If you’re a software developer, chances are you’ve heard of binary indexed trees. They’re a data structure that’s used to store and query data efficiently. But what exactly is a binary indexed tree?
In this blog post, we’ll take a look at what binary indexed trees are, how they work, and when you should use them. We’ll also provide some examples of how to use them in code. By the end, you’ll have a good understanding of binary indexed trees and how they can be used to improve your code.
A binary indexed tree (BIT) is a data structure that stores a collection of items in a tree-like structure. Each item is stored in a node, and each node is connected to other nodes by edges. The nodes are arranged in a way that allows for efficient retrieval and updating of data.
A BIT is a type of Fenwick tree, which is a data structure that was invented by Peter Fenwick in 1994. Fenwick trees are used to store and query data in a way that’s both efficient and space-efficient.
The key feature of a BIT is that it can be used to efficiently query a range of values. For example, if you have a list of numbers and you want to find the sum of all the numbers in a given range, a BIT can do this in O(log n) time. This is much faster than a linear search, which would take O(n) time.
A BIT is a binary tree, meaning that each node has two children. Each node also has an associated value, which is the sum of all the values in the subtree rooted at that node.
The root node has an associated value of 0. The left child of the root node has an associated value of the sum of all the values in the left subtree, and the right child has an associated value of the sum of all the values in the right subtree.
This pattern continues throughout the tree. Each node’s associated value is the sum of all the values in the subtree rooted at that node.
The key to a BIT’s efficiency is in how it stores and retrieves data. To query a range of values, the BIT uses a technique called “range query”. This technique allows the BIT to quickly find the sum of all the values in a given range.
To do this, the BIT starts at the root node and traverses down the tree. At each node, it adds the associated value to a running total. When it reaches the end of the range, it returns the running total.
Here’s an example of how this works:
// Pseudo-code
// Initialize a running total to 0
int total = 0;
// Start at the root node
Node current = root;
// Traverse down the tree
while (current != null) {
// Add the associated value to the running total
total += current.value;
// Move to the left or right child, depending on the range
if (range.start <= current.value) {
current = current.left;
} else {
current = current.right;
}
}
// Return the running total
return total;
BITs are an efficient way to store and query data, so they’re useful in a variety of situations. Here are some examples of when you might want to use a BIT:
BITs are also useful in algorithms that require range queries, such as range minimum queries, range sum queries, and range maximum queries.
Binary indexed trees are a powerful data structure that can be used to store and query data in an efficient manner. They’re especially useful for range queries, as they can quickly find the sum of all the values in a given range.
If you’re looking for an efficient way to store and query data, a BIT might be the right choice for you. With a bit of practice, you’ll be able to use them to improve the performance of your code.