├── polars_pairing ├── _internal.pyi ├── utils.py └── __init__.py ├── requirements.txt ├── docs ├── reference.md ├── stylesheets │ └── extra.css ├── index.md └── images │ ├── hagen_pairing_light.svg │ └── hagen_pairing_dark.svg ├── src ├── lib.rs ├── pairing.rs └── expressions.rs ├── Cargo.toml ├── Makefile ├── .github └── workflows │ ├── documentation.yml │ └── build.yml ├── pyproject.toml ├── mkdocs.yml ├── .gitignore ├── tests ├── test_pairing.py └── test_unpairing.py ├── README.md └── LICENSE /polars_pairing/_internal.pyi: -------------------------------------------------------------------------------- 1 | __version__: str -------------------------------------------------------------------------------- /requirements.txt: -------------------------------------------------------------------------------- 1 | polars 2 | maturin 3 | ruff 4 | pytest 5 | mypy 6 | -------------------------------------------------------------------------------- /docs/reference.md: -------------------------------------------------------------------------------- 1 | # Reference 2 | 3 | ::: polars_pairing.PairingFunctions 4 | -------------------------------------------------------------------------------- /docs/stylesheets/extra.css: -------------------------------------------------------------------------------- 1 | /* Header shadow */ 2 | .md-header--shadow { 3 | box-shadow: none; 4 | } 5 | 6 | /* Title */ 7 | .md-header__topic:first-child { 8 | font-family: var(--md-code-font-family); 9 | } 10 | 11 | /* Figure caption */ 12 | figcaption { 13 | font-size: 0.68rem; 14 | } 15 | -------------------------------------------------------------------------------- /src/lib.rs: -------------------------------------------------------------------------------- 1 | mod expressions; 2 | mod pairing; 3 | 4 | #[cfg(target_os = "linux")] 5 | use jemallocator::Jemalloc; 6 | 7 | #[global_allocator] 8 | #[cfg(target_os = "linux")] 9 | static ALLOC: Jemalloc = Jemalloc; 10 | 11 | use pyo3::types::PyModule; 12 | use pyo3::{pymodule, PyResult, Bound}; 13 | 14 | #[pymodule] 15 | fn _internal(m: &Bound<'_, PyModule>) -> PyResult<()> { 16 | m.add("__version__", env!("CARGO_PKG_VERSION"))?; 17 | Ok(()) 18 | } 19 | -------------------------------------------------------------------------------- /Cargo.toml: -------------------------------------------------------------------------------- 1 | [package] 2 | name = "polars-pairing" 3 | version = "0.1.0" 4 | edition = "2021" 5 | 6 | [lib] 7 | name = "polars_pairing" 8 | crate-type = ["cdylib"] 9 | 10 | [dependencies] 11 | pyo3 = { version = "0.21.2", features = ["extension-module", "abi3-py38"] } 12 | pyo3-polars = { version = "0.15.0", features = ["derive"] } 13 | serde = { version = "1", features = ["derive"] } 14 | polars = { version = "0.41.3", features=["dtype-struct"], default-features = false } 15 | num-integer = { version = "0.1.46" } 16 | 17 | [target.'cfg(target_os = "linux")'.dependencies] 18 | jemallocator = { version = "0.5", features = ["disable_initial_exec_tls"] } 19 | -------------------------------------------------------------------------------- /Makefile: -------------------------------------------------------------------------------- 1 | SHELL=/bin/bash 2 | 3 | .venv: ## Set up virtual environment 4 | python3 -m venv .venv 5 | .venv/bin/pip install -r requirements.txt 6 | 7 | install: .venv 8 | unset CONDA_PREFIX && \ 9 | source .venv/bin/activate && maturin develop 10 | 11 | install-release: .venv 12 | unset CONDA_PREFIX && \ 13 | source .venv/bin/activate && maturin develop --release 14 | 15 | pre-commit: .venv 16 | cargo fmt --all && cargo clippy --all-features 17 | .venv/bin/python -m ruff check . --fix --exit-non-zero-on-fix 18 | .venv/bin/python -m ruff format polars_pairing tests 19 | .venv/bin/mypy polars_pairing tests 20 | 21 | test: .venv 22 | .venv/bin/python -m pytest tests 23 | 24 | run: install 25 | source .venv/bin/activate && python run.py 26 | 27 | run-release: install-release 28 | source .venv/bin/activate && python run.py 29 | 30 | -------------------------------------------------------------------------------- /.github/workflows/documentation.yml: -------------------------------------------------------------------------------- 1 | name: documentation 2 | on: 3 | push: 4 | branches: 5 | - master 6 | - main 7 | permissions: 8 | contents: write 9 | jobs: 10 | deploy: 11 | runs-on: ubuntu-latest 12 | steps: 13 | - uses: actions/checkout@v4 14 | - name: Configure Git Credentials 15 | run: | 16 | git config user.name github-actions[bot] 17 | git config user.email 41898282+github-actions[bot]@users.noreply.github.com 18 | - uses: actions/setup-python@v5 19 | with: 20 | python-version: 3.x 21 | - run: echo "cache_id=$(date --utc '+%V')" >> $GITHUB_ENV 22 | - uses: actions/cache@v4 23 | with: 24 | key: mkdocs-material-${{ env.cache_id }} 25 | path: .cache 26 | restore-keys: | 27 | mkdocs-material- 28 | - run: pip install mkdocs-material mkdocstrings-python 29 | - run: mkdocs gh-deploy --force -------------------------------------------------------------------------------- /pyproject.toml: -------------------------------------------------------------------------------- 1 | [build-system] 2 | requires = ["maturin>=1.0,<2.0", "polars>=0.20.6"] 3 | build-backend = "maturin" 4 | 5 | [project] 6 | name = "polars-pairing" 7 | description = "Polars plugin with pairing functions for nonnegative integers" 8 | authors = [ 9 | { name = "Antonio Camargo", email = "antoniop.camargo@gmail.com" }, 10 | ] 11 | readme = "README.md" 12 | license = { file = "LICENSE" } 13 | requires-python = ">=3.8" 14 | classifiers = [ 15 | "Programming Language :: Rust", 16 | "Programming Language :: Python :: Implementation :: CPython", 17 | "Programming Language :: Python :: Implementation :: PyPy", 18 | ] 19 | 20 | [project.urls] 21 | Homepage = "https://apcamargo.github.io/polars-pairing" 22 | Repository = "https://github.com/apcamargo/polars-pairing" 23 | 24 | [tool.maturin] 25 | module-name = "polars_pairing._internal" 26 | 27 | [[tool.mypy.overrides]] 28 | module = "polars.utils.udfs" 29 | ignore_missing_imports = true 30 | -------------------------------------------------------------------------------- /src/pairing.rs: -------------------------------------------------------------------------------- 1 | use num_integer::Roots; 2 | 3 | pub fn compute_hagen_pair(l: u64, r: u64) -> Option { 4 | let (shell, step) = if l > r { (l, r) } else { (r, l) }; 5 | let flag = match (shell % 2, step) { 6 | (0, s) if s == l => 0, 7 | (1, s) if s == r => 0, 8 | _ => 1, 9 | }; 10 | Some(shell * shell + step * 2 + flag) 11 | } 12 | 13 | pub fn compute_hagen_unpair(p: u64) -> Option<(u64, u64)> { 14 | let shell = p.sqrt(); 15 | let step = (p - shell.pow(2)) / 2; 16 | if p % 2 == 0 { 17 | Some((step, shell)) 18 | } else { 19 | Some((shell, step)) 20 | } 21 | } 22 | 23 | pub fn compute_szudzik_pair(l: u64, r: u64) -> Option { 24 | if l != l.max(r) { 25 | Some(r.pow(2) + l) 26 | } else { 27 | Some(l.pow(2) + l + r) 28 | } 29 | } 30 | 31 | pub fn compute_szudzik_unpair(p: u64) -> Option<(u64, u64)> { 32 | let sqrt_p = p.sqrt(); 33 | if p - sqrt_p.pow(2) < sqrt_p { 34 | Some((p - sqrt_p.pow(2), sqrt_p)) 35 | } else { 36 | Some((sqrt_p, p - sqrt_p.pow(2) - sqrt_p)) 37 | } 38 | } 39 | 40 | pub fn compute_cantor_pair(l: u64, r: u64) -> Option { 41 | Some(((l + r) * (l + r + 1) / 2) + r) 42 | } 43 | 44 | pub fn compute_cantor_unpair(p: u64) -> Option<(u64, u64)> { 45 | let w = ((8 * p + 1).sqrt() - 1) / 2; 46 | let t = (w * (w + 1)) / 2; 47 | let r = p - t; 48 | let l = w - r; 49 | Some((l, r)) 50 | } 51 | -------------------------------------------------------------------------------- /mkdocs.yml: -------------------------------------------------------------------------------- 1 | site_name: polars-pairing 2 | site_url: https://apcamargo.github.io/polars-pairing 3 | repo_name: apcamargo/polars-pairing 4 | repo_url: https://github.com/apcamargo/polars-pairing 5 | nav: 6 | - Home: index.md 7 | - Reference: reference.md 8 | theme: 9 | name: material 10 | features: 11 | - content.code.annotate 12 | - content.code.copy 13 | - navigation.instant 14 | - navigation.sections 15 | - navigation.tabs 16 | - navigation.tabs.sticky 17 | - toc.integrate 18 | palette: 19 | - media: "(prefers-color-scheme)" 20 | toggle: 21 | icon: material/theme-light-dark 22 | name: Switch to light mode 23 | - media: "(prefers-color-scheme: light)" 24 | scheme: default 25 | primary: indigo 26 | accent: indigo 27 | toggle: 28 | icon: material/weather-sunny 29 | name: Switch to dark mode 30 | - media: "(prefers-color-scheme: dark)" 31 | scheme: slate 32 | primary: black 33 | accent: indigo 34 | toggle: 35 | icon: material/weather-night 36 | name: Switch to system preference 37 | font: 38 | text: Inter 39 | code: JetBrains Mono 40 | icon: 41 | logo: material/book-open-page-variant 42 | repo: fontawesome/brands/github 43 | markdown_extensions: 44 | - attr_list 45 | - footnotes 46 | - md_in_html 47 | - pymdownx.highlight: 48 | anchor_linenums: true 49 | line_spans: __span 50 | pygments_lang_class: true 51 | - pymdownx.inlinehilite 52 | - pymdownx.snippets 53 | - pymdownx.superfences 54 | plugins: 55 | - search 56 | - mkdocstrings: 57 | handlers: 58 | python: 59 | paths: [.] 60 | options: 61 | docstring_style: numpy 62 | show_root_heading: true 63 | show_source: true 64 | show_symbol_type_heading: true 65 | show_symbol_type_toc: true 66 | extra_css: 67 | - stylesheets/extra.css 68 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | # Editors 2 | .vscode/ 3 | 4 | # Mac/OSX 5 | .DS_Store 6 | 7 | # Windows 8 | Thumbs.db 9 | 10 | # Byte-compiled / optimized / DLL files 11 | __pycache__/ 12 | *.py[cod] 13 | *.class 14 | 15 | # C extensions 16 | *.so 17 | 18 | # Distribution / packaging 19 | .Python 20 | build/ 21 | develop-eggs/ 22 | dist/ 23 | downloads/ 24 | eggs/ 25 | .eggs/ 26 | lib/ 27 | lib64/ 28 | parts/ 29 | sdist/ 30 | var/ 31 | wheels/ 32 | share/python-wheels/ 33 | *.egg-info/ 34 | .installed.cfg 35 | *.egg 36 | MANIFEST 37 | 38 | # PyInstaller 39 | # Usually these files are written by a python script from a template 40 | # before PyInstaller builds the exe, so as to inject date/other infos into it. 41 | *.manifest 42 | *.spec 43 | 44 | # Installer logs 45 | pip-log.txt 46 | pip-delete-this-directory.txt 47 | 48 | # Unit test / coverage reports 49 | htmlcov/ 50 | .tox/ 51 | .nox/ 52 | .coverage 53 | .coverage.* 54 | .cache 55 | nosetests.xml 56 | coverage.xml 57 | *.cover 58 | .hypothesis/ 59 | .pytest_cache/ 60 | 61 | # Translations 62 | *.mo 63 | *.pot 64 | 65 | # Django stuff: 66 | *.log 67 | local_settings.py 68 | db.sqlite3 69 | 70 | # Flask stuff: 71 | instance/ 72 | .webassets-cache 73 | 74 | # Scrapy stuff: 75 | .scrapy 76 | 77 | # Sphinx documentation 78 | docs/_build/ 79 | 80 | # PyBuilder 81 | target/ 82 | 83 | # Jupyter Notebook 84 | .ipynb_checkpoints 85 | 86 | # IPython 87 | profile_default/ 88 | ipython_config.py 89 | 90 | # pyenv 91 | .python-version 92 | 93 | # celery beat schedule file 94 | celerybeat-schedule 95 | 96 | # SageMath parsed files 97 | *.sage.py 98 | 99 | # Environments 100 | .env 101 | .venv 102 | env/ 103 | venv/ 104 | ENV/ 105 | env.bak/ 106 | venv.bak/ 107 | 108 | # Spyder project settings 109 | .spyderproject 110 | .spyproject 111 | 112 | # Rope project settings 113 | .ropeproject 114 | 115 | # mkdocs documentation 116 | /site 117 | 118 | # mypy 119 | .mypy_cache/ 120 | .dmypy.json 121 | dmypy.json 122 | 123 | # Pyre type checker 124 | .pyre/ 125 | 126 | # Generated by Cargo 127 | # will have compiled files and executables 128 | /target/ 129 | 130 | # Remove Cargo.lock from gitignore if creating an executable, leave it for libraries 131 | # More information here https://doc.rust-lang.org/cargo/guide/cargo-toml-vs-cargo-lock.html 132 | Cargo.lock 133 | 134 | # These are backup files generated by rustfmt 135 | **/*.rs.bk -------------------------------------------------------------------------------- /tests/test_pairing.py: -------------------------------------------------------------------------------- 1 | import polars as pl 2 | from polars.testing import assert_frame_equal 3 | 4 | import polars_pairing as plp 5 | 6 | 7 | def test_pairing_default(): 8 | df = pl.DataFrame( 9 | { 10 | "n1": [2, 2, 3, 4, 1_501, None], 11 | "n2": [-6, 7, 8, 9, 10_740, 11], 12 | } 13 | ) 14 | result = df.select(plp.col("n1").pairing.pair(plp.col("n2")).alias("pair")) 15 | expected_df = pl.DataFrame( 16 | { 17 | "pair": [None, 54, 70, 90, 115_350_602, None], 18 | }, 19 | schema=pl.Schema({"pair": pl.UInt64}), 20 | ) 21 | assert_frame_equal(result, expected_df) 22 | 23 | 24 | def test_pairing_hagen(): 25 | df = pl.DataFrame( 26 | { 27 | "n1": [2, 2, 3, 4, 1_501, None], 28 | "n2": [-6, 7, 8, 9, 10_740, 11], 29 | } 30 | ) 31 | result = df.select( 32 | plp.col("n1").pairing.pair(plp.col("n2"), method="hagen").alias("pair") 33 | ) 34 | expected_df = pl.DataFrame( 35 | { 36 | "pair": [None, 54, 70, 90, 115_350_602, None], 37 | }, 38 | schema=pl.Schema({"pair": pl.UInt64}), 39 | ) 40 | assert_frame_equal(result, expected_df) 41 | 42 | 43 | def test_pairing_szudzik(): 44 | df = pl.DataFrame( 45 | { 46 | "n1": [2, 2, 3, 4, 1_501, None], 47 | "n2": [-6, 7, 8, 9, 10_740, 11], 48 | } 49 | ) 50 | result = df.select( 51 | plp.col("n1").pairing.pair(plp.col("n2"), method="szudzik").alias("pair") 52 | ) 53 | expected_df = pl.DataFrame( 54 | { 55 | "pair": [None, 51, 67, 85, 115_349_101, None], 56 | }, 57 | schema=pl.Schema({"pair": pl.UInt64}), 58 | ) 59 | assert_frame_equal(result, expected_df) 60 | 61 | 62 | def test_pairing_cantor(): 63 | df = pl.DataFrame( 64 | { 65 | "n1": [2, 2, 3, 4, 1_501, None], 66 | "n2": [-6, 7, 8, 9, 10_740, 11], 67 | } 68 | ) 69 | result = df.select( 70 | plp.col("n1").pairing.pair(plp.col("n2"), method="cantor").alias("pair") 71 | ) 72 | expected_df = pl.DataFrame( 73 | { 74 | "pair": [None, 52, 74, 100, 74_937_901, None], 75 | }, 76 | schema=pl.Schema({"pair": pl.UInt64}), 77 | ) 78 | assert_frame_equal(result, expected_df) 79 | 80 | 81 | def test_pairing_method_uppercase(): 82 | df = pl.DataFrame( 83 | { 84 | "n1": [2, 2, 3, 4, 1_501, None], 85 | "n2": [-6, 7, 8, 9, 10_740, 11], 86 | } 87 | ) 88 | result = df.select( 89 | plp.col("n1").pairing.pair(plp.col("n2"), method="Szudzik").alias("pair") 90 | ) 91 | expected_df = pl.DataFrame( 92 | { 93 | "pair": [None, 51, 67, 85, 115_349_101, None], 94 | }, 95 | schema=pl.Schema({"pair": pl.UInt64}), 96 | ) 97 | assert_frame_equal(result, expected_df) 98 | -------------------------------------------------------------------------------- /polars_pairing/utils.py: -------------------------------------------------------------------------------- 1 | from __future__ import annotations 2 | 3 | import re 4 | from typing import TYPE_CHECKING, Any, Sequence 5 | 6 | import polars as pl 7 | 8 | if TYPE_CHECKING: 9 | try: 10 | from polars._typing import IntoExpr, PolarsDataType # type: ignore [no-redef] 11 | except ImportError: 12 | from polars.type_aliases import ( # type: ignore [no-redef] 13 | IntoExpr, 14 | PolarsDataType, 15 | ) 16 | from pathlib import Path 17 | 18 | 19 | def parse_into_expr( 20 | expr: IntoExpr, 21 | *, 22 | str_as_lit: bool = False, 23 | list_as_lit: bool = True, 24 | dtype: PolarsDataType | None = None, 25 | ) -> pl.Expr: 26 | """ 27 | Parse a single input into an expression. 28 | 29 | Parameters 30 | ---------- 31 | expr 32 | The input to be parsed as an expression. 33 | str_as_lit 34 | Interpret string input as a string literal. If set to `False` (default), 35 | strings are parsed as column names. 36 | list_as_lit 37 | Interpret list input as a lit literal, If set to `False`, 38 | lists are parsed as `Series` literals. 39 | dtype 40 | If the input is expected to resolve to a literal with a known dtype, pass 41 | this to the `lit` constructor. 42 | 43 | Returns 44 | ------- 45 | polars.Expr 46 | """ 47 | if isinstance(expr, pl.Expr): 48 | pass 49 | elif isinstance(expr, str) and not str_as_lit: 50 | expr = pl.col(expr) 51 | elif isinstance(expr, list) and not list_as_lit: 52 | expr = pl.lit(pl.Series(expr), dtype=dtype) 53 | else: 54 | expr = pl.lit(expr, dtype=dtype) 55 | 56 | return expr 57 | 58 | 59 | def register_plugin( 60 | *, 61 | symbol: str, 62 | is_elementwise: bool, 63 | kwargs: dict[str, Any] | None = None, 64 | args: list[IntoExpr], 65 | lib: str | Path, 66 | returns_scalar: bool = False, 67 | ) -> pl.Expr: 68 | if parse_version(pl.__version__) < parse_version("0.20.16"): 69 | expr = parse_into_expr(args[0]) 70 | assert isinstance(lib, str) 71 | return expr.register_plugin( 72 | lib=lib, 73 | symbol=symbol, 74 | args=args[1:], 75 | kwargs=kwargs, 76 | is_elementwise=is_elementwise, 77 | returns_scalar=returns_scalar, 78 | ) 79 | from polars.plugins import register_plugin_function 80 | 81 | return register_plugin_function( 82 | args=args, 83 | plugin_path=lib, 84 | function_name=symbol, 85 | kwargs=kwargs, 86 | is_elementwise=is_elementwise, 87 | returns_scalar=returns_scalar, 88 | ) 89 | 90 | 91 | def parse_version(version: Sequence[str | int]) -> tuple[int, ...]: 92 | # Simple version parser; split into a tuple of ints for comparison. 93 | # vendored from Polars 94 | if isinstance(version, str): 95 | version = version.split(".") 96 | return tuple(int(re.sub(r"\D", "", str(v))) for v in version) 97 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # polars-pairing 2 | 3 | This plugin provides pairing functions to Polars. These functions encode a pair of natural numbers into a single natural number. 4 | 5 | ## Installation 6 | 7 | ```bash 8 | pip install polars-pairing 9 | ``` 10 | 11 | ## Functionality and usage 12 | 13 | This plugin implements three pairing functions (Hagen[^1], Szudzik[^2], and Cantor[^3]), along with their corresponding unpairing functions, as Polars expressions. The pairing functions take two integer columns as input and produce a single column containing the encoded pair. Conversely, the unpairing functions accept a single column as input and generate a struct column containing the values of the original pair. 14 | 15 | ```python 16 | >>> import polars as pl 17 | >>> import polars_pairing as plp 18 | 19 | >>> df = pl.DataFrame({ 20 | ... "a": [1, 2, 38, 4], 21 | ... "b": [5, 6, 77, 80] 22 | ... }) 23 | >>> df.select(plp.col("a").pairing.pair(plp.col("b")).alias("pair")) 24 | shape: (4, 1) 25 | ┌──────┐ 26 | │ pair │ 27 | │ --- │ 28 | │ u64 │ 29 | ╞══════╡ 30 | │ 28 │ 31 | │ 40 │ 32 | │ 6006 │ 33 | │ 6408 │ 34 | └──────┘ 35 | 36 | # By default, the Hagen pairing function is used, but you can specify alternative functions: 37 | >>> df.select( 38 | ... plp.col("a").pairing.pair(plp.col("b"), method="hagen").alias("hagen_pair"), 39 | ... plp.col("a").pairing.pair(plp.col("b"), method="szudzik").alias("szudzik_pair"), 40 | ... plp.col("a").pairing.pair(plp.col("b"), method="cantor").alias("cantor_pair") 41 | ... ) 42 | shape: (4, 3) 43 | ┌────────────┬──────────────┬─────────────┐ 44 | │ hagen_pair ┆ szudzik_pair ┆ cantor_pair │ 45 | │ --- ┆ --- ┆ --- │ 46 | │ u64 ┆ u64 ┆ u64 │ 47 | ╞════════════╪══════════════╪═════════════╡ 48 | │ 28 ┆ 26 ┆ 26 │ 49 | │ 40 ┆ 38 ┆ 42 │ 50 | │ 6006 ┆ 5967 ┆ 6747 │ 51 | │ 6408 ┆ 6404 ┆ 3650 │ 52 | └────────────┴──────────────┴─────────────┘ 53 | 54 | # The corresponding unpairing functions are also available: 55 | >>> df = pl.DataFrame({"p": [28, 40, 6006, 6408]}) 56 | >>> df.select(plp.col("p").pairing.unpair().alias("unpair")) 57 | shape: (4, 1) 58 | ┌───────────┐ 59 | │ unpair │ 60 | │ --- │ 61 | │ struct[2] │ 62 | ╞═══════════╡ 63 | │ {1,5} │ 64 | │ {2,6} │ 65 | │ {38,77} │ 66 | │ {4,80} │ 67 | └───────────┘ 68 | 69 | >>> df = pl.DataFrame({"p": [26, 38, 5967, 6404]}) 70 | >>> df.select( 71 | ... plp.col("p").pairing.unpair(method="szudzik").alias("unpair") 72 | ... ).unnest("unpair") 73 | shape: (4, 2) 74 | ┌──────┬───────┐ 75 | │ Left ┆ Right │ 76 | │ --- ┆ --- │ 77 | │ u64 ┆ u64 │ 78 | ╞══════╪═══════╡ 79 | │ 1 ┆ 5 │ 80 | │ 2 ┆ 6 │ 81 | │ 38 ┆ 77 │ 82 | │ 4 ┆ 80 │ 83 | └──────┴───────┘ 84 | ``` 85 | 86 | [^1]: Hagen, D. R. Superior Pairing Function. (2018). 87 | [^2]: Szudzik, Matthew. "An elegant pairing function." *Wolfram Research (ed.) Special NKS 2006 Wolfram Science Conference*. 2006. 88 | [^3]: Cantor, G. Ein Beitrag zur Mannigfaltigkeitslehre. *Journal für die reine und angewandte Mathematik* **1878**, 242–258 (1878). -------------------------------------------------------------------------------- /tests/test_unpairing.py: -------------------------------------------------------------------------------- 1 | import polars as pl 2 | from polars.testing import assert_frame_equal 3 | 4 | import polars_pairing as plp 5 | 6 | 7 | def test_unpairing_default(): 8 | df = pl.DataFrame( 9 | { 10 | "pair": [None, 51, 67, 85, 115_349_101, None], 11 | } 12 | ) 13 | result = df.select(plp.col("pair").pairing.unpair().alias("unpair")).unnest( 14 | "unpair" 15 | ) 16 | expected_df = pl.DataFrame( 17 | { 18 | "Left": [None, 7, 8, 9, 10_740, None], 19 | "Right": [None, 1, 1, 2, 750, None], 20 | }, 21 | schema=pl.Schema({"Left": pl.UInt64, "Right": pl.UInt64}), 22 | ) 23 | assert_frame_equal(result, expected_df) 24 | 25 | 26 | def test_unpairing_hagen(): 27 | df = pl.DataFrame( 28 | { 29 | "pair": [None, 51, 67, 85, 115_349_101, None], 30 | } 31 | ) 32 | result = df.select(plp.col("pair").pairing.unpair("hagen").alias("unpair")).unnest( 33 | "unpair" 34 | ) 35 | expected_df = pl.DataFrame( 36 | { 37 | "Left": [None, 7, 8, 9, 10_740, None], 38 | "Right": [None, 1, 1, 2, 750, None], 39 | }, 40 | schema=pl.Schema({"Left": pl.UInt64, "Right": pl.UInt64}), 41 | ) 42 | assert_frame_equal(result, expected_df) 43 | 44 | 45 | def test_unpairing_szudzik(): 46 | df = pl.DataFrame( 47 | { 48 | "pair": [None, 51, 67, 85, 115_349_101, None], 49 | } 50 | ) 51 | result = df.select( 52 | plp.col("pair").pairing.unpair(method="szudzik").alias("unpair") 53 | ).unnest("unpair") 54 | expected_df = pl.DataFrame( 55 | { 56 | "Left": [None, 2, 3, 4, 1_501, None], 57 | "Right": [None, 7, 8, 9, 10_740, None], 58 | }, 59 | schema=pl.Schema({"Left": pl.UInt64, "Right": pl.UInt64}), 60 | ) 61 | assert_frame_equal(result, expected_df) 62 | 63 | 64 | def test_unpairing_cantor(): 65 | df = pl.DataFrame( 66 | { 67 | "pair": [None, 52, 74, 100, 74_937_901, None], 68 | } 69 | ) 70 | result = df.select( 71 | plp.col("pair").pairing.unpair(method="cantor").alias("unpair") 72 | ).unnest("unpair") 73 | expected_df = pl.DataFrame( 74 | { 75 | "Left": [None, 2, 3, 4, 1_501, None], 76 | "Right": [None, 7, 8, 9, 10_740, None], 77 | }, 78 | schema=pl.Schema({"Left": pl.UInt64, "Right": pl.UInt64}), 79 | ) 80 | assert_frame_equal(result, expected_df) 81 | 82 | 83 | def test_unpairing_uppercase(): 84 | df = pl.DataFrame( 85 | { 86 | "pair": [None, 51, 67, 85, 115_349_101, None], 87 | } 88 | ) 89 | result = df.select( 90 | plp.col("pair").pairing.unpair(method="Szudzik").alias("unpair") 91 | ).unnest("unpair") 92 | expected_df = pl.DataFrame( 93 | { 94 | "Left": [None, 2, 3, 4, 1_501, None], 95 | "Right": [None, 7, 8, 9, 10_740, None], 96 | }, 97 | schema=pl.Schema({"Left": pl.UInt64, "Right": pl.UInt64}), 98 | ) 99 | assert_frame_equal(result, expected_df) 100 | -------------------------------------------------------------------------------- /.github/workflows/build.yml: -------------------------------------------------------------------------------- 1 | name: build 2 | 3 | on: 4 | workflow_dispatch: 5 | 6 | concurrency: 7 | group: ${{ github.workflow }}-${{ github.ref }} 8 | cancel-in-progress: true 9 | 10 | permissions: 11 | contents: read 12 | 13 | # Make sure CI fails on all warnings, including Clippy lints 14 | env: 15 | RUSTFLAGS: "-Dwarnings" 16 | 17 | jobs: 18 | linux_tests: 19 | runs-on: ubuntu-latest 20 | strategy: 21 | matrix: 22 | target: [x86_64] 23 | python-version: ["3.8", "3.9", "3.10", "3.11", "3.12"] 24 | steps: 25 | - uses: actions/checkout@v4 26 | - uses: actions/setup-python@v5 27 | with: 28 | python-version: ${{ matrix.python-version }} 29 | - name: Set up Rust 30 | run: rustup show 31 | - uses: mozilla-actions/sccache-action@v0.0.3 32 | - run: make .venv 33 | - run: make pre-commit 34 | - run: make install 35 | - run: make test 36 | 37 | linux: 38 | runs-on: ubuntu-latest 39 | strategy: 40 | matrix: 41 | target: [x86_64, x86] 42 | steps: 43 | - uses: actions/checkout@v4 44 | - uses: actions/setup-python@v5 45 | with: 46 | python-version: "3.10" 47 | - name: Build wheels 48 | uses: PyO3/maturin-action@v1 49 | with: 50 | target: ${{ matrix.target }} 51 | args: --release --out dist --find-interpreter 52 | sccache: "true" 53 | manylinux: auto 54 | - name: Upload wheels 55 | uses: actions/upload-artifact@v3 56 | with: 57 | name: wheels 58 | path: dist 59 | 60 | windows: 61 | runs-on: windows-latest 62 | strategy: 63 | matrix: 64 | target: [x64] 65 | steps: 66 | - uses: actions/checkout@v4 67 | - uses: actions/setup-python@v5 68 | with: 69 | python-version: "3.10" 70 | architecture: ${{ matrix.target }} 71 | - name: Build wheels 72 | uses: PyO3/maturin-action@v1 73 | with: 74 | target: ${{ matrix.target }} 75 | args: --release --out dist --find-interpreter 76 | sccache: "true" 77 | - name: Upload wheels 78 | uses: actions/upload-artifact@v3 79 | with: 80 | name: wheels 81 | path: dist 82 | 83 | macos: 84 | runs-on: macos-latest 85 | strategy: 86 | matrix: 87 | target: [x86_64, aarch64] 88 | steps: 89 | - uses: actions/checkout@v4 90 | - uses: actions/setup-python@v5 91 | with: 92 | python-version: "3.10" 93 | - name: Build wheels 94 | uses: PyO3/maturin-action@v1 95 | with: 96 | target: ${{ matrix.target }} 97 | args: --release --out dist --find-interpreter 98 | sccache: "true" 99 | - name: Upload wheels 100 | uses: actions/upload-artifact@v3 101 | with: 102 | name: wheels 103 | path: dist 104 | 105 | sdist: 106 | runs-on: ubuntu-latest 107 | steps: 108 | - uses: actions/checkout@v4 109 | - name: Build sdist 110 | uses: PyO3/maturin-action@v1 111 | with: 112 | command: sdist 113 | args: --out dist 114 | - name: Upload sdist 115 | uses: actions/upload-artifact@v3 116 | with: 117 | name: wheels 118 | path: dist 119 | -------------------------------------------------------------------------------- /docs/index.md: -------------------------------------------------------------------------------- 1 | # Introduction 2 | 3 | ## About `polars-pairing` 4 | 5 | `polars-pairing` is a plugin provides pairing functions to [Polars](https://github.com/pola-rs/polars). These functions encode a pair of natural numbers into a single natural number. 6 | 7 |
8 | ![Image title](images/hagen_pairing_light.svg#only-light) 9 | ![Image title](images/hagen_pairing_dark.svg#only-dark) 10 |
A pairing function performs a one-to-one mapping between (x, y) and i (represented by the circles in the Cartesian coordinate system). Figure by David Hagen.
11 |
12 | 13 | This plugin implements three pairing functions (Hagen[^1], Szudzik[^2], and Cantor[^3]), along with their corresponding unpairing functions, as Polars expressions. The pairing functions take two integer columns as input and produce a single column containing the encoded pair. Conversely, the unpairing functions accept a single column as input and generate a struct column containing the values of the original pair. 14 | 15 | ## Installation 16 | 17 | `polars-pairing` can be installed via Python's `pip`: 18 | 19 | ```bash 20 | pip install polars-pairing 21 | ``` 22 | 23 | ## Example 24 | 25 | ```python 26 | >>> import polars as pl 27 | >>> import polars_pairing as plp 28 | 29 | >>> df = pl.DataFrame({ 30 | ... "a": [1, 2, 38, 4], 31 | ... "b": [5, 6, 77, 80] 32 | ... }) 33 | >>> df.select(plp.col("a").pairing.pair(plp.col("b")).alias("pair")) 34 | shape: (4, 1) 35 | ┌──────┐ 36 | │ pair │ 37 | │ --- │ 38 | │ u64 │ 39 | ╞══════╡ 40 | │ 28 │ 41 | │ 40 │ 42 | │ 6006 │ 43 | │ 6408 │ 44 | └──────┘ 45 | 46 | >>> df.select( 47 | ... plp.col("a") 48 | ... .pairing.pair(plp.col("b"), method="hagen") # (1)! 49 | ... .alias("hagen_pair"), 50 | ... plp.col("a") 51 | ... .pairing.pair(plp.col("b"), method="szudzik") 52 | ... .alias("szudzik_pair"), 53 | ... plp.col("a") 54 | ... .pairing.pair(plp.col("b"), method="cantor") 55 | ... .alias("cantor_pair"), 56 | ... ) 57 | shape: (4, 3) 58 | ┌────────────┬──────────────┬─────────────┐ 59 | │ hagen_pair ┆ szudzik_pair ┆ cantor_pair │ 60 | │ --- ┆ --- ┆ --- │ 61 | │ u64 ┆ u64 ┆ u64 │ 62 | ╞════════════╪══════════════╪═════════════╡ 63 | │ 28 ┆ 26 ┆ 26 │ 64 | │ 40 ┆ 38 ┆ 42 │ 65 | │ 6006 ┆ 5967 ┆ 6747 │ 66 | │ 6408 ┆ 6404 ┆ 3650 │ 67 | └────────────┴──────────────┴─────────────┘ 68 | 69 | >>> df = pl.DataFrame({"p": [28, 40, 6006, 6408]}) 70 | >>> df.select(plp.col("p").pairing.unpair().alias("unpair")) # (2)! 71 | shape: (4, 1) 72 | ┌───────────┐ 73 | │ unpair │ 74 | │ --- │ 75 | │ struct[2] │ 76 | ╞═══════════╡ 77 | │ {1,5} │ 78 | │ {2,6} │ 79 | │ {38,77} │ 80 | │ {4,80} │ 81 | └───────────┘ 82 | 83 | >>> df = pl.DataFrame({"p": [26, 38, 5967, 6404]}) 84 | >>> df.select( 85 | ... plp.col("p").pairing.unpair(method="szudzik").alias("unpair") 86 | ... ).unnest("unpair") 87 | shape: (4, 2) 88 | ┌──────┬───────┐ 89 | │ Left ┆ Right │ 90 | │ --- ┆ --- │ 91 | │ u64 ┆ u64 │ 92 | ╞══════╪═══════╡ 93 | │ 1 ┆ 5 │ 94 | │ 2 ┆ 6 │ 95 | │ 38 ┆ 77 │ 96 | │ 4 ┆ 80 │ 97 | └──────┴───────┘ 98 | ``` 99 | 100 | 1. By default, the Hagen pairing function is used, but you can specify alternative functions. The available methods are: `hagen`, `szudzik`, and `cantor`. 101 | 2. The unpairing functions return a struct column containing the values of the original pair. 102 | 103 | [^1]: Hagen, D. R. Superior Pairing Function. (2018). 104 | [^2]: Szudzik, Matthew. "An elegant pairing function." *Wolfram Research (ed.) Special NKS 2006 Wolfram Science Conference*. 2006. 105 | [^3]: Cantor, G. Ein Beitrag zur Mannigfaltigkeitslehre. *Journal für die reine und angewandte Mathematik* **1878**, 242–258 (1878). 106 | -------------------------------------------------------------------------------- /src/expressions.rs: -------------------------------------------------------------------------------- 1 | #![allow(clippy::unused_unit)] 2 | use crate::pairing::*; 3 | use polars::prelude::*; 4 | use pyo3_polars::derive::polars_expr; 5 | use serde::Deserialize; 6 | 7 | #[derive(Deserialize)] 8 | struct PairingKwargs { 9 | method: String, 10 | } 11 | 12 | fn pair_struct(_input_field: &[Field]) -> PolarsResult { 13 | let left = Field::new("Left", DataType::UInt64); 14 | let right = Field::new("Right", DataType::UInt64); 15 | Ok(Field::new("Pair", DataType::Struct(vec![left, right]))) 16 | } 17 | 18 | #[polars_expr(output_type=UInt64)] 19 | fn pair(inputs: &[Series], kwargs: PairingKwargs) -> PolarsResult { 20 | // Match the method string to the pairing method, otherwise return an error 21 | let method = match kwargs.method.to_ascii_lowercase().as_str() { 22 | "hagen" => "hagen", 23 | "szudzik" => "szudzik", 24 | "cantor" => "cantor", 25 | _ => { 26 | polars_bail!(InvalidOperation: "Invalid pairing method: {}", kwargs.method); 27 | } 28 | }; 29 | // Ensure both inputs are of integer type, otherwise return an error 30 | if !inputs[0].dtype().is_integer() || !inputs[1].dtype().is_integer() { 31 | polars_bail!(InvalidOperation: "The pairing function can only be used on integers."); 32 | } 33 | // Cast the first and second input series to UInt64, unwrapping the result directly 34 | let left_series = inputs[0].cast(&DataType::UInt64).unwrap(); 35 | let right_series = inputs[1].cast(&DataType::UInt64).unwrap(); 36 | // Convert the series to their unsigned 64-bit integer chunked arrays 37 | let left = left_series.u64()?; 38 | let right = right_series.u64()?; 39 | // Apply the pairing function element-wise to left and right 40 | // If the method is "szudzik", use the Szudzik pairing function, 41 | // otherwise use the Cantor pairing function 42 | let out: UInt64Chunked = arity::binary_elementwise_values( 43 | left, 44 | right, 45 | match method { 46 | "hagen" => compute_hagen_pair, 47 | "szudzik" => compute_szudzik_pair, 48 | "cantor" => compute_cantor_pair, 49 | _ => unreachable!(), 50 | }, 51 | ); 52 | // Return the result as a Series wrapped in Ok 53 | Ok(out.into_series()) 54 | } 55 | 56 | #[polars_expr(output_type_func=pair_struct)] 57 | fn unpair(inputs: &[Series], kwargs: PairingKwargs) -> PolarsResult { 58 | // Match the method string to the pairing method, otherwise return an error 59 | let method = match kwargs.method.to_ascii_lowercase().as_str() { 60 | "hagen" => "hagen", 61 | "szudzik" => "szudzik", 62 | "cantor" => "cantor", 63 | _ => { 64 | polars_bail!(InvalidOperation: "Invalid pairing method: {}", kwargs.method); 65 | } 66 | }; 67 | // Ensure the input is an integer, otherwise return an error 68 | if !inputs[0].dtype().is_integer() { 69 | polars_bail!(InvalidOperation: "The unpairing function can only be used on integers."); 70 | } 71 | // Cast the input series to UInt64, unwrapping the result directly 72 | let paired_series = inputs[0].cast(&DataType::UInt64).unwrap(); 73 | // Convert the series to their 64-bit integer chunked arrays 74 | let paired = paired_series.u64()?; 75 | // Apply the unpairing function element-wise to paired 76 | // If the method is "szudzik", use the Szudzik unpairing function, 77 | // otherwise use the Cantor unpairing function 78 | let unpaired: Vec> = paired 79 | .into_iter() 80 | .map(|n| match method { 81 | "hagen" => n.and_then(compute_hagen_unpair), 82 | "szudzik" => n.and_then(compute_szudzik_unpair), 83 | "cantor" => n.and_then(compute_cantor_unpair), 84 | _ => unreachable!(), 85 | }) 86 | .collect(); 87 | // Create an iterator that takes the left value of the pair if it exists or 0 otherwise 88 | let left: Vec> = unpaired.iter().map(|p| p.map(|(l, _)| l)).collect(); 89 | let right: Vec> = unpaired.iter().map(|p| p.map(|(_, r)| r)).collect(); 90 | // Create a StructChunked from the left and right vectors 91 | let out = StructChunked::new( 92 | "Pair", 93 | &[ 94 | UInt64Chunked::from_iter(left) 95 | .with_name("Left") 96 | .into_series(), 97 | UInt64Chunked::from_iter(right) 98 | .with_name("Right") 99 | .into_series(), 100 | ], 101 | )?; 102 | // Return the result as a Series wrapped in Ok 103 | Ok(out.into_series()) 104 | } 105 | -------------------------------------------------------------------------------- /polars_pairing/__init__.py: -------------------------------------------------------------------------------- 1 | from __future__ import annotations 2 | 3 | from pathlib import Path 4 | from typing import TYPE_CHECKING, Iterable, Literal, Protocol, cast 5 | 6 | import polars as pl 7 | 8 | from polars_pairing._internal import __version__ as __version__ 9 | from polars_pairing.utils import parse_version, register_plugin 10 | 11 | if TYPE_CHECKING: 12 | try: 13 | from polars._typing import IntoExpr, PolarsDataType # type: ignore [no-redef] 14 | except ImportError: 15 | from polars.type_aliases import ( # type: ignore [no-redef] 16 | IntoExpr, 17 | PolarsDataType, 18 | ) 19 | 20 | if parse_version(pl.__version__) < parse_version("0.20.16"): 21 | from polars.utils.udfs import _get_shared_lib_location 22 | 23 | lib: str | Path = _get_shared_lib_location(__file__) 24 | else: 25 | lib = Path(__file__).parent 26 | 27 | 28 | @pl.api.register_expr_namespace("pairing") 29 | class PairingFunctions: 30 | def __init__(self, expr: pl.Expr): 31 | self._expr = expr 32 | 33 | def pair( 34 | self, other: IntoExpr, method: Literal["hagen", "szudzik", "cantor"] = "hagen" 35 | ) -> pl.Expr: 36 | """ 37 | Applies a pairing function to uniquely encode two nonnegative integers 38 | into a single nonnegative integer. 39 | 40 | Parameters 41 | ---------- 42 | method 43 | The method to use for pairing. Either "hagen", "szudzik" or "cantor", 44 | defaults to "hagen". 45 | 46 | Examples 47 | -------- 48 | >>> df = pl.DataFrame( 49 | ... { 50 | ... "n1": [2, 2, 3, 1501], 51 | ... "n2": [-6, 7, 8, 10740], 52 | ... } 53 | ... ) 54 | >>> df.select(pl.col("n1").pairing.pair(pl.col("n2")).alias("pair")) 55 | shape: (4, 1) 56 | ┌───────────┐ 57 | │ pair │ 58 | │ --- │ 59 | │ u64 │ 60 | ╞═══════════╡ 61 | │ null │ 62 | │ 54 │ 63 | │ 70 │ 64 | │ 115350602 │ 65 | └───────────┘ 66 | >>> df.select(pl.col("n1").pairing.pair(pl.col("n2"), method="szudzik").alias("pair")) 67 | shape: (4, 1) 68 | ┌───────────┐ 69 | │ pair │ 70 | │ --- │ 71 | │ u64 │ 72 | ╞═══════════╡ 73 | │ null │ 74 | │ 51 │ 75 | │ 67 │ 76 | │ 115349101 │ 77 | └───────────┘ 78 | """ 79 | return register_plugin( 80 | args=[self._expr, other], 81 | kwargs={"method": method}, 82 | symbol="pair", 83 | is_elementwise=True, 84 | lib=lib, 85 | ) 86 | 87 | def unpair( 88 | self, method: Literal["hagen", "szudzik", "cantor"] = "hagen" 89 | ) -> pl.Expr: 90 | """ 91 | Applies an unpairing function to uniquely decode a nonnegative integer 92 | into two nonnegative integers. 93 | 94 | Parameters 95 | ---------- 96 | method 97 | The method to use for unpairing. Either "hagen", "szudzik" or "cantor", 98 | defaults to "hagen". 99 | 100 | Examples 101 | -------- 102 | >>> df = pl.DataFrame({"pair": [None, 607, -4, 18871362]}) 103 | >>> df.select(pl.col("pair").pairing.unpair().alias("unpair")) 104 | shape: (4, 1) 105 | ┌─────────────┐ 106 | │ unpair │ 107 | │ --- │ 108 | │ struct[2] │ 109 | ╞═════════════╡ 110 | │ {null,null} │ 111 | │ {24,15} │ 112 | │ {null,null} │ 113 | │ {513,4344} │ 114 | └─────────────┘ 115 | >>> df.select(pl.col("pair").pairing.unpair(method="szudzik").alias("unpair")) 116 | shape: (4, 1) 117 | ┌─────────────┐ 118 | │ unpair │ 119 | │ --- │ 120 | │ struct[2] │ 121 | ╞═════════════╡ 122 | │ {null,null} │ 123 | │ {24,7} │ 124 | │ {null,null} │ 125 | │ {1026,4344} │ 126 | └─────────────┘ 127 | """ 128 | return register_plugin( 129 | args=[self._expr], 130 | kwargs={"method": method}, 131 | symbol="unpair", 132 | is_elementwise=True, 133 | lib=lib, 134 | ) 135 | 136 | 137 | class DExpr(pl.Expr): 138 | @property 139 | def pairing(self) -> PairingFunctions: 140 | return PairingFunctions(self) 141 | 142 | 143 | class PairingCol(Protocol): 144 | def __call__( 145 | self, 146 | name: str | PolarsDataType | Iterable[str] | Iterable[PolarsDataType], 147 | *more_names: str | PolarsDataType, 148 | ) -> DExpr: ... 149 | 150 | def __getattr__(self, name: str) -> pl.Expr: ... 151 | 152 | @property 153 | def pairing(self) -> PairingFunctions: ... 154 | 155 | 156 | col = cast(PairingCol, pl.col) 157 | 158 | __all__ = ["col"] 159 | -------------------------------------------------------------------------------- /docs/images/hagen_pairing_light.svg: -------------------------------------------------------------------------------- 1 | 2 | 4 | 5 | 6 | 7 | 10 | 11 | 12 | 13 | 19 | 20 | 21 | 22 | 28 | 29 | 30 | 31 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 73 | 74 | 75 | 85 | 86 | 87 | 97 | 98 | 99 | 109 | 110 | 111 | 121 | 122 | 123 | 133 | 134 | 135 | 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It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . --------------------------------------------------------------------------------