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מצא את כל השלשות במערך ממוין שיוצר התקדמות גיאומטרית

בהינתן מערך ממוין של מספרים שלמים חיוביים ברורים הדפס את כל השלשות היוצרות התקדמות גיאומטרית עם יחס משותף אינטגרלי.
התקדמות גיאומטרית היא רצף של מספרים שבו כל איבר אחרי הראשון נמצא על ידי הכפלת הקודם במספר קבוע שאינו אפס הנקרא היחס המשותף. לדוגמה, הרצף 2 6 18 54... הוא התקדמות גיאומטרית עם יחס משותף 3.

דוגמאות:  



  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

הרעיון הוא להתחיל מהאלמנט השני ולתקן כל אלמנט כאלמנט אמצעי ולחפש את שני האלמנטים האחרים בשלישייה (אחד קטן ואחד גדול יותר). כדי שאלמנט arr[j] יהיה באמצע ההתקדמות הגיאומטרית חייבים להתקיים אלמנטים arr[i] ו-arr[k] כך ש- 

  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

להלן יישום הרעיון לעיל

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(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>

תְפוּקָה
1 2 4 1 4 16

מורכבות הזמן מהפתרון לעיל הוא O(n2) לגבי כל j אנו מוצאים את i ו-k בזמן ליניארי.



מרחב עזר: O(1) מכיוון שלא השתמשנו בשטח נוסף.