Dada una matriz ordenada de números enteros positivos distintos, imprima todos los tripletes que forman la progresión geométrica con razón común integral.
Una progresión geométrica es una secuencia de números donde cada término después del primero se encuentra multiplicando el anterior por un número fijo distinto de cero llamado razón común. Por ejemplo, la secuencia 2 6 18 54... es una progresión geométrica con razón común 3.
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Ejemplos:
Input: arr = [1 2 6 10 18 54] Output: 2 6 18 6 18 54 Input: arr = [2 8 10 15 16 30 32 64] Output: 2 8 32 8 16 32 16 32 64 Input: arr = [ 1 2 6 18 36 54] Output: 2 6 18 1 6 36 6 18 54
La idea es comenzar desde el segundo elemento y fijar cada elemento como elemento intermedio y buscar los otros dos elementos en un triplete (uno más pequeño y otro más grande). Para que un elemento arr[j] esté en medio de una progresión geométrica, deben existir elementos arr[i] y arr[k] tales que:
arr[j] / arr[i] = r and arr[k] / arr[j] = r where r is an positive integer and 0 <= i < j and j < k <= n - 1
A continuación se muestra la implementación de la idea anterior.
C++// C++ program to find if there exist three elements in // Geometric Progression or not #include
// Java program to find if there exist three elements in // Geometric Progression or not import java.util.*; class GFG { // The function prints three elements in GP if exists // Assumption: arr[0..n-1] is sorted. static void findGeometricTriplets(int arr[] int n) { // One by fix every element as middle element for (int j = 1; j < n - 1; j++) { // Initialize i and k for the current j int i = j - 1 k = j + 1; // Find all i and k such that (i j k) // forms a triplet of GP while (i >= 0 && k <= n - 1) { // if arr[j]/arr[i] = r and arr[k]/arr[j] = r // and r is an integer (i j k) forms Geometric // Progression while (i >= 0 && arr[j] % arr[i] == 0 && arr[k] % arr[j] == 0 && arr[j] / arr[i] == arr[k] / arr[j]) { // print the triplet System.out.println(arr[i] +' ' + arr[j] + ' ' + arr[k]); // Since the array is sorted and elements // are distinct. k++ ; i--; } // if arr[j] is multiple of arr[i] and arr[k] is // multiple of arr[j] then arr[j] / arr[i] != // arr[k] / arr[j]. We compare their values to // move to next k or previous i. if(i >= 0 && arr[j] % arr[i] == 0 && arr[k] % arr[j] == 0) { if(i >= 0 && arr[j] / arr[i] < arr[k] / arr[j]) i--; else k++; } // else if arr[j] is multiple of arr[i] then // try next k. Else try previous i. else if (i >= 0 && arr[j] % arr[i] == 0) k++; else i--; } } } // Driver code public static void main(String[] args) { // int arr[] = {1 2 6 10 18 54}; // int arr[] = {2 8 10 15 16 30 32 64}; // int arr[] = {1 2 6 18 36 54}; int arr[] = {1 2 4 16}; // int arr[] = {1 2 3 6 18 22}; int n = arr.length; findGeometricTriplets(arr n); } } // This code is contributed by Rajput-Ji
# Python 3 program to find if # there exist three elements in # Geometric Progression or not # The function prints three elements # in GP if exists. # Assumption: arr[0..n-1] is sorted. def findGeometricTriplets(arr n): # One by fix every element # as middle element for j in range(1 n - 1): # Initialize i and k for # the current j i = j - 1 k = j + 1 # Find all i and k such that # (i j k) forms a triplet of GP while (i >= 0 and k <= n - 1): # if arr[j]/arr[i] = r and # arr[k]/arr[j] = r and r # is an integer (i j k) forms # Geometric Progression while (arr[j] % arr[i] == 0 and arr[k] % arr[j] == 0 and arr[j] // arr[i] == arr[k] // arr[j]): # print the triplet print( arr[i] ' ' arr[j] ' ' arr[k]) # Since the array is sorted and # elements are distinct. k += 1 i -= 1 # if arr[j] is multiple of arr[i] # and arr[k] is multiple of arr[j] # then arr[j] / arr[i] != arr[k] / arr[j]. # We compare their values to # move to next k or previous i. if(arr[j] % arr[i] == 0 and arr[k] % arr[j] == 0): if(arr[j] // arr[i] < arr[k] // arr[j]): i -= 1 else: k += 1 # else if arr[j] is multiple of # arr[i] then try next k. Else # try previous i. elif (arr[j] % arr[i] == 0): k += 1 else: i -= 1 # Driver code if __name__ =='__main__': arr = [1 2 4 16] n = len(arr) findGeometricTriplets(arr n) # This code is contributed # by ChitraNayal
// C# program to find if there exist three elements // in Geometric Progression or not using System; class GFG { // The function prints three elements in GP if exists // Assumption: arr[0..n-1] is sorted. static void findGeometricTriplets(int []arr int n) { // One by fix every element as middle element for (int j = 1; j < n - 1; j++) { // Initialize i and k for the current j int i = j - 1 k = j + 1; // Find all i and k such that (i j k) // forms a triplet of GP while (i >= 0 && k <= n - 1) { // if arr[j]/arr[i] = r and arr[k]/arr[j] = r // and r is an integer (i j k) forms Geometric // Progression while (i >= 0 && arr[j] % arr[i] == 0 && arr[k] % arr[j] == 0 && arr[j] / arr[i] == arr[k] / arr[j]) { // print the triplet Console.WriteLine(arr[i] +' ' + arr[j] + ' ' + arr[k]); // Since the array is sorted and elements // are distinct. k++ ; i--; } // if arr[j] is multiple of arr[i] and arr[k] is // multiple of arr[j] then arr[j] / arr[i] != // arr[k] / arr[j]. We compare their values to // move to next k or previous i. if(i >= 0 && arr[j] % arr[i] == 0 && arr[k] % arr[j] == 0) { if(i >= 0 && arr[j] / arr[i] < arr[k] / arr[j]) i--; else k++; } // else if arr[j] is multiple of arr[i] then // try next k. Else try previous i. else if (i >= 0 && arr[j] % arr[i] == 0) k++; else i--; } } } // Driver code static public void Main () { // int arr[] = {1 2 6 10 18 54}; // int arr[] = {2 8 10 15 16 30 32 64}; // int arr[] = {1 2 6 18 36 54}; int []arr = {1 2 4 16}; // int arr[] = {1 2 3 6 18 22}; int n = arr.Length; findGeometricTriplets(arr n); } } // This code is contributed by ajit.
<script> // Javascript program to find if there exist three elements in // Geometric Progression or not // The function prints three elements in GP if exists // Assumption: arr[0..n-1] is sorted. function findGeometricTriplets(arrn) { // One by fix every element as middle element for (let j = 1; j < n - 1; j++) { // Initialize i and k for the current j let i = j - 1 k = j + 1; // Find all i and k such that (i j k) // forms a triplet of GP while (i >= 0 && k <= n - 1) { // if arr[j]/arr[i] = r and arr[k]/arr[j] = r // and r is an integer (i j k) forms Geometric // Progression while (i >= 0 && arr[j] % arr[i] == 0 && arr[k] % arr[j] == 0 && arr[j] / arr[i] == arr[k] / arr[j]) { // print the triplet document.write(arr[i] +' ' + arr[j] + ' ' + arr[k]+'
'); // Since the array is sorted and elements // are distinct. k++ ; i--; } // if arr[j] is multiple of arr[i] and arr[k] is // multiple of arr[j] then arr[j] / arr[i] != // arr[k] / arr[j]. We compare their values to // move to next k or previous i. if(i >= 0 && arr[j] % arr[i] == 0 && arr[k] % arr[j] == 0) { if(i >= 0 && arr[j] / arr[i] < arr[k] / arr[j]) i--; else k++; } // else if arr[j] is multiple of arr[i] then // try next k. Else try previous i. else if (i >= 0 && arr[j] % arr[i] == 0) k++; else i--; } } } // Driver code // int arr[] = {1 2 6 10 18 54}; // int arr[] = {2 8 10 15 16 30 32 64}; // int arr[] = {1 2 6 18 36 54}; let arr = [1 2 4 16]; // int arr[] = {1 2 3 6 18 22}; let n = arr.length; findGeometricTriplets(arr n); // This code is contributed by avanitrachhadiya2155 </script>
Producción
1 2 4 1 4 16
Complejidad del tiempo de la solución anterior es O(n2) como para cada j encontramos i y k en tiempo lineal.
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Espacio auxiliar: O(1) ya que no utilizamos ningún espacio extra.