logo

Konvertējiet bināro koku par apļveida dubultsaišu sarakstu

Izmēģiniet to GfG Practice koku sarakstā' title= #practiceLinkDiv { display: none !important; }

Ņemot vērā a Binārais koks pārvērst to par a Apļveida divkārši saistīts saraksts (Vietā).  

  • Kreisais un labais norādes mezglos ir jāizmanto attiecīgi kā iepriekšējās un nākamās norādes konvertētajā apļveida saistīto sarakstā.
  • Mezglu secībai sarakstā ir jābūt tādai pašai kā Inorder dotajam binārajam kokam.
  • Pirmajam Inorder traversal mezglam ir jābūt apļveida saraksta galvenajam mezglam.

Piemēri:



' title=

Ubuntu uzbūve ir būtiska
Ieteicamā prakse Binārais koks uz CDLL Izmēģiniet to!

Konvertējiet bināro koku par apļveida dubultsaišu sarakstu, izmantojot rekursiju:

Ideja ir izveidot vispārējas nozīmes funkciju, kas savieno divus dotos apļveida divkāršos sarakstus

Lai atrisinātu problēmu, veiciet tālāk norādītās darbības.



  • Rekursīvi pārveidojiet kreiso apakškoku par apļveida DLL. Ļaujiet konvertētajam sarakstam būt pa kreisi Saraksts .
  • Rekursīvi konvertējiet labo apakškoku par apļveida DLL. Ļaujiet konvertētajam sarakstam būt RightList
  • Izveidojiet apļveida sarakstu ar koka saknēm un izveidojiet kreiso un labo sakņu punktus sev. 
  • Savienot pa kreisi Saraksts ar viena saknes mezgla sarakstu. 
  • Savienojiet iepriekš minētajā darbībā izveidoto sarakstu ar RightList .

Piezīme: Iepriekš minētā pieeja šķērso koku pēc pasūtījuma. Mēs varam šķērsot arī inorder veidā. Vispirms mēs varam savienot kreiso apakškoku un sakni, pēc tam atkārtot labo apakškoku un savienot rezultātu ar kreisās saknes savienošanu.

Kā savienot divus apļveida DLL?  

  • Iegūstiet kreisā saraksta pēdējo mezglu. Pēdējā mezgla izgūšana ir O(1) darbība, jo galviņas iepriekšējais rādītājs norāda uz saraksta pēdējo mezglu.
  • Savienojiet to ar labā saraksta pirmo mezglu
  • Iegūstiet otrā saraksta pēdējo mezglu
  • Savienojiet to ar saraksta galvu.

Tālāk ir norādītas iepriekš minētās idejas īstenošanas iespējas:



poo
C++
// C++ Program to convert a Binary Tree // to a Circular Doubly Linked List #include    using namespace std; // To represents a node of a Binary Tree struct Node {  struct Node *left *right;  int data; }; // A function that appends rightList at the end // of leftList. Node* concatenate(Node* leftList Node* rightList) {  // If either of the list is empty  // then return the other list  if (leftList == NULL)  return rightList;  if (rightList == NULL)  return leftList;  // Store the last Node of left List  Node* leftLast = leftList->left;  // Store the last Node of right List  Node* rightLast = rightList->left;  // Connect the last node of Left List  // with the first Node of the right List  leftLast->right = rightList;  rightList->left = leftLast;  // Left of first node points to  // the last node in the list  leftList->left = rightLast;  // Right of last node refers to the first  // node of the List  rightLast->right = leftList;  return leftList; } // Function converts a tree to a circular Linked List // and then returns the head of the Linked List Node* bTreeToCList(Node* root) {  if (root == NULL)  return NULL;  // Recursively convert left and right subtrees  Node* left = bTreeToCList(root->left);  Node* right = bTreeToCList(root->right);  // Make a circular linked list of single node  // (or root). To do so make the right and  // left pointers of this node point to itself  root->left = root->right = root;  // Step 1 (concatenate the left list with the list  // with single node i.e. current node)  // Step 2 (concatenate the returned list with the  // right List)  return concatenate(concatenate(left root) right); } // Display Circular Link List void displayCList(Node* head) {  cout << 'Circular Linked List is :n';  Node* itr = head;  do {  cout << itr->data << ' ';  itr = itr->right;  } while (head != itr);  cout << 'n'; } // Create a new Node and return its address Node* newNode(int data) {  Node* temp = new Node();  temp->data = data;  temp->left = temp->right = NULL;  return temp; } // Driver Program to test above function int main() {  Node* root = newNode(10);  root->left = newNode(12);  root->right = newNode(15);  root->left->left = newNode(25);  root->left->right = newNode(30);  root->right->left = newNode(36);  Node* head = bTreeToCList(root);  displayCList(head);  return 0; } // This code is contributed by Aditya Kumar (adityakumar129) 
C
// C Program to convert a Binary Tree // to a Circular Doubly Linked List #include  #include  // To represents a node of a Binary Tree typedef struct Node {  struct Node *left *right;  int data; } Node; // A function that appends rightList at the end // of leftList. Node* concatenate(Node* leftList Node* rightList) {  // If either of the list is empty  // then return the other list  if (leftList == NULL)  return rightList;  if (rightList == NULL)  return leftList;  // Store the last Node of left List  Node* leftLast = leftList->left;  // Store the last Node of right List  Node* rightLast = rightList->left;  // Connect the last node of Left List  // with the first Node of the right List  leftLast->right = rightList;  rightList->left = leftLast;  // Left of first node points to  // the last node in the list  leftList->left = rightLast;  // Right of last node refers to the first  // node of the List  rightLast->right = leftList;  return leftList; } // Function converts a tree to a circular Linked List // and then returns the head of the Linked List Node* bTreeToCList(Node* root) {  if (root == NULL)  return NULL;  // Recursively convert left and right subtrees  Node* left = bTreeToCList(root->left);  Node* right = bTreeToCList(root->right);  // Make a circular linked list of single node  // (or root). To do so make the right and  // left pointers of this node point to itself  root->left = root->right = root;  // Step 1 (concatenate the left list with the list  // with single node i.e. current node)  // Step 2 (concatenate the returned list with the  // right List)  return concatenate(concatenate(left root) right); } // Display Circular Link List void displayCList(Node* head) {  printf('Circular Linked List is :n');  Node* itr = head;  do {  printf('%d ' itr->data);  itr = itr->right;  } while (head != itr);  printf('n'); } // Create a new Node and return its address Node* newNode(int data) {  Node* temp = (Node*)malloc(sizeof(Node));  temp->data = data;  temp->left = temp->right = NULL;  return temp; } // Driver Program to test above function int main() {  Node* root = newNode(10);  root->left = newNode(12);  root->right = newNode(15);  root->left->left = newNode(25);  root->left->right = newNode(30);  root->right->left = newNode(36);  Node* head = bTreeToCList(root);  displayCList(head);  return 0; } // This code is contributed by Aditya Kumar (adityakumar129) 
Java
// Java Program to convert a Binary Tree to a // Circular Doubly Linked List // Node class represents a Node of a Tree class Node {  int val;  Node left right;  public Node(int val)  {  this.val = val;  left = right = null;  } } // A class to represent a tree class Tree {  Node root;  public Tree() { root = null; }  // concatenate both the lists and returns the head  // of the List  public Node concatenate(Node leftList Node rightList)  {  // If either of the list is empty then  // return the other list  if (leftList == null)  return rightList;  if (rightList == null)  return leftList;  // Store the last Node of left List  Node leftLast = leftList.left;  // Store the last Node of right List  Node rightLast = rightList.left;  // Connect the last node of Left List  // with the first Node of the right List  leftLast.right = rightList;  rightList.left = leftLast;  // left of first node refers to  // the last node in the list  leftList.left = rightLast;  // Right of last node refers to the first  // node of the List  rightLast.right = leftList;  // Return the Head of the List  return leftList;  }  // Method converts a tree to a circular  // Link List and then returns the head  // of the Link List  public Node bTreeToCList(Node root)  {  if (root == null)  return null;  // Recursively convert left and right subtrees  Node left = bTreeToCList(root.left);  Node right = bTreeToCList(root.right);  // Make a circular linked list of single node  // (or root). To do so make the right and  // left pointers of this node point to itself  root.left = root.right = root;  // Step 1 (concatenate the left list with the list  // with single node i.e. current node)  // Step 2 (concatenate the returned list with the  // right List)  return concatenate(concatenate(left root) right);  }  // Display Circular Link List  public void display(Node head)  {  System.out.println('Circular Linked List is :');  Node itr = head;  do {  System.out.print(itr.val + ' ');  itr = itr.right;  } while (itr != head);  System.out.println();  } } // Driver Code class Main {  public static void main(String args[])  {  // Build the tree  Tree tree = new Tree();  tree.root = new Node(10);  tree.root.left = new Node(12);  tree.root.right = new Node(15);  tree.root.left.left = new Node(25);  tree.root.left.right = new Node(30);  tree.root.right.left = new Node(36);  // head refers to the head of the Link List  Node head = tree.bTreeToCList(tree.root);  // Display the Circular LinkedList  tree.display(head);  } } 
Python3
# Python3 Program to convert a Binary # Tree to a Circular Doubly Linked List class newNode: def __init__(self data): self.data = data self.left = self.right = None # A function that appends rightList # at the end of leftList. def concatenate(leftList rightList): # If either of the list is empty # then return the other list if (leftList == None): return rightList if (rightList == None): return leftList # Store the last Node of left List leftLast = leftList.left # Store the last Node of right List rightLast = rightList.left # Connect the last node of Left List # with the first Node of the right List leftLast.right = rightList rightList.left = leftLast # Left of first node points to # the last node in the list leftList.left = rightLast # Right of last node refers to # the first node of the List rightLast.right = leftList return leftList # Function converts a tree to a circular # Linked List and then returns the head # of the Linked List def bTreeToCList(root): if (root == None): return None # Recursively convert left and # right subtrees left = bTreeToCList(root.left) right = bTreeToCList(root.right) # Make a circular linked list of single # node (or root). To do so make the # right and left pointers of this node # point to itself root.left = root.right = root # Step 1 (concatenate the left list # with the list with single # node i.e. current node) # Step 2 (concatenate the returned list # with the right List) return concatenate(concatenate(left root) right) # Display Circular Link List def displayCList(head): print('Circular Linked List is :') itr = head first = 1 while (head != itr or first): print(itr.data end=' ') itr = itr.right first = 0 print() # Driver Code if __name__ == '__main__': root = newNode(10) root.left = newNode(12) root.right = newNode(15) root.left.left = newNode(25) root.left.right = newNode(30) root.right.left = newNode(36) head = bTreeToCList(root) displayCList(head) # This code is contributed by PranchalK 
C#
// C# Program to convert a Binary Tree // to a Circular Doubly Linked List using System; // Node class represents a Node of a Tree public class Node {  public int val;  public Node left right;  public Node(int val)  {  this.val = val;  left = right = null;  } } // A class to represent a tree public class Tree {  internal Node root;  public Tree() { root = null; }  // concatenate both the lists  // and returns the head of the List  public virtual Node concatenate(Node leftList  Node rightList)  {  // If either of the list is empty  // then return the other list  if (leftList == null) {  return rightList;  }  if (rightList == null) {  return leftList;  }  // Store the last Node of left List  Node leftLast = leftList.left;  // Store the last Node of right List  Node rightLast = rightList.left;  // Connect the last node of Left List  // with the first Node of the right List  leftLast.right = rightList;  rightList.left = leftLast;  // left of first node refers to  // the last node in the list  leftList.left = rightLast;  // Right of last node refers to  // the first node of the List  rightLast.right = leftList;  // Return the Head of the List  return leftList;  }  // Method converts a tree to a circular  // Link List and then returns the head  // of the Link List  public virtual Node bTreeToCList(Node root)  {  if (root == null) {  return null;  }  // Recursively convert left  // and right subtrees  Node left = bTreeToCList(root.left);  Node right = bTreeToCList(root.right);  // Make a circular linked list of single  // node (or root). To do so make the  // right and left pointers of this node  // point to itself  root.left = root.right = root;  // Step 1 (concatenate the left list with  // the list with single node  // i.e. current node)  // Step 2 (concatenate the returned list  // with the right List)  return concatenate(concatenate(left root) right);  }  // Display Circular Link List  public virtual void display(Node head)  {  Console.WriteLine('Circular Linked List is :');  Node itr = head;  do {  Console.Write(itr.val + ' ');  itr = itr.right;  } while (itr != head);  Console.WriteLine();  } } // Driver Code public class GFG {  public static void Main(string[] args)  {  // Build the tree  Tree tree = new Tree();  tree.root = new Node(10);  tree.root.left = new Node(12);  tree.root.right = new Node(15);  tree.root.left.left = new Node(25);  tree.root.left.right = new Node(30);  tree.root.right.left = new Node(36);  // head refers to the head of the Link List  Node head = tree.bTreeToCList(tree.root);  // Display the Circular LinkedList  tree.display(head);  } } // This code is contributed by Shrikant13 
JavaScript
<script> // javascript Program to convert a Binary Tree to a // Circular Doubly Linked List // Node class represents a Node of a Tree class Node {  constructor(val) {  this.val = val;  this.left = null;  this.right = null;  } } // A class to represent a   var root = null;  // concatenate both the lists and returns the head  // of the List  function concatenate(leftList rightList) {  // If either of the list is empty then  // return the other list  if (leftList == null)  return rightList;  if (rightList == null)  return leftList;  // Store the last Node of left List  var leftLast = leftList.left;  // Store the last Node of right List  var rightLast = rightList.left;  // Connect the last node of Left List  // with the first Node of the right List  leftLast.right = rightList;  rightList.left = leftLast;  // left of first node refers to  // the last node in the list  leftList.left = rightLast;  // Right of last node refers to the first  // node of the List  rightLast.right = leftList;  // Return the Head of the List  return leftList;  }  // Method converts a to a circular  // Link List and then returns the head  // of the Link List  function bTreeToCList(root) {  if (root == null)  return null;  // Recursively convert left and right subtrees  var left = bTreeToCList(root.left);  var right = bTreeToCList(root.right);  // Make a circular linked list of single node  // (or root). To do so make the right and  // left pointers of this node point to itself  root.left = root.right = root;  // Step 1 (concatenate the left list with the list  // with single node i.e. current node)  // Step 2 (concatenate the returned list with the  // right List)  return concatenate(concatenate(left root) right);  }  // Display Circular Link List  function display(head) {  document.write('Circular Linked List is :  
'
); var itr = head; do { document.write(itr.val + ' '); itr = itr.right; } while (itr != head); document.write(); } // Driver Code // Build the root = new Node(10); root.left = new Node(12); root.right = new Node(15); root.left.left = new Node(25); root.left.right = new Node(30); root.right.left = new Node(36); // head refers to the head of the Link List var head = bTreeToCList(root); // Display the Circular LinkedList display(head); // This code contributed by umadevi9616 </script>

Izvade
Circular Linked List is : 25 12 30 10 36 15 

Laika sarežģītība: O(N) Tā kā katrs mezgls tiek apmeklēts ne vairāk kā vienu reizi.
Palīgtelpa: O(log N) Papildu vieta tiek izmantota rekursijas izsaukuma skurstenī, kas var pieaugt līdz maksimālajam logN izmēram, jo ​​tas ir binārs koks.

Pārvērtiet bināro koku par apļveida dubultsaišu sarakstu, izmantojot Inorder Traversal:

Ideja ir veikt binārā koka šķērsošanu kārtībā. Veicot inorder šķērsošanu, sekojiet līdzi iepriekš apmeklētajam mezglam mainīgā teikšanā iepriekj . Katram apmeklētajam mezglam padariet to par nākamo iepriekj un iestatiet šī mezgla iepriekšējo kā iepriekj .

Lai atrisinātu problēmu, veiciet tālāk norādītās darbības.

  • Vispirms konvertējiet bināro koku divkāršā sarakstā, skatiet šo ziņu Konvertējiet doto bināro koku uz divkārši saistītu sarakstu .
  • Tagad pārveidojiet šo divkārši saistīto sarakstu par apļveida dubultsaites sarakstu, savienojot pirmo un pēdējo mezglu.

Zemāk ir aprakstīta iepriekš minētās pieejas īstenošana.

java kārtošanas masīvu saraksts
C++
// A C++ program for in-place conversion of Binary Tree to // CDLL #include    using namespace std; /* A binary tree node has - data  left and right pointers  */ struct Node {  int data;  Node* left;  Node* right; }; // A utility function that converts given binary tree to // a doubly linked list // root --> the root of the binary tree // head --> head of the created doubly linked list Node* BTree2DoublyLinkedList(Node* root Node** head) {  // Base case  if (root == NULL)  return root;  // Initialize previously visited node as NULL. This is  // static so that the same value is accessible in all  // recursive calls  static Node* prev = NULL;  // Recursively convert left subtree  BTree2DoublyLinkedList(root->left head);  // Now convert this node  if (prev == NULL)  *head = root;  else {  root->left = prev;  prev->right = root;  }  prev = root;  // Finally convert right subtree  BTree2DoublyLinkedList(root->right head);  return prev; } // A simple recursive function to convert a given Binary // tree to Circular Doubly Linked List using a utility // function root --> Root of Binary Tree tail --> Pointer to // tail node of created circular doubly linked list Node* BTree2CircularDoublyLinkedList(Node* root) {  Node* head = NULL;  Node* tail = BTree2DoublyLinkedList(root &head);  // make the changes to convert a DLL to CDLL  tail->right = head;  head->left = tail;  // return the head of the created CDLL  return head; } /* Helper function that allocates a new node with the given data and NULL left and right pointers. */ Node* newNode(int data) {  Node* new_node = new Node;  new_node->data = data;  new_node->left = new_node->right = NULL;  return (new_node); } /* Function to print nodes in a given circular doubly linked  * list */ void printList(Node* head) {  if (head == NULL)  return;  Node* ptr = head;  do {  cout << ptr->data << ' ';  ptr = ptr->right;  } while (ptr != head); } /* Driver program to test above functions*/ int main() {  // Let us create the tree shown in above diagram  Node* root = newNode(10);  root->left = newNode(12);  root->right = newNode(15);  root->left->left = newNode(25);  root->left->right = newNode(30);  root->right->left = newNode(36);  // Convert to DLL  Node* head = BTree2CircularDoublyLinkedList(root);  // Print the converted list  printList(head);  return 0; } // This code was contributed by Abhijeet // Kumar(abhijeet19403) 
Java
// A Java program for in-place conversion of Binary Tree to // CDLL // A binary tree node has - data left pointer and right // pointer class Node {  int data;  Node left right;  public Node(int data)  {  this.data = data;  left = right = null;  } } class BinaryTree {  Node root;  // head --> Pointer to head node of created doubly  // linked list  Node head;  // Initialize previously visited node as NULL. This is  // static so that the same value is accessible in all  // recursive calls  static Node prev = null;  // A simple utility recursive function to convert a  // given Binary tree to Doubly Linked List root --> Root  // of Binary Tree  void BTree2DoublyLinkedList(Node root)  {  // Base case  if (root == null)  return;  // Recursively convert left subtree  BTree2DoublyLinkedList(root.left);  // Now convert this node  if (prev == null)  head = root;  else {  root.left = prev;  prev.right = root;  }  prev = root;  // Finally convert right subtree  BTree2DoublyLinkedList(root.right);  }  // A simple function to convert a given binary tree to  // Circular doubly linked list  // using a utility function  void BTree2CircularDoublyLinkedList(Node root)  {  BTree2DoublyLinkedList(root);  // make the changes to convert a DLL to CDLL  prev.right = head;  head.left = prev;  }  /* Function to print nodes in a given doubly linked list  */  void printList(Node node)  {  if (node == null)  return;  Node curr = node;  do {  System.out.print(curr.data + ' ');  curr = curr.right;  } while (curr != node);  }  // Driver program to test above functions  public static void main(String[] args)  {  // Let us create the tree as shown in above diagram  BinaryTree tree = new BinaryTree();  tree.root = new Node(10);  tree.root.left = new Node(12);  tree.root.right = new Node(15);  tree.root.left.left = new Node(25);  tree.root.left.right = new Node(30);  tree.root.right.left = new Node(36);  // convert to DLL  tree.BTree2CircularDoublyLinkedList(tree.root);  // Print the converted List  tree.printList(tree.head);  } } // This code has been contributed by Abhijeet // Kumar(abhijeet19403) 
Python
# A python program for in-place conversion of Binary Tree to DLL # A binary tree node has data left pointers and right pointers class Node: def __init__(self val): self.data = val self.left = None self.right = None # head --> Pointer to head node of created doubly linked list head = None # Initialize previously visited node as NULL. This is # so that the same value is accessible in all recursive # calls prev = None # A simple recursive function to convert a given Binary tree # to Doubly Linked List # root --> Root of Binary Tree def BinaryTree2DoubleLinkedList(root): # Base case if (root == None): return # Recursively convert left subtree BinaryTree2DoubleLinkedList(root.left) # Now convert this node global prev head if (prev == None): head = root else: root.left = prev prev.right = root prev = root # Finally convert right subtree BinaryTree2DoubleLinkedList(root.right) # Function to print nodes in a given doubly linked list def printList(node): while (node != None): print(node.data) node = node.right # Driver program to test above functions # Let us create the tree as shown in above diagram root = Node(10) root.left = Node(12) root.right = Node(15) root.left.left = Node(25) root.left.right = Node(30) root.right.left = Node(36) # convert to DLL BinaryTree2DoubleLinkedList(root) # Print the converted List printList(head) # This code is contributed by adityamaharshi21. 
C#
// A C# program for in-place conversion of Binary Tree to // CDLL using System; public class Node {  public int data;  public Node left right;  public Node(int data)  {  this.data = data;  left = right = null;  } } public class BinaryTree {  Node root;  // head --> Pointer to head node of created doubly  // linked list  Node head;  // Initialize previously visited node as NULL. This is  // static so that the same value is accessible in all  // recursive calls  static Node prev = null;  // A simple utility recursive function to convert a  // given Binary tree to Doubly Linked List root --> Root  // of Binary Tree  void BTree2DoublyLinkedList(Node root)  {  // Base case  if (root == null)  return;  // Recursively convert left subtree  BTree2DoublyLinkedList(root.left);  // Now convert this node  if (prev == null)  head = root;  else {  root.left = prev;  prev.right = root;  }  prev = root;  // Finally convert right subtree  BTree2DoublyLinkedList(root.right);  }  // A simple function to convert a given binary tree to  // Circular doubly linked list  // using a utility function  void BTree2CircularDoublyLinkedList(Node root)  {  BTree2DoublyLinkedList(root);  // make the changes to convert a DLL to CDLL  prev.right = head;  head.left = prev;  }  /* Function to print nodes in a given doubly linked list  */  void printList(Node node)  {  if (node == null)  return;  Node curr = node;  do {  Console.Write(curr.data + ' ');  curr = curr.right;  } while (curr != node);  }  static public void Main()  {  // Let us create the tree as shown in above diagram  BinaryTree tree = new BinaryTree();  tree.root = new Node(10);  tree.root.left = new Node(12);  tree.root.right = new Node(15);  tree.root.left.left = new Node(25);  tree.root.left.right = new Node(30);  tree.root.right.left = new Node(36);  // convert to DLL  tree.BTree2CircularDoublyLinkedList(tree.root);  // Print the converted List  tree.printList(tree.head);  } } // This code is contributed by lokesh(lokeshmvs21). 
JavaScript
// A javascript program for in-place conversion of Binary Tree to DLL // A binary tree node has data left pointers and right pointers class Node {  constructor(val) {  this.data = val;  this.left = null;  this.right = null;  } } var root; // head --> Pointer to head node of created doubly linked list var head; // Initialize previously visited node as NULL. This is // so that the same value is accessible in all recursive // calls var prev = null; // A simple recursive function to convert a given Binary tree // to Doubly Linked List // root --> Root of Binary Tree function BinaryTree2DoubleLinkedList(root) {  // Base case  if (root == null)  return;    // Recursively convert left subtree  BinaryTree2DoubleLinkedList(root.left);    // Now convert this node  if (prev == null)  head = root;  else {  root.left = prev;  prev.right = root;  }  prev = root;    // Finally convert right subtree  BinaryTree2DoubleLinkedList(root.right); } /* Function to print nodes in a given doubly linked list */ function printList(node) {  while (node != null) {  console.log(node.data + ' ');  node = node.right;  } } // Driver program to test above functions // Let us create the tree as shown in above diagram root = new Node(10); root.left = new Node(12); root.right = new Node(15); root.left.left = new Node(25); root.left.right = new Node(30); root.right.left = new Node(36); // convert to DLL BinaryTree2DoubleLinkedList(root); // Print the converted List printList(head); // This code is contributed by ishankhandelwals. 

Izvade
25 12 30 10 36 15 

Laika sarežģītība: O(N) Tā kā katrs mezgls tiek apmeklēts ne vairāk kā vienu reizi.
Palīgtelpa: O(log N) Papildu vieta tiek izmantota rekursīvo funkciju izsaukuma steksā, kas var pieaugt līdz maksimālajam logN izmēram.

Šo pieeju veicināja Abhidžets Kumars