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/src/utils/mod2_sys.rs

/*
 *
 * SPDX-FileCopyrightText: 2025 Dario Moschetti
 * SPDX-FileCopyrightText: 2025 Sebastiano Vigna
 *
 * SPDX-License-Identifier: Apache-2.0 OR LGPL-2.1-or-later
 */

//! Fast solution of random systems of linear equations with coefficients in
//! **F**₂.

#![allow(unexpected_cfgs)]
#![allow(clippy::comparison_chain)]
use std::ptr;

use crate::{
    bit_vec,
    traits::Word,
    traits::{BitVecOps, BitVecOpsMut},
};
use anyhow::{Result, bail, ensure};
use arbitrary_chunks::ArbitraryChunks;

/// An equation on `W::BITS`-dimensional vectors with coefficients in **F**₂.
#[derive(Clone, Debug)]
pub struct Modulo2Equation<W: Word = usize> {
    /// The variables in increasing order.
    vars: Vec<u32>,
    /// The constant term
    c: W,
}

/// A system of [equations](struct.Modulo2Equation.html).
#[derive(Clone, Debug)]
pub struct Modulo2System<W: Word = usize> {
    /// The number of variables.
    num_vars: usize,
    /// The equations in the system.
    equations: Vec<Modulo2Equation<W>>,
}

impl<W: Word> Modulo2Equation<W> {
    /// Creates a new equation with given variables constant term.
    ///
    /// # Safety
    ///
    /// The caller must ensure that the variables are sorted.
    pub unsafe fn from_parts(vars: Vec<u32>, c: W) -> Self {
        debug_assert!(vars.iter().is_sorted());
        Modulo2Equation { vars, c }
    }

    #[inline(always)]
    unsafe fn add_ptr(
        mut left: *const u32,
        left_end: *const u32,
        mut right: *const u32,
        right_end: *const u32,
        mut dst: *mut u32,
    ) -> *mut u32 {
        unsafe {
            while left != left_end && right != right_end {
                let less = *left <= *right;
                let more = *left >= *right;

                let src = if less { left } else { right };
                *dst = *src;

                left = left.add(less as usize);
                right = right.add(more as usize);
                dst = dst.add((less ^ more) as usize);
            }

            let rem_left = left_end.offset_from(left) as usize;
            ptr::copy_nonoverlapping(left, dst, rem_left);
            dst = dst.add(rem_left);
            let rem_right = right_end.offset_from(right) as usize;
            ptr::copy_nonoverlapping(right, dst, rem_right);
            dst = dst.add(rem_right);
            dst
        }
    }

    pub fn add(&mut self, other: &Modulo2Equation<W>) {
        let left_range = self.vars.as_ptr_range();
        let left = left_range.start;
        let left_end = left_range.end;
        let right_range = other.vars.as_ptr_range();
        let right = right_range.start;
        let right_end = right_range.end;
        let mut vars = Vec::with_capacity(self.vars.len() + other.vars.len());
        let dst = vars.as_mut_ptr();

        unsafe {
            let copied =
                Self::add_ptr(left, left_end, right, right_end, dst).offset_from(dst) as usize;
            vars.set_len(copied);
        }

        self.vars = vars;
        self.c ^= other.c;
    }

    /// Checks whether the equation is unsolvable.
    fn is_unsolvable(&self) -> bool {
        self.vars.is_empty() && self.c != W::ZERO
    }

    /// Checks whether the equation is an identity.
    fn is_identity(&self) -> bool {
        self.vars.is_empty() && self.c == W::ZERO
    }

    /// Evaluates the XOR of the values associated to the given variables.
    fn eval_vars(vars: impl AsRef<[u32]>, values: impl AsRef<[W]>) -> W {
        let mut sum = W::ZERO;
        for &var in vars.as_ref() {
            sum ^= values.as_ref()[var as usize];
        }
        sum
    }
}

impl<W: Word> Modulo2System<W> {
    pub fn new(num_vars: usize) -> Self {
        Modulo2System {
            num_vars,
            equations: vec![],
        }
    }

    pub fn num_vars(&self) -> usize {
        self.num_vars
    }

    pub fn num_equations(&self) -> usize {
        self.equations.len()
    }

    /// Creates a new `Modulo2System` from existing equations.
    ///
    /// # Safety
    ///
    /// The caller must ensure that variables in each equation match the number
    /// of variables in the system.
    pub unsafe fn from_parts(num_vars: usize, equations: Vec<Modulo2Equation<W>>) -> Self {
        Modulo2System {
            num_vars,
            equations,
        }
    }

    /// Adds an equation to the system.
    pub fn push(&mut self, equation: Modulo2Equation<W>) {
        self.equations.push(equation);
    }

    /// Checks if a given solution satisfies the system of equations.
    pub fn check(&self, solution: &[W]) -> bool {
        assert_eq!(
            solution.len(),
            self.num_vars,
            "The number of variables in the solution ({}) does not match the number of variables in the system ({})",
            solution.len(),
            self.num_vars
        );
        self.equations
            .iter()
            .all(|eq| eq.c == Modulo2Equation::<W>::eval_vars(&eq.vars, solution))
    }

    /// Puts the system into echelon form.
    fn echelon_form(&mut self) -> Result<()> {
        let equations = &mut self.equations;
        if equations.is_empty() {
            return Ok(());
        }
        'main: for i in 0..equations.len() - 1 {
            ensure!(!equations[i].vars.is_empty());
            for j in i + 1..equations.len() {
                // SAFETY: to add the two equations, multiple references to the vector
                // of equations are needed, one of which is mutable
                let eq_j = unsafe { &*(&equations[j] as *const Modulo2Equation<W>) };
                let eq_i = &mut equations[i];

                let first_var_j = eq_j.vars[0];

                if eq_i.vars[0] == first_var_j {
                    eq_i.add(eq_j);
                    if eq_i.is_unsolvable() {
                        bail!("System is unsolvable");
                    }
                    if eq_i.is_identity() {
                        continue 'main;
                    }
                }

                if eq_i.vars[0] > first_var_j {
                    equations.swap(i, j)
                }
            }
        }
        Ok(())
    }

    /// Solves the system using Gaussian elimination.
    pub fn gaussian_elimination(&mut self) -> Result<Vec<W>> {
        self.echelon_form()?;
        let mut solution = vec![W::ZERO; self.num_vars];
        self.equations
            .iter()
            .rev()
            .filter(|eq| !eq.is_identity())
            .for_each(|eq| {
                solution[eq.vars[0] as usize] =
                    eq.c ^ Modulo2Equation::<W>::eval_vars(&eq.vars, &solution);
            });
        Ok(solution)
    }

    /// Builds the data structures needed for [lazy Gaussian
    /// elimination](https://doi.org/10.1016/j.ic.2020.104517).
    ///
    /// This method returns the variable-to-equation mapping, the weight of each
    /// variable (the number of equations in which it appears), and the priority
    /// of each equation (the number of variables in it). The
    /// variable-to-equation mapping is materialized as a vector of mutable
    /// slices, each of which points inside a provided backing vector. This
    /// approach greatly reduces the number of allocations.
    fn setup<'a>(
        &self,
        backing: &'a mut Vec<usize>,
    ) -> (Vec<&'a mut [usize]>, Vec<usize>, Vec<usize>) {
        let mut weight = vec![0; self.num_vars];
        let mut priority = vec![0; self.equations.len()];

        let mut total_vars = 0;
        for (i, equation) in self.equations.iter().enumerate() {
            priority[i] = equation.vars.len();
            total_vars += equation.vars.len();
            for &var in &equation.vars {
                weight[var as usize] += 1;
            }
        }

        backing.resize(total_vars, 0);
        let mut var_to_eq: Vec<&mut [usize]> = Vec::with_capacity(self.num_vars);

        backing.arbitrary_chunks_mut(&weight).for_each(|chunk| {
            var_to_eq.push(chunk);
        });

        let mut pos = vec![0usize; self.num_vars];
        for (i, equation) in self.equations.iter().enumerate() {
            for &var in &equation.vars {
                let var = var as usize;
                var_to_eq[var][pos[var]] = i;
                pos[var] += 1;
            }
        }

        (var_to_eq, weight, priority)
    }

    /// Solves the system using [lazy Gaussian
    /// elimination](https://doi.org/10.1016/j.ic.2020.104517).
    pub fn lazy_gaussian_elimination(&mut self) -> Result<Vec<W>> {
        let num_vars = self.num_vars;
        let num_equations = self.equations.len();

        if num_equations == 0 {
            return Ok(vec![W::ZERO; num_vars]);
        }

        let mut backing = vec![];
        let (var_to_eqs, mut weight, mut priority);
        (var_to_eqs, weight, priority) = self.setup(&mut backing);

        let mut variables = vec![0; num_vars];
        {
            let mut count = vec![0; num_equations + 1];

            for x in 0..num_vars {
                count[weight[x]] += 1
            }
            for i in 1..count.len() {
                count[i] += count[i - 1]
            }
            for i in (0..num_vars).rev() {
                count[weight[i]] -= 1;
                variables[count[weight[i]]] = i;
            }
        }

        let mut equation_list: Vec<usize> = (0..priority.len())
            .rev()
            .filter(|&x| priority[x] <= 1)
            .collect();

        let mut dense = vec![];
        let mut solved = vec![];
        let mut pivots = vec![];

        let equations = &mut self.equations;
        let mut idle = bit_vec![true; num_vars];

        let mut remaining = equations.len();

        while remaining != 0 {
            if equation_list.is_empty() {
                let mut var = variables.pop().unwrap();
                while weight[var] == 0 {
                    var = variables.pop().unwrap()
                }
                idle.set(var, false);
                var_to_eqs[var].as_ref().iter().for_each(|&eq| {
                    priority[eq] -= 1;
                    if priority[eq] == 1 {
                        equation_list.push(eq)
                    }
                });
            } else {
                remaining -= 1;
                let first = equation_list.pop().unwrap();

                if priority[first] == 0 {
                    let equation = &mut equations[first];
                    if equation.is_unsolvable() {
                        bail!("System is unsolvable")
                    }
                    if equation.is_identity() {
                        continue;
                    }
                    dense.push(equation.to_owned());
                } else if priority[first] == 1 {
                    // SAFETY: to add the equations, multiple references to the vector
                    // of equations are needed, one of which is mutable
                    let equation = unsafe { &*(&equations[first] as *const Modulo2Equation<W>) };

                    let pivot = equation
                        .vars
                        .iter()
                        .copied()
                        .find(|x| idle.get(*x as usize))
                        .expect("Missing expected idle variable in equation");
                    pivots.push(pivot as usize);
                    solved.push(first);
                    weight[pivot as usize] = 0;
                    var_to_eqs[pivot as usize]
                        .as_ref()
                        .iter()
                        .filter(|&&eq_idx| eq_idx != first)
                        .for_each(|&eq| {
                            equations[eq].add(equation);

                            priority[eq] -= 1;
                            if priority[eq] == 1 {
                                equation_list.push(eq)
                            }
                        });
                }
            }
        }

        // SAFETY: the usage of the method is safe, as the equations have the
        // right number of variables
        let mut dense_system = unsafe { Modulo2System::from_parts(num_vars, dense) };
        let mut solution = dense_system.gaussian_elimination()?;

        for i in 0..solved.len() {
            let eq = &equations[solved[i]];
            let pivot = pivots[i];
            assert!(solution[pivot] == W::ZERO);
            solution[pivot] = eq.c ^ Modulo2Equation::<W>::eval_vars(&eq.vars, &solution);
        }

        Ok(solution)
    }
}

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn test_add_ptr() {
        let a = [0, 1, 3, 4];
        let b = [0, 2, 4, 5];
        let mut c = Vec::with_capacity(a.len() + b.len());

        let ra = a.as_ptr_range();
        let rb = b.as_ptr_range();
        let mut dst = c.as_mut_ptr();
        unsafe {
            dst = Modulo2Equation::<u32>::add_ptr(ra.start, ra.end, rb.start, rb.end, dst);
            assert_eq!(dst.offset_from(c.as_ptr()), 4);
            c.set_len(4);
            assert_eq!(c, vec![1, 2, 3, 5]);
        }
    }

    #[test]
    fn test_equation_builder() {
        let eq = unsafe { Modulo2Equation::from_parts(vec![0, 1, 2], 3_usize) };
        assert_eq!(eq.vars.len(), 3);
        assert_eq!(eq.vars.to_vec(), vec![0, 1, 2]);
    }

    #[test]
    fn test_equation_addition() {
        let mut eq0 = unsafe { Modulo2Equation::from_parts(vec![1, 4, 9], 3_usize) };
        let eq1 = unsafe { Modulo2Equation::from_parts(vec![1, 4, 10], 3_usize) };
        eq0.add(&eq1);
        assert_eq!(eq0.vars, vec![9, 10]);
        assert_eq!(eq0.c, 0);
    }

    #[test]
    fn test_system_one_equation() {
        let mut system = Modulo2System::<usize>::new(2);
        let eq = unsafe { Modulo2Equation::from_parts(vec![0], 3_usize) };
        system.push(eq);
        let solution = system.lazy_gaussian_elimination();
        assert!(solution.is_ok());
        assert!(system.check(&solution.unwrap()));
    }

    #[test]
    fn test_impossible_system() {
        let mut system = Modulo2System::<usize>::new(1);
        let eq0 = unsafe { Modulo2Equation::from_parts(vec![0], 0_usize) };
        system.push(eq0);
        let eq1 = unsafe { Modulo2Equation::from_parts(vec![0], 1_usize) };
        system.push(eq1);
        let solution = system.lazy_gaussian_elimination();
        assert!(solution.is_err());
    }

    #[test]
    fn test_redundant_system() {
        let mut system = Modulo2System::<usize>::new(1);
        let eq0 = unsafe { Modulo2Equation::from_parts(vec![0], 0_usize) };
        system.push(eq0);
        let eq1 = unsafe { Modulo2Equation::from_parts(vec![0], 0_usize) };
        system.push(eq1);
        let solution = system.lazy_gaussian_elimination();
        assert!(solution.is_ok());
        assert!(system.check(&solution.unwrap()));
    }

    #[test]
    fn test_small_system() {
        let mut system = Modulo2System::<usize>::new(11);
        let mut eq = unsafe { Modulo2Equation::from_parts(vec![1, 4, 10], 0) };
        system.push(eq);
        eq = unsafe { Modulo2Equation::from_parts(vec![1, 4, 9], 2) };
        system.push(eq);
        eq = unsafe { Modulo2Equation::from_parts(vec![0, 6, 8], 0) };
        system.push(eq);
        eq = unsafe { Modulo2Equation::from_parts(vec![0, 6, 9], 1) };
        system.push(eq);
        eq = unsafe { Modulo2Equation::from_parts(vec![2, 4, 8], 2) };
        system.push(eq);
        eq = unsafe { Modulo2Equation::from_parts(vec![2, 6, 10], 0) };
        system.push(eq);

        let solution = system.lazy_gaussian_elimination();
        assert!(solution.is_ok());
        assert!(system.check(&solution.unwrap()));
    }

    #[test]
    fn test_var_to_vec_builder() {
        let system = unsafe {
            Modulo2System::from_parts(
                11,
                vec![
                    Modulo2Equation::from_parts(vec![1, 4, 10], 0_usize),
                    Modulo2Equation::from_parts(vec![1, 4, 9], 1),
                    Modulo2Equation::from_parts(vec![0, 6, 8], 2),
                    Modulo2Equation::from_parts(vec![0, 6, 9], 3),
                    Modulo2Equation::from_parts(vec![2, 4, 8], 4),
                    Modulo2Equation::from_parts(vec![2, 6, 10], 5),
                ],
            )
        };
        let mut backing: Vec<usize> = vec![];
        let (var_to_eqs, _weight, _priority) = system.setup(&mut backing);
        let expected_res = vec![
            vec![2, 3],
            vec![0, 1],
            vec![4, 5],
            vec![],
            vec![0, 1, 4],
            vec![],
            vec![2, 3, 5],
            vec![],
            vec![2, 4],
            vec![1, 3],
            vec![0, 5],
        ];
        var_to_eqs
            .iter()
            .zip(expected_res.iter())
            .for_each(|(v, e)| v.iter().zip(e.iter()).for_each(|(x, y)| assert_eq!(x, y)));
    }
}

1	/*
2	*
3	* SPDX-FileCopyrightText: 2025 Dario Moschetti
4	* SPDX-FileCopyrightText: 2025 Sebastiano Vigna
5	*
6	* SPDX-License-Identifier: Apache-2.0 OR LGPL-2.1-or-later
7	*/
8
9	//! Fast solution of random systems of linear equations with coefficients in
10	//! F₂.
11
12	#![allow(unexpected_cfgs)]
13	#![allow(clippy::comparison_chain)]
14	use std::ptr;
15
16	use crate::{
17	bit_vec,
18	traits::Word,
19	traits::{BitVecOps, BitVecOpsMut},
20	};
21	use anyhow::{Result, bail, ensure};
22	use arbitrary_chunks::ArbitraryChunks;
23
24	/// An equation on `W::BITS`-dimensional vectors with coefficients in F₂.
25	#[derive(Clone, Debug)]
26	pub struct Modulo2Equation<W: Word = usize> {
27	/// The variables in increasing order.
28	vars: Vec<u32>,
29	/// The constant term
30	c: W,
31	}
32
33	/// A system of [equations](struct.Modulo2Equation.html).
34	#[derive(Clone, Debug)]
35	pub struct Modulo2System<W: Word = usize> {
36	/// The number of variables.
37	num_vars: usize,
38	/// The equations in the system.
39	equations: Vec<Modulo2Equation<W>>,
40	}
41
42	impl<W: Word> Modulo2Equation<W> {
43	/// Creates a new equation with given variables constant term.
44	///
45	/// # Safety
46	///
47	/// The caller must ensure that the variables are sorted.
48	pub unsafe fn from_parts(vars: Vec<u32>, c: W) -> Self {	630,036✔
49	debug_assert!(vars.iter().is_sorted());	1,890,108✔
50	Modulo2Equation { vars, c }
51	}
52
53	#[inline(always)]
54	unsafe fn add_ptr(	1,756,242✔
55	mut left: *const u32,
56	left_end: *const u32,
57	mut right: *const u32,
58	right_end: *const u32,
59	mut dst: *mut u32,
60	) -> *mut u32 {
61	unsafe {
62	while left != left_end && right != right_end {	2,147,483,647✔
63	let less = left <= right;	1,099,765,859✔
64	let more = left >= right;	×
65
66	let src = if less { left } else { right };	1,099,765,859✔
67	dst = src;	×
68
UNCOV 69	left = left.add(less as usize);	×
UNCOV 70	right = right.add(more as usize);	×
UNCOV 71	dst = dst.add((less ^ more) as usize);	×
72	}
73
74	let rem_left = left_end.offset_from(left) as usize;	5,268,726✔
75	ptr::copy_nonoverlapping(left, dst, rem_left);	7,024,968✔
76	dst = dst.add(rem_left);	3,512,484✔
77	let rem_right = right_end.offset_from(right) as usize;	5,268,726✔
78	ptr::copy_nonoverlapping(right, dst, rem_right);	7,024,968✔
79	dst = dst.add(rem_right);	3,512,484✔
80	dst	1,756,242✔
81	}
82	}
83
84	pub fn add(&mut self, other: &Modulo2Equation<W>) {	1,756,241✔
85	let left_range = self.vars.as_ptr_range();	3,512,482✔
86	let left = left_range.start;	3,512,482✔
87	let left_end = left_range.end;	3,512,482✔
88	let right_range = other.vars.as_ptr_range();	3,512,482✔
89	let right = right_range.start;	3,512,482✔
90	let right_end = right_range.end;	3,512,482✔
91	let mut vars = Vec::with_capacity(self.vars.len() + other.vars.len());	8,781,205✔
92	let dst = vars.as_mut_ptr();	5,268,723✔
93
94	unsafe {
95	let copied =	3,512,482✔
96	Self::add_ptr(left, left_end, right, right_end, dst).offset_from(dst) as usize;	14,049,928✔
97	vars.set_len(copied);	3,512,482✔
98	}
99
100	self.vars = vars;	3,512,482✔
101	self.c ^= other.c;	1,756,241✔
102	}
103
104	/// Checks whether the equation is unsolvable.
105	fn is_unsolvable(&self) -> bool {	736,008✔
106	self.vars.is_empty() && self.c != W::ZERO	1,472,148✔
107	}
108
109	/// Checks whether the equation is an identity.
110	fn is_identity(&self) -> bool {	754,757✔
111	self.vars.is_empty() && self.c == W::ZERO	1,509,515✔
112	}
113
114	/// Evaluates the XOR of the values associated to the given variables.
115	fn eval_vars(vars: impl AsRef<[u32]>, values: impl AsRef<[W]>) -> W {	589,403✔
116	let mut sum = W::ZERO;	1,178,806✔
117	for &var in vars.as_ref() {	302,109,293✔
UNCOV 118	sum ^= values.as_ref()[var as usize];	×
119	}
120	sum	589,403✔
121	}
122	}
123
124	impl<W: Word> Modulo2System<W> {
125	pub fn new(num_vars: usize) -> Self {	4✔
126	Modulo2System {
127	num_vars,
128	equations: vec![],	4✔
129	}
130	}
131
UNCOV 132	pub fn num_vars(&self) -> usize {	×
UNCOV 133	self.num_vars	×
134	}
135
136	pub fn num_equations(&self) -> usize {	25✔
137	self.equations.len()	50✔
138	}
139
140	/// Creates a new `Modulo2System` from existing equations.
141	///
142	/// # Safety
143	///
144	/// The caller must ensure that variables in each equation match the number
145	/// of variables in the system.
146	pub unsafe fn from_parts(num_vars: usize, equations: Vec<Modulo2Equation<W>>) -> Self {	215✔
147	Modulo2System {
148	num_vars,
149	equations,
150	}
151	}
152
153	/// Adds an equation to the system.
154	pub fn push(&mut self, equation: Modulo2Equation<W>) {	11✔
155	self.equations.push(equation);	33✔
156	}
157
158	/// Checks if a given solution satisfies the system of equations.
159	pub fn check(&self, solution: &[W]) -> bool {	3✔
160	assert_eq!(	3✔
161	solution.len(),	6✔
162	self.num_vars,	×
UNCOV 163	"The number of variables in the solution ({}) does not match the number of variables in the system ({})",	×
UNCOV 164	solution.len(),	×
UNCOV 165	self.num_vars	×
166	);
167	self.equations	3✔
168	.iter()
169	.all(\|eq\| eq.c == Modulo2Equation::<W>::eval_vars(&eq.vars, solution))	30✔
170	}
171
172	/// Puts the system into echelon form.
173	fn echelon_form(&mut self) -> Result<()> {	59✔
174	let equations = &mut self.equations;	118✔
175	if equations.is_empty() {	118✔
176	return Ok(());	2✔
177	}
178	'main: for i in 0..equations.len() - 1 {	19,211✔
179	ensure!(!equations[i].vars.is_empty());	38,422✔
180	for j in i + 1..equations.len() {	15,020,695✔
181	// SAFETY: to add the two equations, multiple references to the vector
182	// of equations are needed, one of which is mutable
183	let eq_j = unsafe { &(&equations[j] as const Modulo2Equation<W>) };	30,002,968✔
184	let eq_i = &mut equations[i];	30,002,968✔
185
186	let first_var_j = eq_j.vars[0];	30,002,968✔
187
188	if eq_i.vars[0] == first_var_j {	15,001,484✔
189	eq_i.add(eq_j);	2,144,895✔
190	if eq_i.is_unsolvable() {	1,429,930✔
191	bail!("System is unsolvable");	31✔
192	}
UNCOV 193	if eq_i.is_identity() {	×
UNCOV 194	continue 'main;	×
195	}
196	}
197
198	if eq_i.vars[0] > first_var_j {	15,001,453✔
199	equations.swap(i, j)	30,317,403✔
200	}
201	}
202	}
203	Ok(())	26✔
204	}
205
206	/// Solves the system using Gaussian elimination.
207	pub fn gaussian_elimination(&mut self) -> Result<Vec<W>> {	59✔
208	self.echelon_form()?;	149✔
209	let mut solution = vec![W::ZERO; self.num_vars];	28✔
UNCOV 210	self.equations	×
211	.iter()
212	.rev()
213	.filter(\|eq\| !eq.is_identity())	37,760✔
214	.for_each(\|eq\| {	18,880✔
215	solution[eq.vars[0] as usize] =	56,640✔
216	eq.c ^ Modulo2Equation::<W>::eval_vars(&eq.vars, &solution);	56,640✔
217	});
UNCOV 218	Ok(solution)	×
219	}
220
221	/// Builds the data structures needed for [lazy Gaussian
222	/// elimination](https://doi.org/10.1016/j.ic.2020.104517).
223	///
224	/// This method returns the variable-to-equation mapping, the weight of each
225	/// variable (the number of equations in which it appears), and the priority
226	/// of each equation (the number of variables in it). The
227	/// variable-to-equation mapping is materialized as a vector of mutable
228	/// slices, each of which points inside a provided backing vector. This
229	/// approach greatly reduces the number of allocations.
230	fn setup<'a>(	160✔
231	&self,
232	backing: &'a mut Vec<usize>,
233	) -> (Vec<&'a mut [usize]>, Vec<usize>, Vec<usize>) {
234	let mut weight = vec![0; self.num_vars];	480✔
235	let mut priority = vec![0; self.equations.len()];	640✔
236
237	let mut total_vars = 0;	320✔
238	for (i, equation) in self.equations.iter().enumerate() {	630,353✔
239	priority[i] = equation.vars.len();	×
UNCOV 240	total_vars += equation.vars.len();	×
241	for &var in &equation.vars {	4,410,211✔
UNCOV 242	weight[var as usize] += 1;	×
243	}
244	}
245
246	backing.resize(total_vars, 0);	480✔
247	let mut var_to_eq: Vec<&mut [usize]> = Vec::with_capacity(self.num_vars);	640✔
248
249	backing.arbitrary_chunks_mut(&weight).for_each(\|chunk\| {	1,223,490✔
250	var_to_eq.push(chunk);	3,669,030✔
251	});
252
253	let mut pos = vec![0usize; self.num_vars];	480✔
254	for (i, equation) in self.equations.iter().enumerate() {	630,353✔
255	for &var in &equation.vars {	4,410,211✔
UNCOV 256	let var = var as usize;	×
UNCOV 257	var_to_eq[var][pos[var]] = i;	×
UNCOV 258	pos[var] += 1;	×
259	}
260	}
261
262	(var_to_eq, weight, priority)	320✔
263	}
264
265	/// Solves the system using [lazy Gaussian
266	/// elimination](https://doi.org/10.1016/j.ic.2020.104517).
267	pub fn lazy_gaussian_elimination(&mut self) -> Result<Vec<W>> {	159✔
268	let num_vars = self.num_vars;	318✔
269	let num_equations = self.equations.len();	477✔
270
271	if num_equations == 0 {	159✔
272	return Ok(vec![W::ZERO; num_vars]);	×
273	}
274
UNCOV 275	let mut backing = vec![];	×
276	let (var_to_eqs, mut weight, mut priority);	×
UNCOV 277	(var_to_eqs, weight, priority) = self.setup(&mut backing);	×
278
UNCOV 279	let mut variables = vec![0; num_vars];	×
280	{
281	let mut count = vec![0; num_equations + 1];	×
282
283	for x in 0..num_vars {	1,222,999✔
284	count[weight[x]] += 1	×
285	}
286	for i in 1..count.len() {	630,027✔
287	count[i] += count[i - 1]	×
288	}
289	for i in (0..num_vars).rev() {	1,222,999✔
UNCOV 290	count[weight[i]] -= 1;	×
UNCOV 291	variables[count[weight[i]]] = i;	×
292	}
293	}
294
UNCOV 295	let mut equation_list: Vec<usize> = (0..priority.len())	×
296	.rev()
297	.filter(\|&x\| priority[x] <= 1)	630,027✔
298	.collect();
299
UNCOV 300	let mut dense = vec![];	×
301	let mut solved = vec![];	×
302	let mut pivots = vec![];	×
303
304	let equations = &mut self.equations;	×
UNCOV 305	let mut idle = bit_vec![true; num_vars];	×
306
UNCOV 307	let mut remaining = equations.len();	×
308
309	while remaining != 0 {	661,774✔
310	if equation_list.is_empty() {	1,323,430✔
311	let mut var = variables.pop().unwrap();	162,520✔
312	while weight[var] == 0 {	591,296✔
313	var = variables.pop().unwrap()	550,666✔
314	}
315	idle.set(var, false);	121,890✔
316	var_to_eqs[var].as_ref().iter().for_each(\|&eq\| {	347,681✔
317	priority[eq] -= 1;	225,791✔
318	if priority[eq] == 1 {	225,791✔
319	equation_list.push(eq)	127,209✔
320	}
321	});
322	} else {
323	remaining -= 1;	621,085✔
UNCOV 324	let first = equation_list.pop().unwrap();	×
325
UNCOV 326	if priority[first] == 0 {	×
327	let equation = &mut equations[first];	42,086✔
328	if equation.is_unsolvable() {	42,086✔
329	bail!("System is unsolvable")	100✔
330	}
UNCOV 331	if equation.is_identity() {	×
332	continue;	1✔
333	}
334	dense.push(equation.to_owned());	83,768✔
335	} else if priority[first] == 1 {	600,042✔
336	// SAFETY: to add the equations, multiple references to the vector
337	// of equations are needed, one of which is mutable
338	let equation = unsafe { &(&equations[first] as const Modulo2Equation<W>) };	1,200,084✔
339
340	let pivot = equation	1,200,084✔
341	.vars	600,042✔
342	.iter()
343	.copied()
344	.find(\|x\| idle.get(*x as usize))	467,670,243✔
345	.expect("Missing expected idle variable in equation");
346	pivots.push(pivot as usize);	1,800,126✔
347	solved.push(first);	1,800,126✔
348	weight[pivot as usize] = 0;	600,042✔
349	var_to_eqs[pivot as usize]	600,042✔
350	.as_ref()
351	.iter()
352	.filter(\|&&eq_idx\| eq_idx != first)	3,882,676✔
353	.for_each(\|&eq\| {	1,641,317✔
354	equations[eq].add(equation);	3,123,825✔
355
356	priority[eq] -= 1;	1,041,275✔
357	if priority[eq] == 1 {	1,041,275✔
358	equation_list.push(eq)	1,737,117✔
359	}
360	});
361	}
362	}
363	}
364
365	// SAFETY: the usage of the method is safe, as the equations have the
366	// right number of variables
367	let mut dense_system = unsafe { Modulo2System::from_parts(num_vars, dense) };	59✔
368	let mut solution = dense_system.gaussian_elimination()?;	59✔
369
370	for i in 0..solved.len() {	570,514✔
371	let eq = &equations[solved[i]];	1,711,542✔
372	let pivot = pivots[i];	1,141,028✔
373	assert!(solution[pivot] == W::ZERO);	1,141,028✔
374	solution[pivot] = eq.c ^ Modulo2Equation::<W>::eval_vars(&eq.vars, &solution);	570,514✔
375	}
376
377	Ok(solution)	28✔
378	}
379	}
380
381	#[cfg(test)]
382	mod tests {
383	use super::*;
384
385	#[test]
386	fn test_add_ptr() {
387	let a = [0, 1, 3, 4];
388	let b = [0, 2, 4, 5];
389	let mut c = Vec::with_capacity(a.len() + b.len());
390
391	let ra = a.as_ptr_range();
392	let rb = b.as_ptr_range();
393	let mut dst = c.as_mut_ptr();
394	unsafe {
395	dst = Modulo2Equation::<u32>::add_ptr(ra.start, ra.end, rb.start, rb.end, dst);
396	assert_eq!(dst.offset_from(c.as_ptr()), 4);
397	c.set_len(4);
398	assert_eq!(c, vec![1, 2, 3, 5]);
399	}
400	}
401
402	#[test]
403	fn test_equation_builder() {
404	let eq = unsafe { Modulo2Equation::from_parts(vec![0, 1, 2], 3_usize) };
405	assert_eq!(eq.vars.len(), 3);
406	assert_eq!(eq.vars.to_vec(), vec![0, 1, 2]);
407	}
408
409	#[test]
410	fn test_equation_addition() {
411	let mut eq0 = unsafe { Modulo2Equation::from_parts(vec![1, 4, 9], 3_usize) };
412	let eq1 = unsafe { Modulo2Equation::from_parts(vec![1, 4, 10], 3_usize) };
413	eq0.add(&eq1);
414	assert_eq!(eq0.vars, vec![9, 10]);
415	assert_eq!(eq0.c, 0);
416	}
417
418	#[test]
419	fn test_system_one_equation() {
420	let mut system = Modulo2System::<usize>::new(2);
421	let eq = unsafe { Modulo2Equation::from_parts(vec![0], 3_usize) };
422	system.push(eq);
423	let solution = system.lazy_gaussian_elimination();
424	assert!(solution.is_ok());
425	assert!(system.check(&solution.unwrap()));
426	}
427
428	#[test]
429	fn test_impossible_system() {
430	let mut system = Modulo2System::<usize>::new(1);
431	let eq0 = unsafe { Modulo2Equation::from_parts(vec![0], 0_usize) };
432	system.push(eq0);
433	let eq1 = unsafe { Modulo2Equation::from_parts(vec![0], 1_usize) };
434	system.push(eq1);
435	let solution = system.lazy_gaussian_elimination();
436	assert!(solution.is_err());
437	}
438
439	#[test]
440	fn test_redundant_system() {
441	let mut system = Modulo2System::<usize>::new(1);
442	let eq0 = unsafe { Modulo2Equation::from_parts(vec![0], 0_usize) };
443	system.push(eq0);
444	let eq1 = unsafe { Modulo2Equation::from_parts(vec![0], 0_usize) };
445	system.push(eq1);
446	let solution = system.lazy_gaussian_elimination();
447	assert!(solution.is_ok());
448	assert!(system.check(&solution.unwrap()));
449	}
450
451	#[test]
452	fn test_small_system() {
453	let mut system = Modulo2System::<usize>::new(11);
454	let mut eq = unsafe { Modulo2Equation::from_parts(vec![1, 4, 10], 0) };
455	system.push(eq);
456	eq = unsafe { Modulo2Equation::from_parts(vec![1, 4, 9], 2) };
457	system.push(eq);
458	eq = unsafe { Modulo2Equation::from_parts(vec![0, 6, 8], 0) };
459	system.push(eq);
460	eq = unsafe { Modulo2Equation::from_parts(vec![0, 6, 9], 1) };
461	system.push(eq);
462	eq = unsafe { Modulo2Equation::from_parts(vec![2, 4, 8], 2) };
463	system.push(eq);
464	eq = unsafe { Modulo2Equation::from_parts(vec![2, 6, 10], 0) };
465	system.push(eq);
466
467	let solution = system.lazy_gaussian_elimination();
468	assert!(solution.is_ok());
469	assert!(system.check(&solution.unwrap()));
470	}
471
472	#[test]
473	fn test_var_to_vec_builder() {
474	let system = unsafe {
475	Modulo2System::from_parts(
476	11,
477	vec![
478	Modulo2Equation::from_parts(vec![1, 4, 10], 0_usize),
479	Modulo2Equation::from_parts(vec![1, 4, 9], 1),
480	Modulo2Equation::from_parts(vec![0, 6, 8], 2),
481	Modulo2Equation::from_parts(vec![0, 6, 9], 3),
482	Modulo2Equation::from_parts(vec![2, 4, 8], 4),
483	Modulo2Equation::from_parts(vec![2, 6, 10], 5),
484	],
485	)
486	};
487	let mut backing: Vec<usize> = vec![];
488	let (var_to_eqs, _weight, _priority) = system.setup(&mut backing);
489	let expected_res = vec![
490	vec![2, 3],
491	vec![0, 1],
492	vec![4, 5],
493	vec![],
494	vec![0, 1, 4],
495	vec![],
496	vec![2, 3, 5],
497	vec![],
498	vec![2, 4],
499	vec![1, 3],
500	vec![0, 5],
501	];
502	var_to_eqs
503	.iter()
504	.zip(expected_res.iter())
505	.for_each(\|(v, e)\| v.iter().zip(e.iter()).for_each(\|(x, y)\| assert_eq!(x, y)));
506	}
507	}

vigna / sux-rs / 18495149581

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