Proto Van Emde Boas Tree | Set 5 | Queries: Minimum, Maximum

Last Updated : 16 Oct, 2019

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:

VEB minimum query

See the image below for real Proto – VEB minimum operation:

Follow the instructions in order of numberings.
Minimum Query : Proto-VEB

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:

Maximum Query in Proto-VEB

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 { 
  
            // Assiging 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 recusion 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; 
} 

Both Minimum and Maximum query runs in O(log2(u)) time complexity.

Recurrence Relation:

T(u) = 2T() + O(1)

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