Class Linear3SystemSolver
public class Linear3SystemSolver extends Object
Instances of this class contain the data necessary to generate the random system and solve it. At construction time, you provide just the desired number of equations and variables; then, you have two possibilities:
 You can call
generateAndSolve(Iterable, long, LongBigList)
providing a value list; it will generate a random linear system on F_{2} with three variables per equation; the constant term for the kth equation will be the kth element of the provided list. This kind of system is useful for computing aGOV3Function
.  You can call
generateAndSolve(Iterable, long, LongBigList)
with anull
value list; it will generate a random linear system on F_{3} with three variables per equation; to compute the constant term, the system is viewed as a 3hypergraph on the set of variable, and it is oriented—to each equation with associate one of its variables, and distinct equations are associated with distinct variables. The index (0, 1 or 2) of the variable associated to an equation becomes the constant part. This kind of system is useful for computing aGOVMinimalPerfectHashFunction
.
In both cases, the number of elements returned by the provide Iterable
must
be equal to the number of equation passed at construction time.
To guarantee consistent results when reading a GOV3Function
or GOVMinimalPerfectHashFunction
,
the method signatureToEquation(long[], long, int, int[])
can be used to retrieve, starting from
the signature generated by a bit vector, the corresponding equation. While having a function returning the edge starting
from a key would be more objectoriented and avoid hidden dependencies, it would also require
storing the transformation provided at construction time, which would make this class nonthreadsafe.
Just be careful to transform the keys into bit vectors using
the same TransformationStrategy
and the same hash function used to generate the random linear system.
Support for preprocessed keys
This class provides two special access points for classes that have predigested their keys. The methods
generation methods and signatureToEquation(long[], long, int, int[])
use
fixedlength 128bit signatures as key under the form of pairs of longs. The intended usage is that of
turning the keys into such a signature using SpookyHash and
then operating directly on the signatures. This is particularly useful in bucketed constructions, where
the keys are replaced by their signatures in the first place. Note that the signatures are actually
rehashed using Hashes.spooky4(long, long, long, long[])
—this is necessary to vary the linear system
whenever it is unsolvable (or the associated hypergraph is not orientable).
Warning: you cannot mix the bitvectorbased and the signaturebased constructors and static methods. It is your responsibility to pair them correctly.
Implementation details
We use Jenkins's SpookyHash to compute two 64bit hash values.
The XOR trick
Before proceeding with the actual solution of the linear system, we perform a peeling of the hypergraph associated with the system, which iteratively removes edges that contain a vertex of degree one. Since the list of edges incident to a vertex is accessed during the peeling process only when the vertex has degree one, we can actually store in a single integer the XOR of the indices of all edges incident to the vertex. This approach significantly simplifies the code and reduces memory usage. It is described in detail in “Cacheoblivious peeling of random hypergraphs”, by Djamal Belazzougui, Paolo Boldi, Giuseppe Ottaviano, Rossano Venturini, and Sebastiano Vigna, Proc. Data Compression Conference 2014, 2014.
Rounds and Logging
Building and sorting a large 3regular linear system is difficult, as solving linear systems is superquadratic. This classes uses techniques introduced in the paper quoted in the introduction (and in particular broadword programming and lazy Gaussian elimination) to speed up the process by orders of magnitudes.
Note that we might generate nonorientable hypergraphs, or nonsolvable systems, in which case one has to try again with a different seed.
To help diagnosing problem with the generation process, this class will log at debug level what's happening.
 Author:
 Sebastiano Vigna

Field Summary
Fields Modifier and Type Field Description long
numPeeled
The number of peeled nodes.long[]
solution
The vector of solutions.int
unorientable
The number of generated unorientable graphs.int
unsolvable
The number of generated unsolvable systems. 
Constructor Summary
Constructors Constructor Description Linear3SystemSolver(int numVariables, int numEquations)
Creates a linear 3regular system solver for a given number of variables and equations. 
Method Summary
Modifier and Type Method Description boolean
generateAndSolve(Iterable<long[]> iterable, long seed, LongBigList valueList)
Generates a random 3regular linear system on F_{2} or F_{3} and tries to solve it.boolean
generateAndSolve(Iterable<long[]> signatures, long seed, LongBigList valueList, Codec.Coder coder, int m, int w)
boolean
generateAndSolve(Iterable<long[]> signatures, long seed, LongBigList valueList, Codec.Coder coder, int m, int w, boolean peelOnly)
static void
signatureToEquation(long[] signature, long seed, int numVariables, int[] e)
Turns a signature into an equation.

Field Details

solution
public long[] solutionThe vector of solutions. 
unsolvable
public int unsolvableThe number of generated unsolvable systems. 
unorientable
public int unorientableThe number of generated unorientable graphs. 
numPeeled
public long numPeeledThe number of peeled nodes.


Constructor Details

Linear3SystemSolver
public Linear3SystemSolver(int numVariables, int numEquations)Creates a linear 3regular system solver for a given number of variables and equations. Parameters:
numVariables
 the number of variables.numEquations
 the number of equations.


Method Details

signatureToEquation
public static void signatureToEquation(long[] signature, long seed, int numVariables, int[] e)Turns a signature into an equation.If there are no variables the vector
e
will be filled with 1. Parameters:
signature
 a signature (two longs). Note that if a longer vector is provided, only the first two elements will be used.seed
 the seed for the hash function.numVariables
 the nonzero number of variables in the system.e
 an array to store the resulting equation.

generateAndSolve
Generates a random 3regular linear system on F_{2} or F_{3} and tries to solve it.The constant part is provided by
valueList
. If it isnull
, the system will be on F_{3} and the constant part will be obtained by orientation, otherwise the system will be on F_{2}. Parameters:
iterable
 an iterable returning signatures (two longs). Note that if a longer vectors are returned, only the first two elements will be used.seed
 a 64bit random seed.valueList
 a value list containing the constant part, ornull
if the constant part should be computed by orientation. Returns:
 true if a solution was found.
 See Also:
Linear3SystemSolver

generateAndSolve
public boolean generateAndSolve(Iterable<long[]> signatures, long seed, LongBigList valueList, Codec.Coder coder, int m, int w) 
generateAndSolve
public boolean generateAndSolve(Iterable<long[]> signatures, long seed, LongBigList valueList, Codec.Coder coder, int m, int w, boolean peelOnly)
