            return false;        }         if (helper.universeSize == 2)        {            return helper.clusters[key] != null;        }        else        {            return IsMember(                helper.clusters[helper.High(key)],                helper.Low(key));        }    }   // Function to insert a key in the tree    static void Insert(ProtoVanEmdeBoas helper, int key)    {        if (helper.universeSize == 2)        {            helper.clusters[key] = new ProtoVanEmdeBoas(1);        }        else        {            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 delete a key from the tree    public static void PvebDelete( ProtoVanEmdeBoas helper, int key)    {        // If the key is present then make it null        if (helper.universeSize == 2)        {            if (helper.clusters[key] != null)            {                helper.clusters[key] = null;            }        }        else        {            PvebDelete(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.universeSize);                i < (helper.High(key) + 1) * helper.Root(helper.universeSize);                i++)            {                if (IsMember(helper.clusters[helper.High(key)], i))                {                    isanyinCluster = true;                    break;                }            }             // If no member is present then update summary to zero            if (!isanyinCluster)            {                PvebDelete(helper.summary, helper.High(key));            }        }    } static void Main()    {        ProtoVanEmdeBoas hello = new ProtoVanEmdeBoas(4);         // checking is member or not by calling isMember func.        Console.WriteLine(IsMember(hello, 2));         // inserting        Insert(hello, 2);        Insert(hello, 3);  Console.WriteLine(IsMember(hello, 2));       // deleting    PvebDelete(hello, 2);         // checking is member or not by calling isMember func.     Console.WriteLine(IsMember(hello, 2));}}

Recurrence Relation for Delete:

T(u) = T(u) = 2T()) + O(log2())

Time Complexity : O(log2(u)*log2(log2(u)))