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authorValentin Popov <valentin@popov.link>2024-07-19 15:37:58 +0300
committerValentin Popov <valentin@popov.link>2024-07-19 15:37:58 +0300
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tree15afc392522a9e85dc3332235e311b7d39352ea9 /vendor/smawk/src/lib.rs
parent3d48cd3f81164bbfc1a755dc1d4a9a02f98c8ddd (diff)
downloadfparkan-a990de90fe41456a23e58bd087d2f107d321f3a1.tar.xz
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-//! This crate implements various functions that help speed up dynamic
-//! programming, most importantly the SMAWK algorithm for finding row
-//! or column minima in a totally monotone matrix with *m* rows and
-//! *n* columns in time O(*m* + *n*). This is much better than the
-//! brute force solution which would take O(*mn*). When *m* and *n*
-//! are of the same order, this turns a quadratic function into a
-//! linear function.
-//!
-//! # Examples
-//!
-//! Computing the column minima of an *m* ✕ *n* Monge matrix can be
-//! done efficiently with `smawk::column_minima`:
-//!
-//! ```
-//! use smawk::Matrix;
-//!
-//! let matrix = vec![
-//! vec![3, 2, 4, 5, 6],
-//! vec![2, 1, 3, 3, 4],
-//! vec![2, 1, 3, 3, 4],
-//! vec![3, 2, 4, 3, 4],
-//! vec![4, 3, 2, 1, 1],
-//! ];
-//! let minima = vec![1, 1, 4, 4, 4];
-//! assert_eq!(smawk::column_minima(&matrix), minima);
-//! ```
-//!
-//! The `minima` vector gives the index of the minimum value per
-//! column, so `minima[0] == 1` since the minimum value in the first
-//! column is 2 (row 1). Note that the smallest row index is returned.
-//!
-//! # Definitions
-//!
-//! Some of the functions in this crate only work on matrices that are
-//! *totally monotone*, which we will define below.
-//!
-//! ## Monotone Matrices
-//!
-//! We start with a helper definition. Given an *m* ✕ *n* matrix `M`,
-//! we say that `M` is *monotone* when the minimum value of row `i` is
-//! found to the left of the minimum value in row `i'` where `i < i'`.
-//!
-//! More formally, if we let `rm(i)` denote the column index of the
-//! left-most minimum value in row `i`, then we have
-//!
-//! ```text
-//! rm(0) ≤ rm(1) ≤ ... ≤ rm(m - 1)
-//! ```
-//!
-//! This means that as you go down the rows from top to bottom, the
-//! row-minima proceed from left to right.
-//!
-//! The algorithms in this crate deal with finding such row- and
-//! column-minima.
-//!
-//! ## Totally Monotone Matrices
-//!
-//! We say that a matrix `M` is *totally monotone* when every
-//! sub-matrix is monotone. A sub-matrix is formed by the intersection
-//! of any two rows `i < i'` and any two columns `j < j'`.
-//!
-//! This is often expressed as via this equivalent condition:
-//!
-//! ```text
-//! M[i, j] > M[i, j'] => M[i', j] > M[i', j']
-//! ```
-//!
-//! for all `i < i'` and `j < j'`.
-//!
-//! ## Monge Property for Matrices
-//!
-//! A matrix `M` is said to fulfill the *Monge property* if
-//!
-//! ```text
-//! M[i, j] + M[i', j'] ≤ M[i, j'] + M[i', j]
-//! ```
-//!
-//! for all `i < i'` and `j < j'`. This says that given any rectangle
-//! in the matrix, the sum of the top-left and bottom-right corners is
-//! less than or equal to the sum of the bottom-left and upper-right
-//! corners.
-//!
-//! All Monge matrices are totally monotone, so it is enough to
-//! establish that the Monge property holds in order to use a matrix
-//! with the functions in this crate. If your program is dealing with
-//! unknown inputs, it can use [`monge::is_monge`] to verify that a
-//! matrix is a Monge matrix.
-
-#![doc(html_root_url = "https://docs.rs/smawk/0.3.2")]
-// The s! macro from ndarray uses unsafe internally, so we can only
-// forbid unsafe code when building with the default features.
-#![cfg_attr(not(feature = "ndarray"), forbid(unsafe_code))]
-
-#[cfg(feature = "ndarray")]
-pub mod brute_force;
-pub mod monge;
-#[cfg(feature = "ndarray")]
-pub mod recursive;
-
-/// Minimal matrix trait for two-dimensional arrays.
-///
-/// This provides the functionality needed to represent a read-only
-/// numeric matrix. You can query the size of the matrix and access
-/// elements. Modeled after [`ndarray::Array2`] from the [ndarray
-/// crate](https://crates.io/crates/ndarray).
-///
-/// Enable the `ndarray` Cargo feature if you want to use it with
-/// `ndarray::Array2`.
-pub trait Matrix<T: Copy> {
- /// Return the number of rows.
- fn nrows(&self) -> usize;
- /// Return the number of columns.
- fn ncols(&self) -> usize;
- /// Return a matrix element.
- fn index(&self, row: usize, column: usize) -> T;
-}
-
-/// Simple and inefficient matrix representation used for doctest
-/// examples and simple unit tests.
-///
-/// You should prefer implementing it yourself, or you can enable the
-/// `ndarray` Cargo feature and use the provided implementation for
-/// [`ndarray::Array2`].
-impl<T: Copy> Matrix<T> for Vec<Vec<T>> {
- fn nrows(&self) -> usize {
- self.len()
- }
- fn ncols(&self) -> usize {
- self[0].len()
- }
- fn index(&self, row: usize, column: usize) -> T {
- self[row][column]
- }
-}
-
-/// Adapting [`ndarray::Array2`] to the `Matrix` trait.
-///
-/// **Note: this implementation is only available if you enable the
-/// `ndarray` Cargo feature.**
-#[cfg(feature = "ndarray")]
-impl<T: Copy> Matrix<T> for ndarray::Array2<T> {
- #[inline]
- fn nrows(&self) -> usize {
- self.nrows()
- }
- #[inline]
- fn ncols(&self) -> usize {
- self.ncols()
- }
- #[inline]
- fn index(&self, row: usize, column: usize) -> T {
- self[[row, column]]
- }
-}
-
-/// Compute row minima in O(*m* + *n*) time.
-///
-/// This implements the [SMAWK algorithm] for efficiently finding row
-/// minima in a totally monotone matrix.
-///
-/// The SMAWK algorithm is from Agarwal, Klawe, Moran, Shor, and
-/// Wilbur, *Geometric applications of a matrix searching algorithm*,
-/// Algorithmica 2, pp. 195-208 (1987) and the code here is a
-/// translation [David Eppstein's Python code][pads].
-///
-/// Running time on an *m* ✕ *n* matrix: O(*m* + *n*).
-///
-/// # Examples
-///
-/// ```
-/// use smawk::Matrix;
-/// let matrix = vec![vec![4, 2, 4, 3],
-/// vec![5, 3, 5, 3],
-/// vec![5, 3, 3, 1]];
-/// assert_eq!(smawk::row_minima(&matrix),
-/// vec![1, 1, 3]);
-/// ```
-///
-/// # Panics
-///
-/// It is an error to call this on a matrix with zero columns.
-///
-/// [pads]: https://github.com/jfinkels/PADS/blob/master/pads/smawk.py
-/// [SMAWK algorithm]: https://en.wikipedia.org/wiki/SMAWK_algorithm
-pub fn row_minima<T: PartialOrd + Copy, M: Matrix<T>>(matrix: &M) -> Vec<usize> {
- // Benchmarking shows that SMAWK performs roughly the same on row-
- // and column-major matrices.
- let mut minima = vec![0; matrix.nrows()];
- smawk_inner(
- &|j, i| matrix.index(i, j),
- &(0..matrix.ncols()).collect::<Vec<_>>(),
- &(0..matrix.nrows()).collect::<Vec<_>>(),
- &mut minima,
- );
- minima
-}
-
-#[deprecated(since = "0.3.2", note = "Please use `row_minima` instead.")]
-pub fn smawk_row_minima<T: PartialOrd + Copy, M: Matrix<T>>(matrix: &M) -> Vec<usize> {
- row_minima(matrix)
-}
-
-/// Compute column minima in O(*m* + *n*) time.
-///
-/// This implements the [SMAWK algorithm] for efficiently finding
-/// column minima in a totally monotone matrix.
-///
-/// The SMAWK algorithm is from Agarwal, Klawe, Moran, Shor, and
-/// Wilbur, *Geometric applications of a matrix searching algorithm*,
-/// Algorithmica 2, pp. 195-208 (1987) and the code here is a
-/// translation [David Eppstein's Python code][pads].
-///
-/// Running time on an *m* ✕ *n* matrix: O(*m* + *n*).
-///
-/// # Examples
-///
-/// ```
-/// use smawk::Matrix;
-/// let matrix = vec![vec![4, 2, 4, 3],
-/// vec![5, 3, 5, 3],
-/// vec![5, 3, 3, 1]];
-/// assert_eq!(smawk::column_minima(&matrix),
-/// vec![0, 0, 2, 2]);
-/// ```
-///
-/// # Panics
-///
-/// It is an error to call this on a matrix with zero rows.
-///
-/// [SMAWK algorithm]: https://en.wikipedia.org/wiki/SMAWK_algorithm
-/// [pads]: https://github.com/jfinkels/PADS/blob/master/pads/smawk.py
-pub fn column_minima<T: PartialOrd + Copy, M: Matrix<T>>(matrix: &M) -> Vec<usize> {
- let mut minima = vec![0; matrix.ncols()];
- smawk_inner(
- &|i, j| matrix.index(i, j),
- &(0..matrix.nrows()).collect::<Vec<_>>(),
- &(0..matrix.ncols()).collect::<Vec<_>>(),
- &mut minima,
- );
- minima
-}
-
-#[deprecated(since = "0.3.2", note = "Please use `column_minima` instead.")]
-pub fn smawk_column_minima<T: PartialOrd + Copy, M: Matrix<T>>(matrix: &M) -> Vec<usize> {
- column_minima(matrix)
-}
-
-/// Compute column minima in the given area of the matrix. The
-/// `minima` slice is updated inplace.
-fn smawk_inner<T: PartialOrd + Copy, M: Fn(usize, usize) -> T>(
- matrix: &M,
- rows: &[usize],
- cols: &[usize],
- minima: &mut [usize],
-) {
- if cols.is_empty() {
- return;
- }
-
- let mut stack = Vec::with_capacity(cols.len());
- for r in rows {
- // TODO: use stack.last() instead of stack.is_empty() etc
- while !stack.is_empty()
- && matrix(stack[stack.len() - 1], cols[stack.len() - 1])
- > matrix(*r, cols[stack.len() - 1])
- {
- stack.pop();
- }
- if stack.len() != cols.len() {
- stack.push(*r);
- }
- }
- let rows = &stack;
-
- let mut odd_cols = Vec::with_capacity(1 + cols.len() / 2);
- for (idx, c) in cols.iter().enumerate() {
- if idx % 2 == 1 {
- odd_cols.push(*c);
- }
- }
-
- smawk_inner(matrix, rows, &odd_cols, minima);
-
- let mut r = 0;
- for (c, &col) in cols.iter().enumerate().filter(|(c, _)| c % 2 == 0) {
- let mut row = rows[r];
- let last_row = if c == cols.len() - 1 {
- rows[rows.len() - 1]
- } else {
- minima[cols[c + 1]]
- };
- let mut pair = (matrix(row, col), row);
- while row != last_row {
- r += 1;
- row = rows[r];
- if (matrix(row, col), row) < pair {
- pair = (matrix(row, col), row);
- }
- }
- minima[col] = pair.1;
- }
-}
-
-/// Compute upper-right column minima in O(*m* + *n*) time.
-///
-/// The input matrix must be totally monotone.
-///
-/// The function returns a vector of `(usize, T)`. The `usize` in the
-/// tuple at index `j` tells you the row of the minimum value in
-/// column `j` and the `T` value is minimum value itself.
-///
-/// The algorithm only considers values above the main diagonal, which
-/// means that it computes values `v(j)` where:
-///
-/// ```text
-/// v(0) = initial
-/// v(j) = min { M[i, j] | i < j } for j > 0
-/// ```
-///
-/// If we let `r(j)` denote the row index of the minimum value in
-/// column `j`, the tuples in the result vector become `(r(j), M[r(j),
-/// j])`.
-///
-/// The algorithm is an *online* algorithm, in the sense that `matrix`
-/// function can refer back to previously computed column minima when
-/// determining an entry in the matrix. The guarantee is that we only
-/// call `matrix(i, j)` after having computed `v(i)`. This is
-/// reflected in the `&[(usize, T)]` argument to `matrix`, which grows
-/// as more and more values are computed.
-pub fn online_column_minima<T: Copy + PartialOrd, M: Fn(&[(usize, T)], usize, usize) -> T>(
- initial: T,
- size: usize,
- matrix: M,
-) -> Vec<(usize, T)> {
- let mut result = vec![(0, initial)];
-
- // State used by the algorithm.
- let mut finished = 0;
- let mut base = 0;
- let mut tentative = 0;
-
- // Shorthand for evaluating the matrix. We need a macro here since
- // we don't want to borrow the result vector.
- macro_rules! m {
- ($i:expr, $j:expr) => {{
- assert!($i < $j, "(i, j) not above diagonal: ({}, {})", $i, $j);
- assert!(
- $i < size && $j < size,
- "(i, j) out of bounds: ({}, {}), size: {}",
- $i,
- $j,
- size
- );
- matrix(&result[..finished + 1], $i, $j)
- }};
- }
-
- // Keep going until we have finished all size columns. Since the
- // columns are zero-indexed, we're done when finished == size - 1.
- while finished < size - 1 {
- // First case: we have already advanced past the previous
- // tentative value. We make a new tentative value by applying
- // smawk_inner to the largest square submatrix that fits under
- // the base.
- let i = finished + 1;
- if i > tentative {
- let rows = (base..finished + 1).collect::<Vec<_>>();
- tentative = std::cmp::min(finished + rows.len(), size - 1);
- let cols = (finished + 1..tentative + 1).collect::<Vec<_>>();
- let mut minima = vec![0; tentative + 1];
- smawk_inner(&|i, j| m![i, j], &rows, &cols, &mut minima);
- for col in cols {
- let row = minima[col];
- let v = m![row, col];
- if col >= result.len() {
- result.push((row, v));
- } else if v < result[col].1 {
- result[col] = (row, v);
- }
- }
- finished = i;
- continue;
- }
-
- // Second case: the new column minimum is on the diagonal. All
- // subsequent ones will be at least as low, so we can clear
- // out all our work from higher rows. As in the fourth case,
- // the loss of tentative is amortized against the increase in
- // base.
- let diag = m![i - 1, i];
- if diag < result[i].1 {
- result[i] = (i - 1, diag);
- base = i - 1;
- tentative = i;
- finished = i;
- continue;
- }
-
- // Third case: row i-1 does not supply a column minimum in any
- // column up to tentative. We simply advance finished while
- // maintaining the invariant.
- if m![i - 1, tentative] >= result[tentative].1 {
- finished = i;
- continue;
- }
-
- // Fourth and final case: a new column minimum at tentative.
- // This allows us to make progress by incorporating rows prior
- // to finished into the base. The base invariant holds because
- // these rows cannot supply any later column minima. The work
- // done when we last advanced tentative (and undone by this
- // step) can be amortized against the increase in base.
- base = i - 1;
- tentative = i;
- finished = i;
- }
-
- result
-}
-
-#[cfg(test)]
-mod tests {
- use super::*;
-
- #[test]
- fn smawk_1x1() {
- let matrix = vec![vec![2]];
- assert_eq!(row_minima(&matrix), vec![0]);
- assert_eq!(column_minima(&matrix), vec![0]);
- }
-
- #[test]
- fn smawk_2x1() {
- let matrix = vec![
- vec![3], //
- vec![2],
- ];
- assert_eq!(row_minima(&matrix), vec![0, 0]);
- assert_eq!(column_minima(&matrix), vec![1]);
- }
-
- #[test]
- fn smawk_1x2() {
- let matrix = vec![vec![2, 1]];
- assert_eq!(row_minima(&matrix), vec![1]);
- assert_eq!(column_minima(&matrix), vec![0, 0]);
- }
-
- #[test]
- fn smawk_2x2() {
- let matrix = vec![
- vec![3, 2], //
- vec![2, 1],
- ];
- assert_eq!(row_minima(&matrix), vec![1, 1]);
- assert_eq!(column_minima(&matrix), vec![1, 1]);
- }
-
- #[test]
- fn smawk_3x3() {
- let matrix = vec![
- vec![3, 4, 4], //
- vec![3, 4, 4],
- vec![2, 3, 3],
- ];
- assert_eq!(row_minima(&matrix), vec![0, 0, 0]);
- assert_eq!(column_minima(&matrix), vec![2, 2, 2]);
- }
-
- #[test]
- fn smawk_4x4() {
- let matrix = vec![
- vec![4, 5, 5, 5], //
- vec![2, 3, 3, 3],
- vec![2, 3, 3, 3],
- vec![2, 2, 2, 2],
- ];
- assert_eq!(row_minima(&matrix), vec![0, 0, 0, 0]);
- assert_eq!(column_minima(&matrix), vec![1, 3, 3, 3]);
- }
-
- #[test]
- fn smawk_5x5() {
- let matrix = vec![
- vec![3, 2, 4, 5, 6],
- vec![2, 1, 3, 3, 4],
- vec![2, 1, 3, 3, 4],
- vec![3, 2, 4, 3, 4],
- vec![4, 3, 2, 1, 1],
- ];
- assert_eq!(row_minima(&matrix), vec![1, 1, 1, 1, 3]);
- assert_eq!(column_minima(&matrix), vec![1, 1, 4, 4, 4]);
- }
-
- #[test]
- fn online_1x1() {
- let matrix = vec![vec![0]];
- let minima = vec![(0, 0)];
- assert_eq!(online_column_minima(0, 1, |_, i, j| matrix[i][j]), minima);
- }
-
- #[test]
- fn online_2x2() {
- let matrix = vec![
- vec![0, 2], //
- vec![0, 0],
- ];
- let minima = vec![(0, 0), (0, 2)];
- assert_eq!(online_column_minima(0, 2, |_, i, j| matrix[i][j]), minima);
- }
-
- #[test]
- fn online_3x3() {
- let matrix = vec![
- vec![0, 4, 4], //
- vec![0, 0, 4],
- vec![0, 0, 0],
- ];
- let minima = vec![(0, 0), (0, 4), (0, 4)];
- assert_eq!(online_column_minima(0, 3, |_, i, j| matrix[i][j]), minima);
- }
-
- #[test]
- fn online_4x4() {
- let matrix = vec![
- vec![0, 5, 5, 5], //
- vec![0, 0, 3, 3],
- vec![0, 0, 0, 3],
- vec![0, 0, 0, 0],
- ];
- let minima = vec![(0, 0), (0, 5), (1, 3), (1, 3)];
- assert_eq!(online_column_minima(0, 4, |_, i, j| matrix[i][j]), minima);
- }
-
- #[test]
- fn online_5x5() {
- let matrix = vec![
- vec![0, 2, 4, 6, 7],
- vec![0, 0, 3, 4, 5],
- vec![0, 0, 0, 3, 4],
- vec![0, 0, 0, 0, 4],
- vec![0, 0, 0, 0, 0],
- ];
- let minima = vec![(0, 0), (0, 2), (1, 3), (2, 3), (2, 4)];
- assert_eq!(online_column_minima(0, 5, |_, i, j| matrix[i][j]), minima);
- }
-
- #[test]
- fn smawk_works_with_partial_ord() {
- let matrix = vec![
- vec![3.0, 2.0], //
- vec![2.0, 1.0],
- ];
- assert_eq!(row_minima(&matrix), vec![1, 1]);
- assert_eq!(column_minima(&matrix), vec![1, 1]);
- }
-
- #[test]
- fn online_works_with_partial_ord() {
- let matrix = vec![
- vec![0.0, 2.0], //
- vec![0.0, 0.0],
- ];
- let minima = vec![(0, 0.0), (0, 2.0)];
- assert_eq!(
- online_column_minima(0.0, 2, |_, i: usize, j: usize| matrix[i][j]),
- minima
- );
- }
-}