1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233
mod divisors; mod factorize; // Provide submodule functions as crate functions pub use divisors::divisors; pub use factorize::factorize; #[macro_use] extern crate failure; use common_macros::hash_map; use std::collections::HashMap; /// Error to encapsulate invalid ACM construction parameters. #[derive(Fail, Debug)] #[fail(display = "{} incongruent to {} modulus {}.", _0, _1, _2)] pub struct ACMError(u32, u32, u32); /// Arithmetic congruence monoid implementation. #[derive(Debug)] pub struct ArithmeticCongruenceMonoid { a: u32, b: u32, factorizations: HashMap<u32, Vec<Vec<u32>>>, } type ACM = ArithmeticCongruenceMonoid; impl ArithmeticCongruenceMonoid { /// Attempts to construct a new ACM with components `a` and `b`, /// requiring that `a % b == a.pow(2) % b`. /// /// # Examples /// ``` /// // A valid ACM (1 % 4 == 1 == 1*1 % 4) /// assert!(acm::ArithmeticCongruenceMonoid::new(1, 4).is_ok()); /// /// // An invalid ACM (2 % 4 == 2 != 0 == 2*2 % 4) /// assert!(acm::ArithmeticCongruenceMonoid::new(2, 4).is_err()); /// ``` pub fn new(a: u32, b: u32) -> Result<ACM, ACMError> { if a % b == a.pow(2) % b { Ok(ACM { a: a % b, b, factorizations: hash_map! { 1 => vec![vec![]] }, }) } else { Err(ACMError(a, a * a, b)) } } /// Returns the `a` component of the ACM. pub fn a(&self) -> u32 { self.a } /// Returns the `b` component of the ACM. pub fn b(&self) -> u32 { self.b } /// Returns the nearst ACM element equal to or below `n`. /// /// # Examples /// ``` /// let acm = acm::ArithmeticCongruenceMonoid::new(1, 4).unwrap(); /// assert_eq!(acm.element(5), 5); /// assert_eq!(acm.element(6), 5); /// ``` pub fn element(&self, n: u32) -> u32 { n - (n - self.a) % self.b } /// Generate `n` ACM elements starting at nearest element below or equal to `s`. /// /// # Examples /// ``` /// let acm = acm::ArithmeticCongruenceMonoid::new(1, 4).unwrap(); /// assert_eq!(acm.elements(5, 1), [1, 5, 9, 13, 17]); /// ``` pub fn elements(&self, n: u32, s: u32) -> Vec<u32> { let s = self.element(s); (0..n).map(|i| s + i * self.b).collect() } /// Returns `true` if `n` is an element of the ACM. /// /// # Examples /// ``` /// let acm = acm::ArithmeticCongruenceMonoid::new(1, 4).unwrap(); /// assert!( acm.contains(5)); /// assert!(!acm.contains(6)); /// ``` pub fn contains(&self, n: u32) -> bool { n == 1 || n % self.b == self.a } /// Returns the ACM element divisors of an integer `n`. /// /// # Examples /// ``` /// let acm = acm::ArithmeticCongruenceMonoid::new(1, 4).unwrap(); /// assert_eq!(acm.divisors(225), [1, 9, 5, 25, 45, 225]); /// ``` pub fn divisors(&self, n: u32) -> Vec<u32> { let mut ds = vec![]; for d in divisors(n) { if self.contains(d) { ds.push(d); } } ds } /// Returns a reference to the vector of ACM atom factorizations of an integer `n`. /// If `n` is not an element of the ACM then the vector will be empty. /// Because factorization results are stored internally to the ACM in order to reduce /// computational costs, using [`factorize`] requires that the ACM binding be declared mutable. /// /// # Examples /// ``` /// let mut acm = acm::ArithmeticCongruenceMonoid::new(3, 6).unwrap(); /// assert_eq!(acm.factorize(1), &[[]]); /// assert_eq!(acm.factorize(2), &[[]; 0]); /// assert_eq!(acm.factorize(3), &[[3]]); /// assert_eq!(acm.factorize(9), &[[3, 3]]); /// assert_eq!(acm.factorize(225), &[[15, 15], [3, 75]]); /// ``` /// [`factorize`]: ./struct.ArithmeticCongruenceMonoid.html#methods.factorize pub fn factorize(&mut self, n: u32) -> &Vec<Vec<u32>> { if self.factorizations.contains_key(&n) { return self.factorizations.get(&n).unwrap(); } self.factorizations.insert(n, vec![]); let factorization_is_empty = |acm: &ACM, n: u32| acm.factorizations.get(&n).unwrap().is_empty(); let add_factorization = |acm: &mut ACM, n: u32, factorization: Vec<u32>| { acm.factorizations.get_mut(&n).unwrap().push(factorization) }; if self.contains(n) { let n_divisors = self.divisors(n); for (d, q) in n_divisors .iter() .skip(1) .map(|d| (*d, n / d)) .filter(|(_d, q)| n_divisors.contains(q)) { if q == 1 && factorization_is_empty(self, n) { add_factorization(self, n, vec![n]); } else if let Some(d_fs) = self.factorize(d).first() { if d_fs.len() == 1 { for mut q_f in self.factorize(q).clone().into_iter() { if q_f.is_empty() || &d >= q_f.last().unwrap() { q_f.push(d); add_factorization(self, n, q_f); } } } } } } self.factorizations.get(&n).unwrap() } /// Returns `true` if `n` is atomic under the ACM (is an ACM element, and cannot be expressed /// as a product of smaller ACM atoms). /// Because of underlying usage of [`factorize`], using [`atomic`] requires that the ACM binding be /// declared mutable. /// /// # Examples /// ``` /// let mut acm = acm::ArithmeticCongruenceMonoid::new(1, 4).unwrap(); /// assert!( acm.contains(5) && acm.atomic(5)); /// assert!(!acm.contains(15) && !acm.atomic(15)); /// assert!( acm.contains(25) && !acm.atomic(25)); /// ``` /// [`factorize`]: ./struct.ArithmeticCongruenceMonoid.html#method.factorize /// [`atomic`]: ./struct.ArithmeticCongruenceMonoid.html#method.atomic pub fn atomic(&mut self, n: u32) -> bool { if !self.contains(n) { return false; } let n_fs = self.factorize(n); n_fs.len() == 1 && n_fs.first().unwrap().len() == 1 } /// Returns a vector of the first `n` atoms of the ACM. /// Because of underlying usage of [`atomic`], using [`atoms`] requires that the ACM binding be /// declared mutable. /// /// # Examples /// ``` /// let mut acm = acm::ArithmeticCongruenceMonoid::new(3, 6).unwrap(); /// assert_eq!(acm.atoms(5, acm.a()), [3, 15, 21, 33, 39]); /// ``` /// [`atomic`]: ./struct.ArithmeticCongruenceMonoid.html#method.atomic /// [`atoms`]: ./struct.ArithmeticCongruenceMonoid.html#method.atoms pub fn atoms(&mut self, n: u32, s: u32) -> Vec<u32> { let s = self.element(s); (s..) .step_by(self.b as usize) .filter(|x| self.atomic(*x)) .take(n as usize) .collect() } /// Returns a vector of the density (distance between) the first `n` atoms of the ACM. /// Because of underlying usage of [`atoms`], using [`atom_density`] requires that the ACM /// binding be declared mutable. /// /// # Examples /// ``` /// let mut acm = acm::ArithmeticCongruenceMonoid::new(1, 4).unwrap(); /// assert_eq!(acm.atom_density(10, acm.a()), [4, 4, 4, 4, 8, 4, 4, 4, 8]); /// ``` /// [`atoms`]: ./struct.ArithmeticCongruenceMonoid.html#method.atoms /// [`atom_density`]: ./struct.ArithmeticCongruenceMonoid.html#method.atom_density pub fn atom_density(&mut self, n: u32, s: u32) -> Vec<u32> { let s = self.element(s); let atoms = self.atoms(n, s); atoms.iter() .zip(atoms.iter().skip(1)) .map(|(a1, a2)| a2 - a1) .collect::<Vec<u32>>() } }