Minimum number of towers required such that every house is in the range of at least one tower

Given a map of the city and the network range, the task is to determine the minimum number of the tower so that every house is within range of at least one tower. Each tower must be installed on top of an existing house.

Examples:

Input: range : 1
       house : 1 2 3 4 5
Output: 2

Input: range : 2
       house : 7 2 4 6 5 9 12 11 
Output: 3


All cities can be covered by inserting 2 towers i.e. at house 2 and 4.


All cities can be covered by inserting 3 towers i.e. at house 4, 9 and 12.

Algorithm:-

  1. First, sort all the element.
  2. Count only once and then traverse till its middle house.
  3. After this again traverse till tower range.
  4. Again repeat 1, 2, 3 step till all the houses are covered.

Below is the implementation of above approach:

C++

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// C++ implementation of above approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to count the number of tower
int number_of_tower(int house[], int range, int n)
{
  
    // first we sort the house numbers
    sort(house, house + n);
  
    // for count number of twoers
    int numOfTower = 0;
  
    // for iterate all houses
    int i = 0;
    while (i < n) {
  
        // count number of towers
        numOfTower++;
  
        // find find the middle location
        int loc = house[i] + range;
  
        // traverse till middle location
        while (i < n && house[i] <= loc)
            i++;
  
        // this is point to middle
        // house where we insert the tower
        --i;
  
        // now find the last location
        loc = house[i] + range;
  
        // traverse till last house of the range
        while (i < n && house[i] <= loc)
            i++;
    }
  
    // return the number of tower
    return numOfTower;
}
  
// Driver code
int main()
{
    // given elements
    int house[] = { 7, 2, 4, 6, 5, 9, 12, 11 };
    int range = 2;
    int n = sizeof(house) / sizeof(house[0]);
  
    // print number of towers
    cout << number_of_tower(house, range, n);
}

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Java

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// Java implementation of above approach
  
import java.util.Arrays;
  
public class Improve {
  
    // Function to count the number of tower
    static int number_of_tower(int house[], int range, int n)
    {
  
        // first we sort the house numbers
        Arrays.sort(house);
  
        // for count number of towers
        int numOfTower = 0;
  
        // for iterate all houses
        int i = 0;
        while (i < n) {
  
            // count number of towers
            numOfTower++;
  
            // find find the middle location
            int loc = house[i] + range;
  
            // traverse till middle location
            while (i < n && house[i] <= loc)
                i++;
  
            // this is point to middle
            // house where we insert the tower
            --i;
  
            // now find the last location
            loc = house[i] + range;
  
            // traverse till last house of the range
            while (i < n && house[i] <= loc)
                i++;
        }
  
        // return the number of tower
        return numOfTower;
    }
  
  
      
    public static void main(String args[])
    {
        // given elements
        int house[] = { 7, 2, 4, 6, 5, 9, 12, 11 };
        int range = 2;
        int n = house.length;
  
        // print number of towers
        System.out.println(number_of_tower(house, range, n));
  
          
  
    }
    // This code is contributed by ANKITRAI1
}
  

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Python 3

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# Python 3 implementation of 
# above approach
  
# Function to count the
# number of tower
def number_of_tower(house, r, n):
  
    # first we sort the house numbers
    house.sort()
  
    # for count number of twoers
    numOfTower = 0
  
    # for iterate all houses
    i = 0
    while (i < n) :
  
        # count number of towers
        numOfTower += 1
  
        # find find the middle location
        loc = house[i] + r
  
        # traverse till middle location
        while (i < n and house[i] <= loc):
            i += 1
  
        # this is point to middle
        # house where we insert the tower
        i -= 1
  
        # now find the last location
        loc = house[i] + r
  
        # traverse till last house 
        # of the range
        while (i < n and house[i] <= loc):
            i += 1
  
    # return the number of tower
    return numOfTower
  
# Driver code
if __name__ == "__main__":
      
    # given elements
    house = [ 7, 2, 4, 6, 5, 9, 12, 11 ]
    r = 2
    n = len(house)
  
    # print number of towers
    print(number_of_tower(house, r, n))
  
# This code is contributed
# by ChitraNayal

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C#

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// C# implementation of above approach
  
using System;
  
public class Improve {
  
    // Function to count the number of tower
    static int number_of_tower(int []house, int range, int n)
    {
  
        // first we sort the house numbers
        Array.Sort(house);
  
        // for count number of towers
        int numOfTower = 0;
  
        // for iterate all houses
        int i = 0;
        while (i < n) {
  
            // count number of towers
            numOfTower++;
  
            // find find the middle location
            int loc = house[i] + range;
  
            // traverse till middle location
            while (i < n && house[i] <= loc)
                i++;
  
            // this is point to middle
            // house where we insert the tower
            --i;
  
            // now find the last location
            loc = house[i] + range;
  
            // traverse till last house of the range
            while (i < n && house[i] <= loc)
                i++;
        }
  
        // return the number of tower
        return numOfTower;
    }
  
  
      
    public static void Main()
    {
        // given elements
        int []house = { 7, 2, 4, 6, 5, 9, 12, 11 };
        int range = 2;
        int n = house.Length;
  
        // print number of towers
        Console.WriteLine(number_of_tower(house, range, n));
  
          
 // This code is contributed by inder_verma..
    }
      
}

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PHP

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<?php
//PHP implementation of above approach 
  
// Function to count the number of tower 
function  number_of_tower($house, $range, $n
  
    // first we sort the house numbers 
    sort($house); 
  
    // for count number of twoers 
    $numOfTower = 0; 
  
    // for iterate all houses 
    $i = 0; 
    while ($i < $n) { 
  
        // count number of towers 
        $numOfTower++; 
  
        // find find the middle location 
        $loc = $house[$i] + $range
  
        // traverse till middle location 
        while ($i < $n && $house[$i] <= $loc
            $i++; 
  
        // this is point to middle 
        // house where we insert the tower 
        --$i
  
        // now find the last location 
        $loc = $house[$i] + $range
  
        // traverse till last house of the range 
        while ($i < $n && $house[$i] <= $loc
            $i++; 
    
  
    // return the number of tower 
    return $numOfTower
  
// Driver code 
  
    // given elements 
    $house = array( 7, 2, 4, 6, 5, 9, 12, 11 ); 
    $range = 2; 
    $n = sizeof($house) / sizeof($house[0]); 
  
    // print number of towers 
    echo  number_of_tower($house, $range, $n); 
  
// This code is contributed by Sachin.
?>

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Output:

3

Time Complexity: O(nlogn)
Space Complextity: O(1)



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