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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.
// 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
///
/// 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.
///
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.
///
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
);
}
}