An implementation of hash table for classification of combinatorial objects

Transcription

1 An implementation of hash table for classification of combinatorial objects Mariya Dzhumalieva-Stoeva, Venelin Monev Faculty of Mathematics and Informatics, Veliko Tarnovo University Mariya Dzhumalieva-Stoeva, Venelin Monev ( Faculty Anof implementation Mathematics and of hash Informatics, table Veliko Tarnovo University ) 1 / 15

2 Outline The classification problem Implementation of hash tables Experimental results Mariya Dzhumalieva-Stoeva, Venelin Monev ( Faculty Anof implementation Mathematics and of hash Informatics, table Veliko Tarnovo University ) 2 / 15

3 Classification problem Ω - finite set of combinatorial objects, concerned with an equivalence relation =. Classification problem Finding a subset T Ω, such that: 1 a Ω, b T : a = b, 2 a, b T a b. isomorphism problem isomorph free generation recorded objects Mariya Dzhumalieva-Stoeva, Venelin Monev ( Faculty Anof implementation Mathematics and of hash Informatics, table Veliko Tarnovo University ) 3 / 15

4 Canonical representative map Definition A canonical representative map is a function ρ : Ω Ω which satisfies the following properties: 1 for all X Ω it holds that ρ(x ) = X, 2 for all X,Y Ω it holds that X = Y implies ρ(x ) = ρ(y ). The object X is in canonical form if ρ(x ) = X. Mariya Dzhumalieva-Stoeva, Venelin Monev ( Faculty Anof implementation Mathematics and of hash Informatics, table Veliko Tarnovo University ) 4 / 15

5 Basic algorithm Procedure RejectionByRecordCan Input: U -set of binary matrices Output: T -set of binary matrices in canonical form 1] T 2] for( a U ) do 3] obtain ρ(a) 4] if( ρ(a) / T ) 5] T T {ρ(a)} 6] end if 7] end for 8] T T end procedure Mariya Dzhumalieva-Stoeva, Venelin Monev ( Faculty Anof implementation Mathematics and of hash Informatics, table Veliko Tarnovo University ) 5 / 15

6 Basic algorithm look up fast whether ρ(a) / T store element ρ(a) to the set T possible implementation single-linked list hash table Mariya Dzhumalieva-Stoeva, Venelin Monev ( Faculty Anof implementation Mathematics and of hash Informatics, table Veliko Tarnovo University ) 6 / 15

7 Hash table Each element is represented in the hash table via two fields: key and value. The key is calculated by specific hash function: Definition A hash function h is any mathematical function of the form h : Ω {0, 1,..., s 1}, A hash table of size s can be implemented as one dimensional array HT [0... s 1] of slots. The key h(a) is the index of an element a. The slot HT [h(a)] contains an essential information. Mariya Dzhumalieva-Stoeva, Venelin Monev ( Faculty Anof implementation Mathematics and of hash Informatics, table Veliko Tarnovo University ) 7 / 15

8 Hash table Main points: to chose an appropriate hash function to select a method for collision resolution Collision - two distinct elements a 1 a 2 have the same hash value h(a 1 ) = h(a 2 ). Mariya Dzhumalieva-Stoeva, Venelin Monev ( Faculty Anof implementation Mathematics and of hash Informatics, table Veliko Tarnovo University ) 8 / 15

9 Implementation of the hash table The hash table is separated into two structures: one-dimensional integer array HT [0..size 1] an object a is represented in the slot HT [h(a)] each slot contains position of the corresponding file record random access file of records each record consists of pointer field and matrix field collision resolution is implemented by chaining via linked list structure Mariya Dzhumalieva-Stoeva, Venelin Monev ( Faculty Anof implementation Mathematics and of hash Informatics, table Veliko Tarnovo University ) 9 / 15

10 Procedure HashTableLookUp Input: Output: A - matrix in canonical form ρ(a) boolean value stored: true or false 1] key h(a); 2] if( slot HT[key] is empty ) 3] append A to file; 4] HT[key] position of A in the file; 5] return stored true; 6] else... Mariya Dzhumalieva-Stoeva, Venelin Monev ( Faculty Anof implementation Mathematics and of hash Informatics, table Veliko Tarnovo University ) 10 / 15

11 Procedure HashTableLookUp... 6] else 7] currentrec record in position HT[key]; 8] while( currentrec.next end of list ); 9] if( currentrec.matrix is equal to A ) 10] return stored false; 11] currentrec currentrec.next; 12] end while 13] if( currentrec.next is end of list ) 14] append A to file; 15] currentrec.next position of A in the file; 16] return stored true; 17] end if 18] end else Mariya Dzhumalieva-Stoeva, Venelin Monev ( Faculty Anof implementation Mathematics and of hash Informatics, table Veliko Tarnovo University ) 11 / 15

12 Remarks The algorithm is implemented in C programming language on a computer with CPU Intel Core i7-6700, 3,40GHz, 16 GB RAM memory, having an ordinary HDD. The input file contains set of random binary matrices. The used hash function is FNV-1a, which hashes strings. Each matrix is written row by row and converted to string of printable ASCII characters. Mariya Dzhumalieva-Stoeva, Venelin Monev ( Faculty Anof implementation Mathematics and of hash Informatics, table Veliko Tarnovo University ) 12 / 15

13 Experimental results matrices size n m memory cells collisions Running time sec n, m sec n, m sec n, m sec sec sec sec sec 19 min Mariya Dzhumalieva-Stoeva, Venelin Monev ( Faculty Anof implementation Mathematics and of hash Informatics, table Veliko Tarnovo University ) 13 / 15

14 Conclusion The procedure HashTableLookUp is called for each input matrix. The number of collisions depends on the hash table sizes and the type of the objects. The operation deleting is not implemented. There are no empty records in the output file. More than one file could be used for the implementation of the hash table. The algorithm can be easily implemented for the non binary case. Mariya Dzhumalieva-Stoeva, Venelin Monev ( Faculty Anof implementation Mathematics and of hash Informatics, table Veliko Tarnovo University ) 14 / 15

15 References I. Bouyukliev (2007) About the code equivalence. Advances in Coding Theory and Cryptology, T. Shaska, W.C. Huffman, D. Joyner, V. Ustimenko, Series on Coding Theory and Cryptology, World Scientific Publishing, Hackensack, NJ, I. Bouyukliev and M. Dzhumalieva-Stoeva (2014) Representing Equivalence Problems For Combinatorial Objects. Serdica Journal of Computing 8,No 4, , Th. H. Cormen, Ch. E. Leiserson, R. L. Rivest, Cl. Stein (2009) Introduction to algorithms, Third edition. The MIT Press Cambridge, Massachusetts London, England, P. Kaski and P. R. J. Östergård (2006) St. Skiena (2008) The Algorithm Design Manual, Second Edition. Springer-Verlag London, Fowler-Noll-Vo hash funciton FNV-1a alternate algorithm. ( Mariya Dzhumalieva-Stoeva, Venelin Monev ( Faculty Anof implementation Mathematics and of hash Informatics, table Veliko Tarnovo University ) 15 / 15