bit-ds 0.1.1

A Python library for the Binary Indexed Tree data structure.

Binary-Indexed-Tree (Fenwick Tree)

A high-performance Rust (via PyO3) implementation of a Binary Indexed Tree (Fenwick Tree), designed for efficient prefix sum queries and point updates in Python.

Features

• O(log n) point updates and prefix sum queries.
• 0-based indexing (for intuitive Python compatibility).
• Supports negative values in the tree.
• Rust backend for blazing-fast operations.

Why Use a Fenwick Tree?

Fenwick Trees are ideal for problems requiring dynamic prefix sums or frequent updates, where a static array would be too slow. Common use cases include:

1. Real-time prefix sums:
   • Compute sums over arbitrary ranges in logarithmic time (e.g., financial tracking, scoring systems).
2. Coordinate compression in algorithms:
   • Used in competitive programming for problems like inversion counting or range updates.
3. Efficient updates:
   • Unlike prefix sum arrays (which require O(n) updates), Fenwick Trees handle updates in O(log n).
4. Low memory overhead:
   • Uses only O(n) space, unlike segment trees which require O(4n).

Comparison to Alternatives

Structure	Build Time	Update Time	Prefix Sum Time	Space	Use Case
Binary Indexed Tree	O(n log n)	O(log n)	O(log n)	O(n)	Dynamic prefix sums
Prefix Sum Array	O(n)	O(n)	O(1)	O(n)	Static arrays (no updates)
Segment Tree	O(n)	O(log n)	O(log n)	O(4n)	Flexible range operations but heavier

pip install bit_ds

Usage

There are 2 classes defined by the library:

1. BIT: A class for creating a Binary Indexed Tree (Fenwick Tree) using a list of integers.
2. NdBIT: A class for creating a multi-dimensional Binary Indexed Tree (Fenwick Tree) using a numpy list of integers.

Requirements

• Python 3.12+

Optional:

• numpy (for NdBIT)
• Rust 1.70+ (for building from source)

Basic Operations (0-based Indexing)

from bit_ds import BIT

# Initialize a BIT using a list of integers
bit = BIT([1, 2, 3, 4, 5])

# Point update: Add 5 to index 2 (0-based)
bit.update(2, 5)

# Prefix sum: Sum from index 0 to 5 (inclusive, 0-based) 
print(bit.query(2))  # Output: 6 

# Range sum: Sum from index 3 to 7 (inclusive, 0-based)
print(bit.range_query(2, 4))  # Output: 12

API Reference (0-based Indexing)

BIT(input: list)

• Creates a new BIT instance using an input list of integers.

Methods

Method	Description	Time Complexity
update(index: int, delta: int)	Updates the value at index (0-based) by delta.	O(log n)
query(index: int) -> int	Returns the sum from [0, index] (inclusive).	O(log n)
range_query(l: int, r: int) -> int	Returns the sum from [l, r] (inclusive).	O(log n)

Key Notes

1. 0-based Indexing:
   • update(0, x) affects the first element.
   • query(3) returns the sum from the first to the fourth element.
2. Range Queries:
   • range_query(l, r) is equivalent to query(r) - query(l-1) (handles bounds automatically).
3. Negative Deltas:
   • Use update(i, -5) to subtract values.

Benchmarks

WIP

Contributing

Pull requests welcome! For major changes, open an issue first.

License

MIT