Vertical Sum in Binary Tree | Set 2 (Space Optimized)
Given a Binary Tree, find vertical sum of the nodes that are in same vertical line. Print all sums through different vertical lines.
Examples:
1 / \ 2 3 / \ / \ 4 5 6 7
The tree has 5 vertical lines
Vertical-Line-1 has only one node 4 => vertical sum is 4
Vertical-Line-2: has only one node 2=> vertical sum is 2
Vertical-Line-3: has three nodes: 1,5,6 => vertical sum is 1+5+6 = 12
Vertical-Line-4: has only one node 3 => vertical sum is 3
Vertical-Line-5: has only one node 7 => vertical sum is 7
So expected output is 4, 2, 12, 3 and 7
We have discussed Hashing Based Solution in Set 1. Hashing based solution requires a Hash Table to be maintained. We know that hashing requires more space than the number of entries in it. In this post, Doubly Linked List based solution is discussed. The solution discussed here requires only n nodes of linked list where n is total number of vertical lines in binary tree. Below is algorithm.
verticalSumDLL(root) 1) Create a node of doubly linked list node with value 0. Let the node be llnode. 2) verticalSumDLL(root, llnode) verticalSumDLL(tnode, llnode) 1) Add current node's data to its vertical line llnode.data = llnode.data + tnode.data; 2) Recursively process left subtree // If left child is not empty if (tnode.left != null) { if (llnode.prev == null) { llnode.prev = new LLNode(0); llnode.prev.next = llnode; } verticalSumDLLUtil(tnode.left, llnode.prev); } 3) Recursively process right subtree if (tnode.right != null) { if (llnode.next == null) { llnode.next = new LLNode(0); llnode.next.prev = llnode; } verticalSumDLLUtil(tnode.right, llnode.next); }
C++
// C++ program of space optimized solution // to find vertical sum of binary tree. #include <bits/stdc++.h> using namespace std; // Tree node structure struct TNode { int key; struct TNode *left, *right; }; // Function to create new tree node TNode* newTNode( int key) { TNode* temp = new TNode; temp->key = key; temp->left = temp->right = NULL; return temp; } // Doubly linked list structure struct LLNode { int key; struct LLNode *prev, *next; }; // Function to create new Linked List Node LLNode* newLLNode( int key) { LLNode* temp = new LLNode; temp->key = key; temp->prev = temp->next = NULL; return temp; } // Function that creates Linked list and store // vertical sum in it. void verticalSumDLLUtil(TNode* root, LLNode* sumNode) { // Update sum of current line by adding value // of current tree node. sumNode->key = sumNode->key + root->key; // Recursive call to left subtree. if (root->left) { if (sumNode->prev == NULL) { sumNode->prev = newLLNode(0); sumNode->prev->next = sumNode; } verticalSumDLLUtil(root->left, sumNode->prev); } // Recursive call to right subtree. if (root->right) { if (sumNode->next == NULL) { sumNode->next = newLLNode(0); sumNode->next->prev = sumNode; } verticalSumDLLUtil(root->right, sumNode->next); } } // Function to print vertical sum of Tree. // It uses verticalSumDLLUtil() to calculate sum. void verticalSumDLL(TNode* root) { // Create Linked list node for // line passing through root. LLNode* sumNode = newLLNode(0); // Compute vertical sum of different lines. verticalSumDLLUtil(root, sumNode); // Make doubly linked list pointer point // to first node in list. while (sumNode->prev != NULL) { sumNode = sumNode->prev; } // Print vertical sum of different lines // of binary tree. while (sumNode != NULL) { cout << sumNode->key << " " ; sumNode = sumNode->next; } } int main() { /* 1 / \ / \ 2 3 / \ / \ / \ / \ 4 5 6 7 */ TNode* root = newTNode(1); root->left = newTNode(2); root->right = newTNode(3); root->left->left = newTNode(4); root->left->right = newTNode(5); root->right->left = newTNode(6); root->right->right = newTNode(7); cout << "Vertical Sums are\n" ; verticalSumDLL(root); return 0; } // This code is contributed by Sania Kumari Gupta // (kriSania804) |
C
// C program of space optimized solution // to find vertical sum of binary tree. #include <stdio.h> #include <stdlib.h> // Tree node structure typedef struct TNode { int key; struct TNode *left, *right; } TNode; // Function to create new tree node TNode* newTNode( int key) { TNode* temp = (TNode*) malloc ( sizeof (TNode)); temp->key = key; temp->left = temp->right = NULL; return temp; } // Doubly linked list structure typedef struct LLNode { int key; struct LLNode *prev, *next; } LLNode; // Function to create new Linked List Node LLNode* newLLNode( int key) { LLNode* temp = (LLNode*) malloc ( sizeof (LLNode)); temp->key = key; temp->prev = temp->next = NULL; return temp; } // Function that creates Linked list and store // vertical sum in it. void verticalSumDLLUtil(TNode* root, LLNode* sumNode) { // Update sum of current line by adding value // of current tree node. sumNode->key = sumNode->key + root->key; // Recursive call to left subtree. if (root->left) { if (sumNode->prev == NULL) { sumNode->prev = newLLNode(0); sumNode->prev->next = sumNode; } verticalSumDLLUtil(root->left, sumNode->prev); } // Recursive call to right subtree. if (root->right) { if (sumNode->next == NULL) { sumNode->next = newLLNode(0); sumNode->next->prev = sumNode; } verticalSumDLLUtil(root->right, sumNode->next); } } // Function to print vertical sum of Tree. // It uses verticalSumDLLUtil() to calculate sum. void verticalSumDLL(TNode* root) { // Create Linked list node for // line passing through root. LLNode* sumNode = newLLNode(0); // Compute vertical sum of different lines. verticalSumDLLUtil(root, sumNode); // Make doubly linked list pointer point // to first node in list. while (sumNode->prev != NULL) { sumNode = sumNode->prev; } // Print vertical sum of different lines // of binary tree. while (sumNode != NULL) { printf ( "%d " , sumNode->key); sumNode = sumNode->next; } } int main() { /* 1 / \ / \ 2 3 / \ / \ / \ / \ 4 5 6 7 */ TNode* root = newTNode(1); root->left = newTNode(2); root->right = newTNode(3); root->left->left = newTNode(4); root->left->right = newTNode(5); root->right->left = newTNode(6); root->right->right = newTNode(7); printf ( "Vertical Sums are\n" ); verticalSumDLL(root); return 0; } // This code is contributed by Sania Kumari Gupta // (kriSania804) |
Java
// Java implementation of space optimized solution // to find vertical sum. public class VerticalSumBinaryTree { // Prints vertical sum of different vertical // lines in tree. This method mainly uses // verticalSumDLLUtil(). static void verticalSumDLL(TNode root) { // Create a doubly linked list node to // store sum of lines going through root. // Vertical sum is initialized as 0. LLNode llnode = new LLNode( 0 ); // Compute vertical sum of different lines verticalSumDLLUtil(root, llnode); // llnode refers to sum of vertical line // going through root. Move llnode to the // leftmost line. while (llnode.prev != null ) llnode = llnode.prev; // Prints vertical sum of all lines starting // from leftmost vertical line while (llnode != null ) { System.out.print(llnode.data + " " ); llnode = llnode.next; } } // Constructs linked list static void verticalSumDLLUtil(TNode tnode, LLNode llnode) { // Add current node's data to its vertical line llnode.data = llnode.data + tnode.data; // Recursively process left subtree if (tnode.left != null ) { if (llnode.prev == null ) { llnode.prev = new LLNode( 0 ); llnode.prev.next = llnode; } verticalSumDLLUtil(tnode.left, llnode.prev); } // Process right subtree if (tnode.right != null ) { if (llnode.next == null ) { llnode.next = new LLNode( 0 ); llnode.next.prev = llnode; } verticalSumDLLUtil(tnode.right, llnode.next); } } // Driver code public static void main(String[] args) { // Let us construct the tree shown above TNode root = new TNode( 1 ); root.left = new TNode( 2 ); root.right = new TNode( 3 ); root.left.left = new TNode( 4 ); root.left.right = new TNode( 5 ); root.right.left = new TNode( 6 ); root.right.right = new TNode( 7 ); System.out.println( "Vertical Sums are" ); verticalSumDLL(root); } // Doubly Linked List Node static class LLNode { int data; LLNode prev, next; public LLNode( int d) { data = d; } } // Binary Tree Node static class TNode { int data; TNode left, right; public TNode( int d) { data = d; } } } |
Python3
# Python3 program of space optimized # solution to find vertical sum of # binary tree. # Tree node structure class TNode: def __init__( self , key): self .key = key self .left = None self .right = None # Doubly linked list structure class LLNode: def __init__( self , key): self .key = key self .prev = None self . next = None # Function that creates Linked list and store # vertical sum in it. def verticalSumDLLUtil(root: TNode, sumNode: LLNode) - > None : # Update sum of current line by adding # value of current tree node. sumNode.key = sumNode.key + root.key # Recursive call to left subtree. if (root.left): if (sumNode.prev = = None ): sumNode.prev = LLNode( 0 ) sumNode.prev. next = sumNode verticalSumDLLUtil(root.left, sumNode.prev) # Recursive call to right subtree. if (root.right): if (sumNode. next = = None ): sumNode. next = LLNode( 0 ) sumNode. next .prev = sumNode verticalSumDLLUtil(root.right, sumNode. next ) # Function to print vertical sum of Tree. # It uses verticalSumDLLUtil() to calculate sum. def verticalSumDLL(root: TNode) - > None : # Create Linked list node for # line passing through root. sumNode = LLNode( 0 ) # Compute vertical sum of different lines. verticalSumDLLUtil(root, sumNode) # Make doubly linked list pointer point # to first node in list. while (sumNode.prev ! = None ): sumNode = sumNode.prev # Print vertical sum of different lines # of binary tree. while (sumNode ! = None ): print (sumNode.key, end = " " ) sumNode = sumNode. next # Driver code if __name__ = = "__main__" : ''' 1 / \ / \ 2 3 / \ / \ / \ / \ 4 5 6 7 ''' root = TNode( 1 ) root.left = TNode( 2 ) root.right = TNode( 3 ) root.left.left = TNode( 4 ) root.left.right = TNode( 5 ) root.right.left = TNode( 6 ) root.right.right = TNode( 7 ) print ( "Vertical Sums are" ) verticalSumDLL(root) # This code is contributed by sanjeev2552 |
C#
// C# implementation of space optimized // solution to find vertical sum. using System; class GFG { // Prints vertical sum of different vertical // lines in tree. This method mainly uses // verticalSumDLLUtil(). public static void verticalSumDLL(TNode root) { // Create a doubly linked list node to // store sum of lines going through root. // Vertical sum is initialized as 0. LLNode llnode = new LLNode(0); // Compute vertical sum of different lines verticalSumDLLUtil(root, llnode); // llnode refers to sum of vertical line // going through root. Move llnode to the // leftmost line. while (llnode.prev != null ) { llnode = llnode.prev; } // Prints vertical sum of all lines // starting from leftmost vertical line while (llnode != null ) { Console.Write(llnode.data + " " ); llnode = llnode.next; } } // Constructs linked list public static void verticalSumDLLUtil(TNode tnode, LLNode llnode) { // Add current node's data to // its vertical line llnode.data = llnode.data + tnode.data; // Recursively process left subtree if (tnode.left != null ) { if (llnode.prev == null ) { llnode.prev = new LLNode(0); llnode.prev.next = llnode; } verticalSumDLLUtil(tnode.left, llnode.prev); } // Process right subtree if (tnode.right != null ) { if (llnode.next == null ) { llnode.next = new LLNode(0); llnode.next.prev = llnode; } verticalSumDLLUtil(tnode.right, llnode.next); } } // Doubly Linked List Node public class LLNode { public int data; public LLNode prev, next; public LLNode( int d) { data = d; } } // Binary Tree Node public class TNode { public int data; public TNode left, right; public TNode( int d) { data = d; } } // Driver code public static void Main( string [] args) { // Let us construct the tree shown above TNode root = new TNode(1); root.left = new TNode(2); root.right = new TNode(3); root.left.left = new TNode(4); root.left.right = new TNode(5); root.right.left = new TNode(6); root.right.right = new TNode(7); Console.WriteLine( "Vertical Sums are" ); verticalSumDLL(root); } } // This code is contributed by Shrikant13 |
Javascript
<script> // Javascript implementation of space optimized // solution to find vertical sum. // Prints vertical sum of different vertical // lines in tree. This method mainly uses // verticalSumDLLUtil(). function verticalSumDLL(root) { // Create a doubly linked list node to // store sum of lines going through root. // Vertical sum is initialized as 0. var llnode = new LLNode(0); // Compute vertical sum of different lines verticalSumDLLUtil(root, llnode); // llnode refers to sum of vertical line // going through root. Move llnode to the // leftmost line. while (llnode.prev != null ) { llnode = llnode.prev; } // Prints vertical sum of all lines // starting from leftmost vertical line while (llnode != null ) { document.write(llnode.data + " " ); llnode = llnode.next; } } // Constructs linked list function verticalSumDLLUtil(tnode, llnode) { // Add current node's data to // its vertical line llnode.data = llnode.data + tnode.data; // Recursively process left subtree if (tnode.left != null ) { if (llnode.prev == null ) { llnode.prev = new LLNode(0); llnode.prev.next = llnode; } verticalSumDLLUtil(tnode.left, llnode.prev); } // Process right subtree if (tnode.right != null ) { if (llnode.next == null ) { llnode.next = new LLNode(0); llnode.next.prev = llnode; } verticalSumDLLUtil(tnode.right, llnode.next); } } // Doubly Linked List Node class LLNode { constructor(d) { this .data = d; this .prev = null ; this .next = null ; } } // Binary Tree Node class TNode { constructor(d) { this .data = d; this .left = null ; this .right = null ; } } // Driver code // Let us construct the tree shown above var root = new TNode(1); root.left = new TNode(2); root.right = new TNode(3); root.left.left = new TNode(4); root.left.right = new TNode(5); root.right.left = new TNode(6); root.right.right = new TNode(7); document.write( "Vertical Sums are<br>" ); verticalSumDLL(root); // This code is contributed by itsok </script> |
Output :
Vertical Sums are 4 2 12 3 7
Time Complexity: O(n)
As it is a normal preorder traversal and we need to visit every node atmost once.
Auxiliary Space: O(h)
Here h is the height of the tree and v is the number of vertical lines. The extra space h is required for recursion call stack and v is required to store the elements of the doubly linked list.