Wavelet Matrix for Rust language

It provides the various analytics on very large sequence of unsigned integers in constant time.

Usage

After adding to Cargo.toml, add this line to lib.rs or main.rs.

extern crate wavelet_matrix;

See crate document top for further examples.

• https://docs.rs/wavelet-matrix/

Benchmarks

• Overall Performance

  • https://github.com/sekineh/wavelet-matrix-rs/blob/master/BENCH.md
  • Bench is performed on Intel Xeon 2800 MHz

• Error evaluation of O(1) SUM

  • https://github.com/sekineh/wavelet-matrix-rs/blob/master/BENCH_SUM.md

Features

Given an unsigned integer sequence T, it provides the following queries.

Basic operations

• .len():
  • Returns the length of T.
  • It's almost no overhead, just returning the stored value.
• .lookup(pos):
  • Returns the value at the position of T, T[pos].
  • It's slower than array lookup but still is O(1). You might want to save the original Vector for faster lookup.

Counting

Counting is performed in O(1).

• .count(start..end, value):
  • Returns the number of the element e which satisfies e == value included in T[start..end]
• .count_prefix(start..end, value, ignore_bit):
  • Returns the number of the element e which satisfies e >> ignore_bit == value >> ignore_bit included in T[start..end]
  • This will be useful for counting the number of IPv4 address that satisfies IPv4 prefix such as 192.168.10.0/24. In this case, the ignore_bit will be 8.
• .count_lt(start..end, value):
  • Returns the number of the element e which satisfies e < value included in T[start..end]
• .count_gt(start..end, value):
  • Returns the number of the element e which satisfies e > value included in T[start..end]
• .count_range(start..end, val_start..val_end):
  • Returns the number of the element e which satisfies val_start <= e < val_end included in T[start..end]

Searching

Searching is performed in O(1) per a next index.

• .search(start..end, value):
  • Returns the iterator that find indexes of the element e which satisfies e == value included in T[start..end]
• .search_prefix(start..end, value, ignore_bit):
  • Returns the iterator that find indexes of the element e which satisfies e >> ignore_bit == value >> ignore_bit included in T[start..end]
• [TODO] implement various conditions other than equal.

Ranking

Ranking is performed in roughly O(1) with regard to the number of elements n.

• .max_k(start..end, val_start..val_end, k):
  • list the (value, count) pairs in descending order.
  • values are constrained to the range val_start..val_end.
• .min_k(start..end, val_start..val_end, k):
  • list the (value, count) pairs in ascending order.
  • values are constrained to the range val_start..val_end.

.top_k() is also performed in O(1) in best case, but may take O(n) in the worst case where every value occurs only once!

• .top_k(start..end, val_start..val_end, k):
  • list the (value, count) pairs in most-frequent-one-first order.
  • values are constrained to the range val_start..val_end.
  • [TODO] clarify the order of same count.

To achieve O(1) performance regardless of the number of unique values, use .top_k_ranges() instead:

• [EXPERIMENTAL] .top_k_ranges(start..end, val_start..val_end, k):
  • list the (v_start..v_end, count) pairs in most-frequent-one-first order.
  • unlike .top_k(), .top_k_ranges() returns the exhaustive range set that covers all of the values.
  • values are constrained to the range val_start..val_end.
  • [TODO] clarify the order of same count.

Statistics

O(1) Median / O(1) Quantile

• .median(start..end):
  • Returns the median value from T[start..end].
  • same as .quantile(start..end, start + (end - start) / 2).
• .quantile(start..end, k):
  • Returns the (k+1)th smallest value from T[start..end].

O(1) Sum / O(1) Average / O(1) Variance

Experiment 1

These methods use .top_k_ranges() to enumerate the most relevant value ranges.

They are not as accurate as the method used in Experiment 2.

• [EXPERIMENTAL] .sum_experiment1(start..end, val_start..val_end, k):
  • Approximately calculate the sum of T[start..end] using up to k wavelet tree nodes.
  • Only values included in the range val_start..val_end are processed.
  • To get the exact result, specify k = m + 1 where m is the number of values which are unique.
• [EXPERIMENTAL] .mean_experiment1(start..end, val_start..val_end, k):
  • Approximately calculate the average of T[start..end] using up to k wavelet tree nodes.
  • Only values included in the range val_start..val_end are processed.
  • To get the exact result, specify k = m + 1 where m is the number of values which are unique.
• [EXPERIMENTAL] .variance_experiment1(start..end, val_start..val_end, k):
  • Approximately calculate the variance of T[start..end] using up to k wavelet tree nodes.
  • Only values included in the range val_start..val_end are processed.
  • To get the exact result, specify k = m + 1 where m is the number of values which are unique.

Experiment 2

Improvement over Experiment 1. They use custom node enumerator to minimize the error.

• [EXPERIMENTAL] .sum_experiment2(start..end, val_start..val_end, k):

Experiment 3

Improvement over Experiment 2. They use Range<u64> to tell how accurate the computed value is.

• [EXPERIMENTAL] .sum_experiment3(start..end, val_start..val_end, k):

Classical WaveletMatrix operations

• .rank(pos, value): counts value included in T[0..pos].
  • Note: pos is exclusive. When pos is 0, .rank() always returns 0.
• .select(rank, value): return the position of the (rank+1)-th value
  • Note: When found nothing, it returns .len() instead of None.

TODO

• Add Benchmark.

  • Implement same queries using trivial algorithm
  • Compare wm's queries against trivial one.
  • Make a nice plot.

• Profiling

• Optimize underlying rsdic structure.

• Add travis CI.

• Add u128 support or arbitrary-length integer support

• The fastest implementation on the planet

Credits

• A Go package for myriad array operations using wavelet trees

  • https://github.com/hillbig/waveletTree
  • Basically, the algorithm is deeply derived from the above Go implementation.

• excellent slides in Japanese

  • https://www.slideshare.net/pfi/ss-15916040
  • https://www.slideshare.net/bonprosoft/the-wavelet-matrix

• The original inventor's pdf:

  • http://www.dcc.uchile.cl/~gnavarro/ps/spire12.4.pdf