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

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

Procedure for finding minimum:

1. 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.
2. 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++

 // 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 return the minimum key from the treeint 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 treeint 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 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 << boolalpha;     insert(hello, 2);     insert(hello, 3);     cout << minimum(hello) << endl;     cout << maximum(hello) << endl;}

## Java

 // Java implementationimport 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 clusters;     public Proto_Van_Emde_Boas(int size)    {        universe_size = size;        if (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));            clusters = new ArrayList(                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));    }}

## Python3

 # Python implementation of the programimport 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 treedef 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 minimumdef 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 maximumdef 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 codehello = Proto_Van_Emde_Boas(4) # insertinginsert(hello, 2);insert(hello, 3); print(minimum(hello))print(maximum(hello))

## Javascript

 // 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 treefunction 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 minimumfunction 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 maximumfunction 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 codelet hello = new Proto_Van_Emde_Boas(4); // insertinginsert(hello, 2);insert(hello, 3); console.log(minimum(hello));console.log(maximum(hello));

## C#

 using System;using System.Collections.Generic; public class Proto_Van_Emde_Boas {    public int universe_size;    public Proto_Van_Emde_Boas summary;    public List clusters;     public Proto_Van_Emde_Boas(int size)    {        universe_size = size;        if (size <= 2) {            summary = null;            clusters = new List(size);            for (int i = 0; i < size; i++) {                clusters.Add(null);            }        }        else {            summary = new Proto_Van_Emde_Boas(root(size));            clusters                = new List(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)

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