bitree

A small, no_std-friendly Binary Indexed Tree for Rust — the same data structure more commonly known as a Fenwick tree.

A Fenwick tree maintains prefix sums over a mutable sequence. Both point updates and prefix queries run in O(log n) time, and the whole structure fits in a single Vec<T> the same length as the input array.

• A prefix sum array answers range-sum queries in O(1) but costs O(n) per update — good for static data, bad for streaming.
• A segment tree is a strictly more general cousin (arbitrary monoids, range updates), at the cost of ~2× memory and a heavier implementation.

A Fenwick tree sits somewhere in between: O(log n) for both queries and updates, with no memory overhead versus the original array. See Fenwick tree on Wikipedia.

Features

• Highly performant, but only minimal use of unsafe.
• no_std compatible (uses alloc).
• No Copy needed, so works for e.g., arbitrary precision types.
• Dynamic sizing via push / pop, just like (the underlying) Vec.
• Inverse lookup: given a prefix sum, find the slot that contains it.
• Efficiently restores to the original values in O(n).
• Optional serde support behind the serde feature.

API at a glance

Operation	Method	Complexity
Build from values	BITree::from_iter	O(n)
Create empty	BITree::new	O(1)
Prefix sum [0..i)	prefix_sum(i)	O(log n)
Accumulate prefix sum into an existing value	add_prefix_sum / sub_prefix_sum	O(log n)
Point update	add_at(i, v) / sub_at(i, v)	O(log n)
Append / truncate	push(v) / pop()	amortised O(log n)
Invert a prefix sum	binary_search(s) / sub_binary_search	O(log n)
Consume into an iterator over the original values	into_iter() (+ .rev())	O(n) total

Efficient iteration

IntoIterator::into_iter runs an O(n) in-place un-build that inverts the Fenwick construction, then hands you a plain Vec<T> iterator. That iterator is both DoubleEndedIterator and ExactSizeIterator, so you get forward and reverse traversal at O(1) per element:

use bitree::BITree;

let bitree = BITree::from_iter([1, 6, 3, 9, 2]);

let forward: Vec<_> = bitree.clone().into_iter().collect();
assert_eq!(forward, vec![1, 6, 3, 9, 2]);

let reversed: Vec<_> = bitree.into_iter().rev().collect();
assert_eq!(reversed, vec![2, 9, 3, 6, 1]);

Demo

use bitree::BITree;

// Build from a slice — values are the per-slot counts.
let mut bitree = BITree::from_iter([1usize, 6, 3, 9, 2]);

// Prefix sums in O(log n). prefix_sum(i) covers [0..i).
assert_eq!(bitree.prefix_sum(0), 0);
assert_eq!(bitree.prefix_sum(3), 10); // 1 + 6 + 3
assert_eq!(bitree.prefix_sum(5), 21); // full sum

// Point update: add 5 to slot 1. Logical array becomes [1, 11, 3, 9, 2].
bitree.add_at(1, 5);
assert_eq!(bitree.prefix_sum(3), 15);

// Grow and shrink dynamically.
bitree.push(4);
assert_eq!(bitree.prefix_sum(6), 30);
bitree.pop();
assert_eq!(bitree.prefix_sum(5), 26);

// Inverse lookup: binary-search the prefix sums [0, 1, 7, 10, 19, 21].
// `Ok(k)` means target == prefix_sum(k); `Err(k)` is the insertion point.
let bitree = BITree::from_iter([1usize, 6, 3, 9, 2]);
assert_eq!(bitree.binary_search(7), Ok(2));  // exact boundary
assert_eq!(bitree.binary_search(9), Err(3)); // falls inside slot 2

// Or drive the walk in-place to inspect the remainder yourself.
let mut remaining = 9usize;
let pos = bitree.sub_binary_search(&mut remaining);
assert_eq!((pos, remaining), (2, 2)); // 1 + 6 = 7, then 2 more into slot 2.

// Restore the original values, cheaply.
let values: Vec<_> = bitree.into();
assert_eq!(forward, vec![1, 6, 3, 9, 2]);

Comparison with related crates

Reproduce with cd examples/benchmark && cargo run --release (n = 2²⁰ u64 elements). build and restore operations consume their inputs and use whatever methods are available in the crate.

crate	build (ns/el)	prefix_sum (ns)	add_at (ns)	restore (ns/el)
bitree	0.8	16.3	24.0	0.6
ftree	0.8	16.1	24.3	3.9
segment-tree	0.9	17.1	23.9	3.9
fenwick	11.1	20.0	23.9	6.8

The differences in prefix_sum and add_at are within a noise range; bitree's in-place inverse build makes restore roughly an order of magnitude faster than reconstructing the values from prefix-sum differences.

Acknowledgements

This crate derives from the ftree crate.