public class Linear3SystemSolver extends ObjectA class implementing generation and solution of a random 3-regular linear system on F2 or F3 using the techniques described by Marco Genuzio, Giuseppe Ottaviano and Sebastiano Vigna in “Fast Scalable Construction of (Minimal Perfect Hash) Functions”, 15th International Symposium on Experimental Algorithms — SEA 2016, Lecture Notes in Computer Science, Springer, 2016. It can be seen as a generalization of the 3-hypergraph edge sorting procedure that is necessary for the Majewski-Wormald-Havas-Czech technique.
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 F2 with three variables per equation; the constant term for the k-th equation will be the k-th element of the provided list. This kind of system is useful for computing a
- You can call
generateAndSolve(Iterable, long, LongBigList)with a
nullvalue list; it will generate a random linear system on F3 with three variables per equation; to compute the constant term, the system is viewed as a 3-hypergraph 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 a
In both cases, the number of elements returned by the provide
Iterablemust be equal to the number of equation passed at construction time.
To guarantee consistent results when reading a
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 object-oriented and avoid hidden dependencies, it would also require storing the transformation provided at construction time, which would make this class non-thread-safe. Just be careful to transform the keys into bit vectors using the same
TransformationStrategyand 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 pre-digested their keys. The methods generation methods and
signatureToEquation(long, long, int, int)use fixed-length 128-bit 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 bitvector-based and the signature-based constructors and static methods. It is your responsibility to pair them correctly.
We use Jenkins's SpookyHash to compute two 64-bit 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 “Cache-oblivious 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 3-regular 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 non-orientable hypergraphs, or non-solvable 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.
- Sebastiano Vigna
- You can call
Constructors Constructor Description
Linear3SystemSolver(int numVariables, int numEquations)Creates a linear 3-regular system solver for a given number of variables and equations.
Modifier and Type Method Description
generateAndSolve(Iterable<long> iterable, long seed, LongBigList valueList)Generates a random 3-regular linear system on F2 or F3 and tries to solve it.
generateAndSolve(Iterable<long> signatures, long seed, LongBigList valueList, Codec.Coder coder, int m, int w)
generateAndSolve(Iterable<long> signatures, long seed, LongBigList valueList, Codec.Coder coder, int m, int w, boolean peelOnly)
signatureToEquation(long signature, long seed, int numVariables, int e)Turns a signature into an equation.
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
ewill be filled with -1.
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.
public boolean generateAndSolve(Iterable<long> iterable, long seed, LongBigList valueList)Generates a random 3-regular linear system on F2 or F3 and tries to solve it.
The constant part is provided by
valueList. If it is
null, the system will be on F3 and the constant part will be obtained by orientation, otherwise the system will be on F2.
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 64-bit random seed.
valueList- a value list containing the constant part, or
nullif the constant part should be computed by orientation.
- true if a solution was found.
- See Also:
public boolean generateAndSolve(Iterable<long> signatures, long seed, LongBigList valueList, Codec.Coder coder, int m, int w)