Designing a Path-oriented Indexing Structure for a Graph Structured Data

Transcription

1 Designing a Path-oriented Indexing Structure for a Graph Structured Data Stanislav Bartoň Masaryk university, Brno June, 2005 tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data / 34

2 Introduction An example of an RDF Graph RDF Graph Knowledge Base Schemas locationthesaurus location fname creates exhibited String Artist Artifact Museum lname working_hours String sculpts Enumeration Sculptor Sculpture paints Painter Painting technique String Cubist Flemish technique "oil on canvas" "Pablo" paints &r2 fname paints technique "oil on canvas" &r "Picasso" lname last_modified &r3 exhibited title &r4 title "Descent" "Michelangelo" sculpts &r5 fname title sculpts &r6 "Buonarroti" lname exhibited &r7 location exhibited &r8 working_hours technique "Rembrandt" &r9 paints &r0 "oil on canvas" fname technique "oil on canvas" paints &r file_size 7 title subclassof, subpropertyof (is a) Abraham and Isaac DateTime last_modified ExtResource String title file_size Integer T2:30:34 "Reina Sofia Museum" "Louvre Museum" FRANCE 9, 5 8 typeof (instance) tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 2 / 34

3 ρ operators Introduction Rho-operators Firstly introduced in the context of Semantic Web Designed to study complex relationships between entities defined as Complex Associations Can be generalized into terms of graphs and a problem of searching paths in them tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 3 / 34

4 Introduction Rho-operators ρ-path operator locationthesaurus location Knowledge Base Schemas fname creates String Artist lname String sculpts Sculptor paints Painter Cubist Flemish "Pablo" paints fname paints &r "Picasso" lname "Michelangelo" sculpts fname sculpts "Buonarroti" lname &r6 "Rembrandt" paints fname exhibited Artifact Museum working_hours Enumeration Sculpture Painting technique String technique "oil on canvas" &r2 technique "oil on canvas" last_modified &r3 exhibited title &r4 title "Descent" &r5 title exhibited &r7 location exhibited &r8 working_hours technique &r0 "oil on canvas" DateTime last_modified ExtResource String title file_size Integer T2:30:34 "Reina Sofia Museum" "Louvre Museum" FRANCE 9, 5 8 &r9 paints technique "oil on canvas" &r file_size title 7 subclassof, subpropertyof (is a) Abraham and Isaac typeof (instance) tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 4 / 34

5 ρ-path operator Introduction Rho-operators ρ-path operator definition in graph theory terms: ρ-path(x, y) = {p = (v e v 2 e 2... e n v n+ ) v = x v n+ = y p is acyclic} tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 5 / 34

6 Introduction Rho-operators ρ-connection operator locationthesaurus location Knowledge Base Schemas fname creates String Artist lname String sculpts Sculptor paints Painter Cubist Flemish "Pablo" paints fname paints &r "Picasso" lname "Michelangelo" sculpts fname sculpts "Buonarroti" lname &r6 "Rembrandt" paints fname exhibited Artifact Museum working_hours Enumeration Sculpture Painting technique String technique "oil on canvas" &r2 technique "oil on canvas" last_modified &r3 exhibited title &r4 title "Descent" &r5 title exhibited &r7 location exhibited &r8 working_hours technique &r0 "oil on canvas" DateTime last_modified ExtResource String title file_size Integer T2:30:34 "Reina Sofia Museum" "Louvre Museum" FRANCE 9, 5 8 &r9 paints technique "oil on canvas" &r file_size title 7 subclassof, subpropertyof (is a) Abraham and Isaac typeof (instance) tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 6 / 34

7 ρ-connection operator Introduction Rho-operators ρ-connection operator definition in graph theory terms: ρ-connection(x, y) = {(p, p 2 ) p = (v e v 2 e 2... e n v n+ ), p 2 = w h w 2 h 2... h n w m+ ) v = x w = y v n+ = w m+ p, p 2 are acyclic} tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 7 / 34

8 Introduction Designing the indexing structure Designing the indexing structure Adjacency matrix Great graph description, simple transitive closure computation Can be easily modified to store paths themselves rather then just amounts of them The use of matrix algebra is limited to relatively small graphs due to space and time complexity With graph transformations towards graph simplification transformed graph must have similar properties as the original graph had Graph segmentation (vertex clustering, graph to forest of trees) The transitive closure of segment graph models all paths from the original graph tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 8 / 34

9 Introduction Designing the indexing structure Designing the indexing structure Path type matrix Instead of storing the amounts of paths, keeps the paths themselves + and replaced by path concatenation and set union enables cycle detection during computation Vertex clustering two pass algorithm that divides the graph into several subgraphs of predefined size the vertices that are close to each other are put to same subgraphs very general, non-restrictive technique that can be applied to arbitrary directed graph tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 9 / 34

10 Introduction Path type matrix transitive closure Designing the indexing structure A e B C e 2 e 5 e 3 e 4 D E M = 0 A B C D E A {(e )} B {(e 2)} {(e 3)} C {(e 4)} D {(e 5)} E C A tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 0 / 34

11 Introduction Path type matrix transitive closure Designing the indexing structure M = M 2 = M + M 2 = 0 B@ 0 B@ 0 B@ A B C D E A X {(e )} B X {(e 2 )} {(e 3 )} C {(e 4 )} D {(e 5 )} E CA A B C D E A {(e e 2 )} {(e e 3 )} B {(e 3 e 5 )} {(e 2 e 4 )} C D E A B C D E A {(e )} {(e e 2 )} {(e e 3 )} B {(e 3 e 5 )} {(e 2 )} {(e 3 )} {(e 2 e 4 )} C {(e 4 )} D {(e 5 )} E CA CA tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data / 34

12 Introduction Path type matrix transitive closure Designing the indexing structure M = M 2 = M + M 2 = 0 B@ 0 B@ 0 B@ A B C D E A {(e )} B X X {(e 2 )} {(e 3 )} C {(e 4 )} D {(e 5 )} E CA A B C D E A {(e e 2 )} {(e e 3 )} B {(e 3 e 5 )} {(e 2 e 4 )} C D E A B C D E A {(e )} {(e e 2 )} {(e e 3 )} B {(e 3 e 5 )} {(e 2 )} {(e 3 )} {(e 2 e 4 )} C {(e 4 )} D {(e 5 )} E CA CA tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data / 34

13 Introduction Path type matrix transitive closure Designing the indexing structure M = M 2 = M + M 2 = 0 B@ 0 B@ 0 B@ A B C D E A {(e )} B {(e 2 )} {(e 3 )} C X {(e 4 )} D {(e 5 )} E CA A B C D E A {(e e 2 )} {(e e 3 )} B {(e 3 e 5 )} {(e 2 e 4 )} C D E A B C D E A {(e )} {(e e 2 )} {(e e 3 )} B {(e 3 e 5 )} {(e 2 )} {(e 3 )} {(e 2 e 4 )} C {(e 4 )} D {(e 5 )} E CA CA tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data / 34

14 Introduction Path type matrix transitive closure Designing the indexing structure M = M 2 = M + M 2 = 0 B@ 0 B@ 0 B@ A B C D E A {(e )} B {(e 2 )} {(e 3 )} C {(e 4 )} D {(e 5 )} E CA A B C D E A {(e e 2 )} {(e e 3 )} B {(e 3 e 5 )} {(e 2 e 4 )} C D E A B C D E A {(e )} {(e e 2 )} {(e e 3 )} B {(e 3 e 5 )} {(e 2 )} {(e 3 )} {(e 2 e 4 )} C {(e 4 )} D {(e 5 )} E CA CA Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data / 34

15 Introduction Path type matrix transitive closure Designing the indexing structure M = M 2 = M + M 2 = 0 B@ 0 B@ 0 B@ A B C D E A {(e )} B {(e 2 )} {(e 3 )} X C {(e 4 )} D {(e 5 )} CA E X A B C D E A {(e e 2 )} {(e e 3 )} B {(e 3 e 5 )} {(e 2 e 4 )} C D E A B C D E A {(e )} {(e e 2 )} {(e e 3 )} B {(e 3 e 5 )} {(e 2 )} {(e 3 )} {(e 2 e 4 )} C {(e 4 )} D {(e 5 )} E CA CA tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data / 34

16 Graph segmentation Graph theory Necessary definitions Segment S in a graph G : S = (V S, E S ) : V S V E S = {e E RIGHT VERTEX (e) V S LEFT VERTEX (e) V S } S S 2 S 4 S 3 S 5 S 6 Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 2 / 34

17 Graph segmentation Graph theory Necessary definitions Segment S in a graph G : S = (V S, E S ) : V S V E S = {e E RIGHT VERTEX (e) V S LEFT VERTEX (e) V S } EDGES OUT(S) = {e e E S LEFT VERTEX (e) V S RIGHT VERTEX (e) V S } EDGES IN(S) = {e e E S RIGHT VERTEX (e) V S LEFT VERTEX (e) V S } S S 2 S 4 S 3 S 5 S 6 Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 2 / 34

18 Graph segmentation Graph theory Necessary definitions Segment S in a graph G : S = (V S, E S ) : V S V E S = {e E RIGHT VERTEX (e) V S LEFT VERTEX (e) V S } EDGES OUT(S) = {e e E S LEFT VERTEX (e) V S RIGHT VERTEX (e) V S } EDGES IN(S) = {e e E S RIGHT VERTEX (e) V S LEFT VERTEX (e) V S } Segmentation S(G)= {S S is a segment of G} S, S S(G), S S : V S V S = V S = V S S(G) S S 2 S 4 S 3 S 5 S 6 Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 2 / 34

19 Sequence of segments Graph theory Necessary definitions Sequence of segments (S... S m ) = S,... S l S(G), i m : EDGES OUT (S i ) EDGES IN(S i+ ) S S 2 S 3 S 4 a c e g h i b d f j tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 3 / 34

20 Sequence of segments Graph theory Necessary definitions Sequence of segments (S... S m ) = S,... S l S(G), i m : EDGES OUT (S i ) EDGES IN(S i+ ) Connecting path p = (e e 2... e n ) in a segment sequence (S... S m ): p (S... S m ) : e EDGES OUT (S ) EDGES IN(S 2 ) e n EDGES OUT (S m ) EDGES IN(S l ) i 2, i 3,... i m : < i 2 < i 3 <... < i m < n : {e 2,... e i2 } E S2 {e i2,... e i3 } E S3... {e il 2,... e im } E Sm S S 2 S 3 S 4 a c e g h i b d f j Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 3 / 34

21 Paths Graph theory Necessary definitions Acyclic path p = (v e v 2 e 2... e n v n+ ) in G : i n, j n +, i j : e i E v i, v j V v i = LEFT VERTEX (e i ) v i+ = RIGHT VERTEX (e i ) v i v j Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 4 / 34

22 Paths Graph theory Necessary definitions Acyclic path p = (v e v 2 e 2... e n v n+ ) in G : i n, j n +, i j : e i E v i, v j V v i = LEFT VERTEX (e i ) v i+ = RIGHT VERTEX (e i ) v i v j Proper segment sequence for a path p = (v e v 2 e 2... e n v n+ ) : S(p) = (S... S m ) : S(p) is a segment sequence i < i 2 <... < i l n + : {v,... v i } V S {v i,... v i2 } V S2... {v il,... v n+ } V Sl Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 4 / 34

23 Segment graph Graph theory Necessary definitions Segment graph of G: SG(G) = (S(G), X ), X = {h h = (S i, S j ) i, j k EDGES OUT (S i ) EDGES IN(S j ) } S S 2 S 4 S 3 S S 2 S3 S 5 S 4 S 6 S 5 S 6 Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 5 / 34

24 Graph theory Basic hypotheses Representing paths in G by segment sequences in S(G) Lemma If a graph G = (V, E) has a segmentation S(G) that forms a graph SG(G), any path p = (v e v 2 e 2... e n v n+ ) in G can be represented by its proper segment sequence in S(G) and this representation is unique. Lemma If a graph G = (V, E) has a segmentation S(G) that forms a graph SG(G), a segment sequence in S(G) represents either some path in G or an empty path. Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 6 / 34

25 Preliminary evaluation Graph theory Preliminary evaluation If we generate all possible segment sequences in S(G), we get all possible paths in G Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 7 / 34

26 Preliminary evaluation Graph theory Preliminary evaluation If we generate all possible segment sequences in S(G), we get all possible paths in G + Fast generation of dense path representations Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 7 / 34

27 Preliminary evaluation Graph theory Preliminary evaluation If we generate all possible segment sequences in S(G), we get all possible paths in G + Fast generation of dense path representations - Problem with disconnected segment sequences Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 7 / 34

28 Preliminary evaluation Graph theory Preliminary evaluation If we generate all possible segment sequences in S(G), we get all possible paths in G + Fast generation of dense path representations - Problem with disconnected segment sequences - A huge amount of paths can be found in dense graphs between almost any two vertices, the number of such paths grows exponentially with the maximal length of a path allowed Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 7 / 34

29 Preliminary evaluation Graph theory Preliminary evaluation If we generate all possible segment sequences in S(G), we get all possible paths in G + Fast generation of dense path representations - Problem with disconnected segment sequences - A huge amount of paths can be found in dense graphs between almost any two vertices, the number of such paths grows exponentially with the maximal length of a path allowed A need for l-ρ-index which is a variation of rho-index, where only those paths between two vertices with length l and some paths having length > l are indexed Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 7 / 34

30 Preliminary evaluation Graph theory Preliminary evaluation If we generate all possible segment sequences in S(G), we get all possible paths in G + Fast generation of dense path representations - Problem with disconnected segment sequences - A huge amount of paths can be found in dense graphs between almost any two vertices, the number of such paths grows exponentially with the maximal length of a path allowed A need for l-ρ-index which is a variation of rho-index, where only those paths between two vertices with length l and some paths having length > l are indexed Computational overhead bound with a weight computation of each segment sequence stored An upper bound on a maximal number of connecting paths to be computed to find the one with a lowest weight tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 7 / 34

31 Graph theory Weights in G and S(G) - Definitions Proposing weights to vertices Weight of vertex v: w(v) <, > Weight of path p = (v e v 2 e 2... e n v n+ ) : w(p) = n+ w(v i ) Set of weights of segment sequence: (S... S m ) = {w(p) p (S... S m )} Weight of segment sequence (S... S m ) = min( (S... S m ) ) i= tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 8 / 34

32 Graph theory Facts about weights in G and S(G) Proposing weights to vertices The relation between (v e v 2 e 2... e n v n+ ) and (S... S m )!! m n + = the segment sequence representation is always shorter or of the same length as the path it represents tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 9 / 34

33 Graph theory Facts about weights in G and S(G) Proposing weights to vertices The relation between (v e v 2 e 2... e n v n+ ) and (S... S m )!! m n + = the segment sequence representation is always shorter or of the same length as the path it represents!! (S... S m ) w(p) = the proper segment sequence for a path p has always lower or the same weight as the path it represents tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 9 / 34

34 Graph theory Proposing weights to vertices Proposing the limit l Lemma If a graph G = (V, E) has a segmentation S(G) that forms a graph SG(G) then for a limit l, segment sequences in S(G) having weight l represent all paths in G that have weight l. Proof. Lets assume that there is a path p with w(p) l and that it is not present in the result represented by segment sequences with (S... S m ) l. This would imply that the S(p) > w(p) but this is contradictory to the previous facts. Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 20 / 34

35 Graph theory Upper bound Upper bound on the minimal number of connecting paths Lemma (An upper bound on a maximal number of connecting paths to be computed to find the one with a lowest weight) A connecting path for a segment sequence (S... S m ) with the lowest weight is a path in CPs with the lowest weight. CPs is a set of connecting paths that have for each combination of common edges for each two neighboring segments in (S... S m ) minimal weight. The upper bound is represented by the number of combinations of common edges picked from m sets of common edges. Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 2 / 34

36 Graph theory Iteration step Recursively applying the graph segmentation What if the segment graph SG(G) of the indexed graph is not small enough to be described by a path type matrix? Intuitively, the graph segmentation can be applied again to the segment graph SG(G) forming the SG(SG(G)). But what happens to the vertices weights? Segments do not have weights assigned, since the segment s shortest traversal is context dependent. How to propose the vertices weights to upper levels of the indexing structure? tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 22 / 34

37 Graph theory Iteration step Assigning weight to a segment!!! Segment sequence (EABX ) is disconnected!!! (ABC) = 3, (ABX ) = 4 = w(b) = or 2? D D A B A B X E X E F C Y C Y 7 F tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 23 / 34

38 Graph theory Iteration step Altering the weight definitions for an iteration step G = (V, E), G = SG(G) = (S(G), E ), G = (SG(SG(G)) = (S(S(G)), E ) Weight of vertex v V : w(v) <, > Weight of path p = (v e v 2 e 2... e n v n+ ) : w(p) = n+ w(v i ), p G Connecting segment sequence (A... A m ) G for (S... S m ) G denotes a path (A e A 2... e k A k ) in G where e (EDGES OUT (S ) EDGES IN(S 2 )), e k (EDGES OUT (S m ) EDGES IN(S m )) and (S... S m ) is a proper segment sequence for (A e A 2... e k A k ). i= tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 24 / 34

39 Graph theory Iteration step Altering the weight definitions Set of weights{ of segment sequence: {w(p) p (S... S (S... S m ) = m )}, S S(G) { (A... A k ) (A... A k ) (S... S m )}, S S(S(G)) Weight of segment sequence (S... S m ) = min( (S... S m ) ) tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 25 / 34

40 ρ-index s structure The rho-index The rho-index structure ρ-index comprises of: Each segment is represented by its path type matrix EDGES IN and EDGES OUT are also stored for each segment Path type matrix of a segment graph at the topmost level Stanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 26 / 34

41 The rho-index Outline of a ρ-index s structure The rho-index structure Level 3 Top level matrix Level Level Matrices for segments Matrices fot segments Original graph tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 27 / 34

42 Creating ρ index The rho-index Creating the rho-index Creating algorithm: Segmentation of the indexed graph G using the vertex clustering transformation 2 Creation of path type matrix for each segment, subsequent transitive closure computation 3 Creation of a segment graph SG(G) 4 If the segment graph is not small enough repeat previous steps tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 28 / 34

43 The rho-index Creating the rho-index Path search algorithm - Breadth First Graph G End Start Accesible area from Start Level 3 Level 2 Level tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 29 / 34

44 The rho-index Creating the rho-index Path search algorithm - Depth First Graph G End Accesible area from Start Start Step Step 2 Level 3 Level 2 Level 2 Level Level tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 30 / 34

45 The rho-index Creating the rho-index Practical experience with the approximative ρ-index The approximative (k, l)-ρ-index implementation: Limiting parameters pseudo k and l k - limits the number of segment sequences stored in one matrix field l - limits the degree of computation of the transitive closure of the matrices representing segments and the top matrix Insufficiency of the implemented (k, l) parameters lead to the design of the correct l - variant of the ρ-index by proposing weights of vertices and segment sequences to the design of the indexing structure tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 3 / 34

46 The rho-index Creating the rho-index Practical experience with the approximative ρ-index Index efficiency is very dependent on a size of the cluster used to segment the graph using the same (k, l) parameters lead into different number of paths indexed the lower the size of the cluster the more precise results gained The unlimited variant of the ρ-index can be achieved using a small size of a cluster small number of stored segment sequences in each matrix field enables complete transitive closure computation for each segment tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 32 / 34

47 Future research The rho-index Future research Implementation and full evaluation of the l-ρ-index variation including optimized depth first search algorithm Explore the impact of a graph segmentation strategy to the indexing structure Optimization of the vertex clustering technique Further research of other segmentation techniques (k, l) ρ index - another variation of ρ-index where only the first k paths of length l are indexed Explore the possibilities of distributing the ρ-index tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 33 / 34

48 The rho-index Future research Thank you for your attention. tanislav Bartoň (Masaryk university, Brno)Designing a Path-oriented Indexing Structure for a Graph Structured June, 2005 Data 34 / 34