Trobeu tots els triplets en una matriu ordenada que forma la progressió geomètrica
Donada una matriu ordenada de nombres enters positius diferents, imprimiu tots els triplets que formen la progressió geomètrica amb una relació comuna integral.
Una progressió geomètrica és una seqüència de nombres on cada terme després del primer es troba multiplicant l'anterior per un nombre fix diferent de zero anomenat raó comuna. Per exemple, la seqüència 2 6 18 54... és una progressió geomètrica amb una raó comuna 3.
Exemples:
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 és partir del segon element i fixar cada element com a element mitjà i buscar els altres dos elements en un triplet (un més petit i un altre més gran). Perquè un element arr[j] sigui el mig de la progressió geomètrica ha d'existir elements arr[i] i arr[k] tals 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 continuació es mostra la implementació de la idea anterior
C++ // C++ program to find if there exist three elements in // Geometric Progression or not #include using namespace std ; // The function prints three elements in GP if exists // Assumption: arr[0..n-1] is sorted. 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 ( arr [ j ] % arr [ i ] == 0 && arr [ k ] % arr [ j ] == 0 && arr [ j ] / arr [ i ] == arr [ k ] / arr [ j ]) { // print the triplet cout < < arr [ i ] < < ' ' < < arr [ j ] < < ' ' < < arr [ k ] < < endl ; // 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 ( arr [ j ] % arr [ i ] == 0 && arr [ k ] % arr [ j ] == 0 ) { if ( 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 ( arr [ j ] % arr [ i ] == 0 ) k ++ ; else i -- ; } } } // Driver code int 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 = sizeof ( arr ) / sizeof ( arr [ 0 ]); findGeometricTriplets ( arr n ); return 0 ; }
Java // 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 # 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# // 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.
JavaScript < 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 ( arr n ) { // 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>
Sortida
1 2 4 1 4 16
Complexitat temporal de la solució anterior és O(n 2 ) ja que per a cada j estem trobant i i k en temps lineal.
Espai auxiliar: O(1) ja que no hem utilitzat cap espai addicional.
