-- Homogeneous Linear Diaphantine Equation solver -- -- Copyright (C) 2009 John D. Ramsdell -- -- This program is free software: you can redistribute it and/or modify -- it under the terms of the GNU General Public License as published by -- the Free Software Foundation, either version 3 of the License, or -- (at your option) any later version. -- This program is distributed in the hope that it will be useful, -- but WITHOUT ANY WARRANTY; without even the implied warranty of -- MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the -- GNU General Public License for more details. -- You should have received a copy of the GNU General Public License -- along with this program. If not, see <http://www.gnu.org/licenses/>. -- | -- Module : Algebra.CommutativeMonoid.HomLinDiaphEq -- Copyright : (C) 2009 John D. Ramsdell -- License : GPL -- -- Homogeneous Linear Diaphantine Equation solver. -- -- The solver uses the algorithm of Contejean and Devie as specified -- by David Papp and Bela Vizari in \"Effective Solutions of Linear -- Diophantine Equation Systems with an Application to Chemistry\", -- Rutcor Research Report RRR 28-2004, September, 2004, -- <http://rutcor.rutgers.edu/pub/rrr/reports2004/28_2004.ps>, after -- modification so as to ensure every basis vector is considered. -- -- The algorithm for systems of homogeneous linear Diophantine -- equations follows. Let e[k] be the kth basis vector for 1 <= k <= -- n. To find the minimal, non-negative solutions M to the system of -- equations sum(i=1,n,a[i]*v[i]) = 0, the algorithm of Contejean and -- Devie is: -- -- 1. [init] A := {e[k] | 1 <= k <= n}; M := {} -- -- 2. [new minimal results] M := M + {a in A | a is a solution} -- -- 3. [unnecessary branches] A := {a in A | all m in M : some -- 1 <= k <= n : m[k] < a[k]} -- -- 4. [test] If A = {}, stop -- -- 5. [breadth-first search] A := {a + e[k] | a in A, 1 <= k <= n, -- \<sum(i=1,n,a[i]*v[i]),v[k]> \< 0}; go to step 2 module Algebra.CommutativeMonoid.HomLinDiaphEq (homLinDiaphEq) where import Data.Array import Data.Set (Set) import qualified Data.Set as S {-- Debugging hack import System.IO.Unsafe z :: Show a => a -> b -> b z x y = seq (unsafePerformIO (print x)) y --} type Vector a = Array Int a vector :: Int -> [a] -> Vector a vector n elems = listArray (0, n - 1) elems -- | The 'homLinDiaphEq' function takes a list of integers that -- specifies a homogeneous linear Diophantine equation, and returns -- the equation's minimal, non-negative solutions. homLinDiaphEq :: [Int] -> [[Int]] homLinDiaphEq [] = [] homLinDiaphEq v = newMinimalResults (vector n v) (basis n) S.empty where n = length v -- Construct the basis vectors for an n-dimensional space basis :: Int -> Set (Vector Int) basis n = foldl (flip S.insert) S.empty [ z // [(k, 1)] | k <- indices z ] where z = vector n $ replicate n 0 -- The main loop has been reorganized to ensure every basis vector is -- considered. The breadth-first search step is now the last step. -- Add elements of a that solve the equation to m and the output newMinimalResults :: Vector Int -> Set (Vector Int) -> Set (Vector Int) -> [[Int]] newMinimalResults v a m = loop m (S.toList a) -- Test each element in a where loop m [] = nextSearch v a m -- Generate new a and try again loop m (x:xs) | prod v x == 0 && S.notMember x m = elems x:loop (S.insert x m) xs -- Answer found | otherwise = loop m xs -- Generate the next set of test vectors--if there aren't any, your done nextSearch :: Vector Int -> Set (Vector Int) -> Set (Vector Int) -> [[Int]] nextSearch v a m = if S.null a' then [] else newMinimalResults v (breadthFirstSearch v a') m where a' = unnecessaryBranches a m -- Remove unnecessary branches. A test vector is not necessary if all -- of its elements are greater than or equal to the elements of some -- minimal solution. unnecessaryBranches :: Set (Vector Int) -> Set (Vector Int) -> Set (Vector Int) unnecessaryBranches a m = S.filter f a where f x = all (g x) (S.toList m) g x y = not (lessEq y x) -- Compare vectors element-wise. lessEq :: Vector Int -> Vector Int -> Bool lessEq x y = all (\i-> x!i <= y!i) (indices x) -- Breadth-first search using the algorithm of Contejean and Devie breadthFirstSearch :: Vector Int -> Set (Vector Int) -> Set (Vector Int) breadthFirstSearch v a = S.fold f S.empty a where f x acc = foldl (flip S.insert) acc [ x // [(k, x!k + 1)] | k <- indices x, prod v x * v!k < 0 ] -- Contejean-Devie contribution -- Inner product prod :: Vector Int -> Vector Int -> Int prod x y = sum [ x!i * y!i | i <- indices x ]