vers-vecs 1.1.0

A collection of succinct data structures supported by fast implementations of rank and select queries.
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# Vers - Very Efficient Rank and Select


Vers (vers-vecs on crates.io)
contains pure-Rust implementations of several data structures backed by rank and select operations.
The library was originally a grad student project for a semester course,
but since it performs quite well (especially if worst-case run-times are important) compared to publicly available implementations,
I've decided to publish it.

## Data Structures

- a bit-vector with no overhead.
- a succinct bit-vector supporting fast rank and select queries.
- an Elias-Fano encoding of monotone sequences; supporting constant time predecessor/successor queries.
- two Range Minimum Query vector structures for constant-time range minimum queries.

# Intrinsics

This crate uses compiler intrinsics for bit-manipulation. The intrinsics are supported by
all modern x86_64 CPUs, but not by other architectures. Since the data structures depend
very heavily on these intrinsics, they are forcibly enabled, which means the crate will not
compile on non-x86_64 architectures, and will not work correctly on very old x86_64 CPUs.

The intrinsics in question are `popcnt` (supported since SSE4.2 resp. SSE4a on AMD, 2007-2008),
`pdep` (supported with BMI2 since Intel Haswell resp. AMD Excavator, in hardware since AMD Zen 3, 2011-2013),
and `tzcnt` (supported with BMI1 since Intel Haswell resp. AMD Jaguar, ca. 2013).

Rust offers library functions for popcount and trailing zero count, but not for parallel deposit,
which is why I cannot fall back to a software implementation for different architectures.

## Safety

This crate uses no unsafe code, with the only exception being compiler intrinsics for
bit-manipulation. The intrinsics cannot fail with the provided inputs (provided they are
supported by the target machine), so even if they were to be implemented incorrectly, no
memory corruption can occur (only incorrect results).

Unsafe code is hidden behind public API.
The crate fails compilation, if the intrinsics are not available.

## Dependencies

The library has no dependencies outside the Rust standard library by default.
It has a plethora of dependencies for benchmarking purposes, but these are not required for normal use.
Optionally, the `serde` feature can be enabled to allow serialization and deserialization of the data structures,
which requires the `serde` crate and its `derive` feature.

## Benchmarks

I benchmarked the implementations against publicly available implementations of the same data structures.
The benchmarking code is available in the `bench` directory.
Since one of the available implementations requires nightly Rust,
compiling this project in any configuration that includes tests or benchmarks requires nightly Rust.

I performed benchmarks on a Ryzen 9 7950X with 32GB of RAM.
The results are shown below.

### Bit-Vector

The bit-vector implementation is among the fastest publicly available implementation for rank and select operations.
Note that the `succinct` crate outperforms Vers' `rank` operation, but does not provide an efficient select operation.
The `sucds` generally performs better than `vers` in average case, but is significantly slower in worst case (see below).

The x-axis is the number of bits in the bit-vector.
An increase in all runtimes can be observed for input sizes exceeding the L2 cache size (16 MB).

| Legend           | Crate                                    | Notes                               |
|------------------|------------------------------------------|-------------------------------------|
| bio              | https://crates.io/crates/bio             | with adaptive block-size            |
| fair bio         | https://crates.io/crates/bio             | with constant block-size            |
| fid              | https://crates.io/crates/fid             |                                     |
| indexed bitvector | https://crates.io/crates/indexed_bitvec |                                     |
| rank9            | https://crates.io/crates/succinct        | Fastest of multiple implementations |
| rsdict           | https://crates.io/crates/rsdict          |                                     |
| vers             | https://github.com/Cydhra/vers           |                                     |

![Bit-Vector Rank Benchmark](images/rank_comparison.svg)
![Bit-Vector Select Benchmark](images/select_comparison.svg)

For small worst-case bit distributions, Vers' implementation is significantly faster than the `sucds` crate.
Since the `sucds` crate relies on dense arrays, the worst-case for `vers` is different from the worst case for `sucds`,
and I was unable to find a worst-case distribution for the latter.
It is unclear if the `sucds` crate is faster than Vers' implementation for worst-case distributions,
or if I was simply unable to find a suitable worst-case distribution.
If worst-case runtimes are important to you, I recommend benchmarking both crates with data representative of your
input bit distribution (since both crates have different worst-case scenarios), or to default to `vers` 
(look at the benchmark code to get an idea what distributions are bad for `vers`).

![Bit-Vector Select Worst Case Benchmark](images/select_worst_case.svg)

### Elias-Fano

The benchmark compares the performance of Vers' implementation against a naive implementation that uses binary search
to find the predecessor.
The x-axis is the number of elements in the sequence.
An increase in the near-constant runtime can be observed for input sizes exceeding the L3 cache size 
(64 MB).
As expected, predecessor and successor queries are the same speed and much faster than binary search (for large inputs).

![Elias-Fano Randomized](images/elias_fano_randomized.svg)

Another benchmark for worst-case input distributions shows that Vers' implementation is still only a factor slower than
on average inputs and asymptotically equal to binary search.
I did not include the successor query in this benchmark, since it is identical to the predecessor query.
Notably, `sucds` runtime grows exponentially with the number of elements in the sequence for worst-case distributions.

![Elias-Fano Worst Case](images/elias_fano_worst_case.svg)

There is an Elias-Fano implementation in the [elias-fano](https://crates.io/crates/elias-fano) crate,
which does not support predecessor/successor queries, but I benchmarked the access times for elements at a given index.
It should be noted that the elias-fano crate is inefficient with random order access, so I benchmarked in-order access
as well.
The x-axis is the number of elements in the sequence, the y-axis is the time for accessing one element.
Scales are logarithmic.

![Elias-Fano comparison with randomized query order](images/elias_fano_comparison_random.svg)
![Elias-Fano comparison with in-order query order](images/elias_fano_comparison_in_order.svg)

### Range Minimum Query

The Range Minimum Query implementations are compared against the 
[range_minimum_query](https://crates.io/crates/range_minimum_query) and 
[librualg](https://crates.io/crates/librualg) crate.
Vers outperforms both crates by a significant margin.
The x-axis is the number of elements in the sequence.
An increase in runtime can be observed for input sizes exceeding the L3 cache size (64 MB).

![RMQ Comparison](images/rmq_comparison.svg)

## License

Licensed under either of

* Apache License, Version 2.0
  ([LICENSE-APACHE](LICENSE-APACHE) or http://www.apache.org/licenses/LICENSE-2.0)
* MIT license
  ([LICENSE-MIT](LICENSE-MIT) or http://opensource.org/licenses/MIT)

at your option.

## Contribution

Unless you explicitly state otherwise, any contribution intentionally submitted
for inclusion in the work by you, as defined in the Apache-2.0 license, shall be
dual licensed as above, without any additional terms or conditions.