Constructing Signature Graphs for Signature Files

1. Constructing Signature Graphs for Signature Files Dr. Yangjun ChenDept. Applied Computer Science University of Winnipeg Canada

2. Motivation • Signature Files as Indexes • Signature Graph and its Construction • Signature Graph and its Construction • Searching a Signature Graph • Maintenance of Signature Graph • Summary and Future Work

3. Motivation • Establish Indexes to speed up query evaluation • B+-trees, inverted files, signature files • Signature files: simple and easy for maintenance • Signature graphs: less time for searching

4. Signature Files as Indexes • Definition A signature for a key word or an attribute value is hash-coded bit string. • Signature construction - Important parameters: m: number of 1s in bit string F: length of bit string D: size of a block (or average number of the key words of an element) - optimal choice of the parameters: Fln2 =mD.

5. Example: (constructing a signature for a word with m = 4 and F = 12) “database”  letter triplets: dat, ata, tab, aba, bas, ase  H(dat) = 5, H(ata) = 1, H(tab) = 8, H(aba) = 1, H(bas) = 10, H(ase) = 8.  100 010 010 100

6. Signature Files as Indexes text: … SGML … database …information … matching word signatures: queries: query signatures: results SGML 010000100110 SGML 010000100110 match with OS database 100010010100 XML 011000100100 no match with OS information  010100011000 informatik 110100100000 false drop object signature 110110111110 (OS)

7. name sex ... ... John male ... ... Ù query: John male query signature: 1010 0101 Example: relation:

8. Signature Graph Consider a signature si of length m. We denote it as si = si[1]si[2] ... si[m], where each si[j]  {0, 1} (j = 1, ..., F). We also use si(j1, ..., jh) to denote a sequence of pairs w.r.t. si: (j1,si[j1])(j2, si[j2]) ... (jh, si[jh]), where 1 jk m for k {1, ..., h}. Definition (signature identifier)Let S = s1.s2 ... .sn denote a signature file. Consider si (1 in). If there exists a sequence: j1, ..., jh such that for any k i (1 kn)we have si(j1, ..., jh) sk(j1, ..., jh), then we say si(j1, ..., jh) identifies the signature si or say si(j1, ..., jh) is an identifier of si.

9. Example: s8(5, 1, 4) = (5, 1)(1, 1)(4, 0) (*For any i  8 we have si(5, 1, 4)s8(5, 1, 4). For instance, s5(5, 1, 4) = (5, 0)(1, 0)(4, 1) s8(5, 1, 4), s2(5, 1, 4) = (5, 1)(1, 1)(4, 1) s8(5, 1, 4), and so on.*) s1(5, 4, 1) = (5, 0)(4, 1)(1, 1) (*For anyi  1 we have si(5, 4, 1)s1(5, 4, 1).*)

10. Signature Graph • Definition (signature graph)A signature graph G for a signature file S = s1.s2 ... .sn, where si sj for i j and |sk| = F for k = 1, ..., n, is a graph G = (V, E) such that 1. each node vV is of the form (p, skip), where p is a pointer to a signature s in S, and skip is a non-negative integer i. If i > 0, it tells that the ith bit of sq will be checked when searching. If i = 0, s will be compared with sq. 2. Let e = (u, v)E. Then, e is labeled with 0 or 1 and skip(u) > 0. Let skip(u) = i. If e is labeled with 0 and i > 0, the ith bit of the signature pointed to by p(v) is 0. If e is labeled with 1 and i > 0, the ith bit of the signature pointed to by p(v) is 1. A node v with skip(u) = 0 does not have any children.

11. p2 5 0 1 1 0 0 p3 p6 4 1 1 1 1 p4 p7 1 2 p8 0 1 4 0 0 p5 p1 3 0 1 0 S1: 1011 0110 S2: 1011 1001 S3: 1010 0111 S4: 0111 0110 S5: 0111 0101 S6: 0101 1100 S7: 1110 0100 S8: 1010 1011

12. p1 p1 p2 p3 p1 p1 p2 p2 p3 p4 0 0 5 4 5 0 0 5 1 4 0 0 1 1 1 1 0 0 1 0 Insert s1 Insert s2 Insert s4 Insert s3 Construction of signature graph: 0 1

13. 1 0 1 0 1 0 p5 p2 p1 p3 p6 p3 p2 p4 5 4 5 4 0 3 1 1 p4 1 0 1 0 1 0 1 0 p5 p1 3 0 0 1 1 1 0 Insert s5 Insert s6 Insert s7 Insert s8 p2 p2 5 5 1 1 0 1 0 1 0 0 0 0 p3 p6 p3 p6 4 4 1 1 1 1 1 1 1 p4 p4 p7 p7 1 1 2 2 p8 0 1 0 1 4 0 0 0 0 p5 p1 p5 p1 3 3 0 0 1 1 0

14. Signature Graph Searching a signature graph Denote sq(i) the i-th position of sq. During the traversal of a signature graph, the inexact matching can be done as follows: (i) Let v be the node encountered and sq (i) be the position to be checked. (ii) If sq (i) = 1, we move to the right child of v (iii)If sq (i) = 0, both the right and left child of v will be visited. (iv)A search along a path stops when a node without any child node or a node is encountered for the second time.

15. Signature Graph marked p2 5 1 0 marked marked 1 0 0 p3 p6 4 1 1 1 marked marked 1 p4 p7 1 2 p8 0 1 4 0 0 p5 p1 3 0 marked 1 0 marked

16. Maintenance of Signature Graph - Insertion of a signature s into G Same as the construction of a signature graph • Deletion of a signature s from G (i) Search G from the root until a node v is encountered, which is marked or skip(v) = 0. (ii) If skip(v) = 0, Compare p(v) and s. If s matches p(v) exactly, do the following; otherwise, nothing will be done. Let v1 ... vk-1 vkv be the path explored. Let u1 be another child of vk (not on the path). Remove vk-1 vk, vk u1 and v; and generate a new edge vk-1 u1. skip(vk) := 0.

17. Maintenance of Signature Graph • - Deletion of a signature s from G (continued) • (iii) If skip(v)  0, Compare p(v’s father) and s. If s matches p(v’s father) exactly, do the following; otherwise, nothing will be done. • Let v1 ... vk-1 vkv be the path explored. • If vkv, replace p(v) with p(vk). Let u1 be another child of vk (not on the path). Let u2 be another parent of vk (not on the path). Replace vk-1 vk with vk u1, and replace vkv with u2v. Remove vk. Note that u2 can be found by searching G from vk with the target signature being p(vk). • If vk= v, replace vk vk with vk-1 u1. Remove vk.

18. Maintenance of Signature Graph Illustration for (ii) To be removed vk-1 vk-1 v1 vk v1 vk … … v v u1 u1 u2 u2

19. Example: p2 5 1 0 1 0 0 p6 p3 4 1 1 1 1 p4 p7 1 2 p8 0 1 0 0 0 p5 p1 3 0 1 0 remove p1 p2 5 1 0 1 0 0 p3 p6 4 1 1 1 1 p7 p5 2 3 p8 1 0 0 0 0 p4 0

20. Maintenance of Signature Graph Illustration for (iii) To be removed vk-1 vk-1 v1 vk v1 vk … … v v u1 u1 u2 u2

21. p2 5 1 0 1 0 0 p3 p6 4 1 1 1 p4 p7 1 2 0 1 0 0 p5 p1 3 0 1 Example: p2 5 1 0 1 0 0 p6 p3 4 1 1 1 1 p4 p7 1 2 p8 0 1 4 0 0 p5 p1 3 0 1 0 remove p8

22. To be removed vk-1 vk-1 v1 v1 v … v … u1 u1 Illustration for (iii)

23. p2 5 1 0 1 0 0 p6 p3 4 1 1 1 1 p4 p7 1 2 p8 0 1 4 0 0 p5 p1 3 0 1 0 p2 5 1 0 1 0 0 p6 p3 4 1 1 1 p4 1 p8 0 1 4 0 p5 p1 3 0 1 0 Example: remove p7

24. Summary and Future Work • - Signature and signature file • - Signature graph • Construction of a signature graphSearch of a signature graphMaintenance of a signature graph • Future work: • Apply signature techniques to evaluation of • path-oriented queries in document databases.