# Succinct Partial Sums and Fenwick Trees

## TL;DR

This paper introduces two space-efficient Fenwick Tree variants that support partial sums and updates with near-optimal space and time complexity, leveraging bit-packing and sampling techniques for practicality and parallelization.

## Contribution

The paper presents the first succinct Fenwick Tree implementations with nk + o(n) bits space and efficient query/update times, improving space efficiency while maintaining performance.

## Key findings

- Achieved nk + o(n) bits space with O(log_b n) query/update time.
- Supported near-optimal time complexity with only a slight increase in space usage.
- Designed methods are practical, based on bit-packing and sampling, enabling simple parallelization.

## Abstract

We consider the well-studied partial sums problem in succint space where one is to maintain an array of n k-bit integers subject to updates such that partial sums queries can be efficiently answered. We present two succint versions of the Fenwick Tree - which is known for its simplicity and practicality. Our results hold in the encoding model where one is allowed to reuse the space from the input data. Our main result is the first that only requires nk + o(n) bits of space while still supporting sum/update in O(log_b n) / O(b log_b n) time where 2 <= b <= log^O(1) n. The second result shows how optimal time for sum/update can be achieved while only slightly increasing the space usage to nk + o(nk) bits. Beyond Fenwick Trees, the results are primarily based on bit-packing and sampling - making them very practical - and they also allow for simple optimal parallelization.