merge trees Algorithm
The merge trees algorithm is a highly efficient and versatile data structure used for solving a wide range of computational geometry problems. It is particularly useful for performing range queries and range updates on one-dimensional or multi-dimensional data, such as finding the sum, maximum, or minimum of elements within a specific range. The core idea behind merge trees is to divide the dataset into smaller segments at each level of the tree, and then merge these segments in a bottom-up fashion, similar to merge sort. Each node in the merge tree represents a continuous segment of the data, and stores aggregated information about the data in that segment, which allows for efficient query processing and updates.
To build a merge tree, the input data is first sorted based on the primary dimension, and then recursively divided into two equal halves, creating a binary tree structure. Each node in the tree represents a continuous segment of the sorted data and stores aggregated information about the data in that segment. When searching for the answer to a query, the algorithm traverses the tree from the root to the leaves, following the path that corresponds to the query range. As the tree is traversed, the algorithm efficiently combines the aggregated information from the relevant nodes to compute the final answer. This enables the merge trees algorithm to perform range queries and updates in logarithmic time complexity, making it highly suitable for processing large datasets and handling dynamic scenarios where the input data changes over time.
/*
Given two binary trees and imagine that when you put one of them to cover the other, some nodes
of the two trees are overlapped while the others are not.
You need to merge them into a new binary tree. The merge rule is that if two nodes overlap, then
sum node values up as the new value of the merged node.
Otherwise, the NOT null node will be used as the node of new tree.
Input:
Tree 1 Tree 2
1 2
/ \ / \
3 2 1 3
/ \ \
5 4 7
Output:
Merged tree:
3
/ \
4 5
/ \ \
5 4 7
*/
#include <iostream>
struct Node {
int value;
Node* left;
Node* right;
Node(int val):
value{val}, left{nullptr}, right{nullptr}{}
};
Node* mergeTrees(Node* t1, Node* t2)
{
if (!t1)
{
return t2;
}
if (!t2)
{
return t1;
}
t1->value += t2->value;
t1->left = mergeTrees(t1->left, t2->left);
t1->right = mergeTrees(t1->right, t2->right);
return t1;
}
void preOrder(Node* t1)
{
if (t1)
{
std::cout << t1->value << " ";
if (t1->left)
{
preOrder(t1->left);
}
else
{
std::cout << "null" << " ";
}
if (t1->right)
{
preOrder(t1->right);
}
else
{
std::cout << "null" << " ";
}
}
}
int main()
{
Node* t1 = new Node(1);
t1->left = new Node(3);
t1->right = new Node(2);
t1->left->left = new Node(5);
std::cout << "Tree 1:";
preOrder(t1);
std::cout << std::endl;
Node* t2 = new Node(2);
t2->left = new Node(1);
t2->right = new Node(3);
t2->left->right = new Node(4);
t2->right->right = new Node(7);
std::cout << "Tree 2:";
preOrder(t2);
std::cout << std::endl;
Node* t3 = mergeTrees(t1, t2);
std::cout << "Tree 3:";
preOrder(t3);
std::cout << std::endl;
return 0;
}
