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Ternary representation of Cantor set
  • Last Updated : 25 Mar, 2020

Given three integers A, B and L, the task is to print the ternary cantor set from range [A, B] upto L levels.

Ternary Cantor Set: A ternary Cantor set is a set built by removing the middle part of a line segment when divided into 3 parts and repeating this process with the remaining shorter segments. Below is an illustration of a cantor set.

Cantor Set

An illustration of a Ternary Cantor Set

Examples:

Input: A = 0, B = 1, L = 2
Output:
Level 0: [0.000000] — [1.000000]
Level 1: [0.000000] — [0.333333] [0.666667] — [1.000000]
Level 2: [0.000000] — [0.111111] [0.222222] — [0.333333] [0.666667] — [0.777778] [0.888889] — [1.000000]
Explanation: For the given range [0, 1], in level 1, it is divided into three parts ([0, 0.33], [0.33, 0.67], [0.67, 1]). From the three parts, the middle part is ignored. This process is continued for every part in the subsequent executions.

Input: A = 0, B = 9, L = 3
Output:
Level_0: [0.000000] — [9.000000]
Level_1: [0.000000] — [3.000000] [6.000000] — [9.000000]
Level_2: [0.000000] — [1.000000] [2.000000] — [3.000000] [6.000000] — [7.000000] [8.000000] — [9.000000]
Level_3: [0.000000] — [0.333333] [0.666667] — [1.000000] [2.000000] — [2.333333] [2.666667] — [3.000000] [6.000000] — [6.333333] [6.666667] — [7.000000] [8.000000] — [8.333333] [8.666667] — [9.000000]



Approach:

  1. Create a linked list data structure for each node of the Set, having the start value, end value and a pointer to the next node.
  2. Initialize the list with the start and end value given as the input.
  3. For the next level:
    • Create a new node where the difference between the start and end values is  \frac{1}{3} rd of the initial, i.e. start value is  \frac{1}{3} rd less than the initial end value.
    • Further, modify the original node, such that the end value is  \frac{1}{3} rd more of the initial start value.
    • Place the pointer to the new node after the original one accordingly

Below is the implementation of the above approach:

C++




// C++ implementation to find the cantor set
// for n levels and
// for a given start_num and end_num
#include <bits/stdc++.h>
using namespace std;
  
// The Linked List Structure for the Cantor Set
typedef struct cantor {
    double start, end;
    struct cantor* next;
} Cantor;
  
// Function to initialize the Cantor Set List
Cantor* startList(Cantor* head,
                double start_num,
                double end_num)
{
    if (head == NULL) {
        head = new Cantor;
        head->start = start_num;
        head->end = end_num;
        head->next = NULL;
    }
    return head;
}
  
// Function to propogate the list
// by adding new nodes for the next levels
Cantor* propagate(Cantor* head)
{
    Cantor* temp = head;
  
    if (temp != NULL) {
        Cantor* newNode
            = new Cantor;
        double diff
            = (((temp->end) - (temp->start)) / 3);
  
        // Modifying the start and end values
        // for the next level
        newNode->end = temp->end;
        temp->end = ((temp->start) + diff);
        newNode->start = (newNode->end) - diff;
  
        // Changing the pointers
        // to the next node
        newNode->next = temp->next;
        temp->next = newNode;
  
        // Recursively call the function
        // to generate the Cantor Set
        // for the entire level
        propagate(temp->next->next);
    }
  
    return head;
}
  
// Function to print a level of the Set
void print(Cantor* temp)
{
    while (temp != NULL) {
        printf("[%lf] -- [%lf]\t",
            temp->start, temp->end);
        temp = temp->next;
    }
    cout << endl;
}
  
// Function to build and display
// the Cantor Set for each level
void buildCantorSet(int A, int B, int L)
{
    Cantor* head = NULL;
    head = startList(head, A, B);
    for (int i = 0; i < L; i++) {
        cout <<"Level_"<< i<<" : ";
        print(head);
        propagate(head);
    }
    cout <<"Level_"<< L<<" : ";
    print(head);
}
  
// Driver code
int main()
{
    int A = 0;
    int B = 9;
    int L = 2;
    buildCantorSet(A, B, L);
  
    return 0;
}
  
// This code is contributed by shivanisingh

C




// C implementation to find the cantor set
// for n levels and
// for a given start_num and end_num
  
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
  
// The Linked List Structure for the Cantor Set
typedef struct cantor {
    double start, end;
    struct cantor* next;
} Cantor;
  
// Function to initialize the Cantor Set List
Cantor* startList(Cantor* head,
                  double start_num,
                  double end_num)
{
    if (head == NULL) {
        head = (Cantor*)malloc(sizeof(Cantor));
        head->start = start_num;
        head->end = end_num;
        head->next = NULL;
    }
    return head;
}
  
// Function to propogate the list
// by adding new nodes for the next levels
Cantor* propagate(Cantor* head)
{
    Cantor* temp = head;
  
    if (temp != NULL) {
        Cantor* newNode
            = (Cantor*)malloc(sizeof(Cantor));
        double diff
            = (((temp->end) - (temp->start)) / 3);
  
        // Modifying the start and end values
        // for the next level
        newNode->end = temp->end;
        temp->end = ((temp->start) + diff);
        newNode->start = (newNode->end) - diff;
  
        // Changing the pointers
        // to the next node
        newNode->next = temp->next;
        temp->next = newNode;
  
        // Recursively call the function
        // to generate the Cantor Set
        // for the entire level
        propagate(temp->next->next);
    }
  
    return head;
}
  
// Function to print a level of the Set
void print(Cantor* temp)
{
    while (temp != NULL) {
        printf("[%lf] -- [%lf]\t",
               temp->start, temp->end);
        temp = temp->next;
    }
    printf("\n");
}
  
// Function to build and display
// the Cantor Set for each level
void buildCantorSet(int A, int B, int L)
{
    Cantor* head = NULL;
    head = startList(head, A, B);
    for (int i = 0; i < L; i++) {
        printf("Level_%d : ", i);
        print(head);
        propagate(head);
    }
    printf("Level_%d : ", L);
    print(head);
}
  
// Driver code
int main()
{
    int A = 0;
    int B = 9;
    int L = 2;
    buildCantorSet(A, B, L);
  
    return 0;
}

Java




// Java implementation to find the cantor set
// for n levels and
// for a given start_num and end_num
  
class GFG
{
  
    // The Linked List Structure for the Cantor Set
    static class Cantor
    {
        double start, end;
        Cantor next;
    };
  
    static Cantor Cantor;
  
    // Function to initialize the Cantor Set List
    static Cantor startList(Cantor head, double start_num, 
                            double end_num)
    {
        if (head == null
        {
            head = new Cantor();
            head.start = start_num;
            head.end = end_num;
            head.next = null;
        }
        return head;
    }
  
    // Function to propogate the list
    // by adding new nodes for the next levels
    static Cantor propagate(Cantor head) 
    {
        Cantor temp = head;
  
        if (temp != null)
        {
            Cantor newNode = new Cantor();
            double diff = (((temp.end) - (temp.start)) / 3);
  
            // Modifying the start and end values
            // for the next level
            newNode.end = temp.end;
            temp.end = ((temp.start) + diff);
            newNode.start = (newNode.end) - diff;
  
            // Changing the pointers
            // to the next node
            newNode.next = temp.next;
            temp.next = newNode;
  
            // Recursively call the function
            // to generate the Cantor Set
            // for the entire level
            propagate(temp.next.next);
        }
  
        return head;
    }
  
    // Function to print a level of the Set
    static void print(Cantor temp)
    {
        while (temp != null
        {
            System.out.printf("[%f] -- [%f]", temp.start, temp.end);
            temp = temp.next;
        }
        System.out.printf("\n");
    }
  
    // Function to build and display
    // the Cantor Set for each level
    static void buildCantorSet(int A, int B, int L)
    {
        Cantor head = null;
        head = startList(head, A, B);
        for (int i = 0; i < L; i++) 
        {
            System.out.printf("Level_%d : ", i);
            print(head);
            propagate(head);
        }
        System.out.printf("Level_%d : ", L);
        print(head);
    }
  
    // Driver code
    public static void main(String[] args) 
    {
        int A = 0;
        int B = 9;
        int L = 2;
        buildCantorSet(A, B, L);
    }
}
  
// This code is contributed by Rajput-Ji

C#




// C# implementation to find the cantor set
// for n levels and
// for a given start_num and end_num
using System;
  
class GFG
{
  
    // The Linked List Structure for the Cantor Set
    class Cantor
    {
        public double start, end;
        public Cantor next;
    };
  
    static Cantor cantor;
  
    // Function to initialize the Cantor Set List
    static Cantor startList(Cantor head, double start_num, 
                            double end_num)
    {
        if (head == null
        {
            head = new Cantor();
            head.start = start_num;
            head.end = end_num;
            head.next = null;
        }
        return head;
    }
  
    // Function to propogate the list
    // by adding new nodes for the next levels
    static Cantor propagate(Cantor head) 
    {
        Cantor temp = head;
  
        if (temp != null)
        {
            Cantor newNode = new Cantor();
            double diff = (((temp.end) - (temp.start)) / 3);
  
            // Modifying the start and end values
            // for the next level
            newNode.end = temp.end;
            temp.end = ((temp.start) + diff);
            newNode.start = (newNode.end) - diff;
  
            // Changing the pointers
            // to the next node
            newNode.next = temp.next;
            temp.next = newNode;
  
            // Recursively call the function
            // to generate the Cantor Set
            // for the entire level
            propagate(temp.next.next);
        }
  
        return head;
    }
  
    // Function to print a level of the Set
    static void print(Cantor temp)
    {
        while (temp != null
        {
            Console.Write("[{0:F6}] -- [{1:F6}]"
                            temp.start, temp.end);
            temp = temp.next;
        }
        Console.Write("\n");
    }
  
    // Function to build and display
    // the Cantor Set for each level
    static void buildCantorSet(int A, int B, int L)
    {
        Cantor head = null;
        head = startList(head, A, B);
        for (int i = 0; i < L; i++) 
        {
            Console.Write("Level_{0} : ", i);
            print(head);
            propagate(head);
        }
        Console.Write("Level_{0} : ", L);
        print(head);
    }
  
    // Driver code
    public static void Main(String[] args) 
    {
        int A = 0;
        int B = 9;
        int L = 2;
        buildCantorSet(A, B, L);
    }
}
  
// This code is contributed by Rajput-Ji
Output:

Level_0 : [0.000000] — [9.000000]
Level_1 : [0.000000] — [3.000000] [6.000000] — [9.000000]
Level_2 : [0.000000] — [1.000000] [2.000000] — [3.000000] [6.000000] — [7.000000] [8.000000] — [9.000000]

References: Cantor Set Wikipedia
Related Article: N-th term of George Cantor set of rational numbers

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