1.18.0 User Documentation for Apache MADlib
All Pairs Shortest Path
Contents

The all pairs shortest paths (APSP) algorithm finds the length (summed weights) of the shortest paths between all pairs of vertices, such that the sum of the weights of the path edges is minimized.

Warning
APSP is an expensive algorithm for run-time because it finds the shortest path between all nodes in the graph. It is recommended that you start with a small graph to get a sense of run-time for your use case, then increase size carefully from there. The worst case run-time for this implementation is O(V^2 * E) where V is the number of vertices and E is the number of edges. In practice, run-time will be generally be much less than this, but it depends on the graph. On a Greenplum cluster, the edge table should be distributed by the source vertex id column for better performance.

APSP
graph_apsp( vertex_table,
vertex_id,
edge_table,
edge_args,
out_table,
grouping_cols
)


Arguments

vertex_table

TEXT. Name of the table containing the vertex data for the graph. Must contain the column specified in the 'vertex_id' parameter below.

vertex_id

TEXT, default = 'id'. Name of the column in 'vertex_table' containing vertex ids. The vertex ids can be of type INTEGER or BIGINT with no duplicates. They do not need to be contiguous.

edge_table

TEXT. Name of the table containing the edge data. The edge table must contain columns for source vertex, destination vertex and edge weight. Column naming convention is described below in the 'edge_args' parameter.

edge_args

TEXT. A comma-delimited string containing multiple named arguments of the form "name=value". The following parameters are supported for this string argument:

• src (INTEGER or BIGINT): Name of the column containing the source vertex ids in the edge table. Default column name is 'src'.
• dest (INTEGER or BIGINT): Name of the column containing the destination vertex ids in the edge table. Default column name is 'dest'.
• weight (FLOAT8): Name of the column containing the edge weights in the edge table. Default column name is 'weight'.

out_table

TEXT. Name of the table to store the result of APSP. It contains a row for every vertex of every group and have the following columns (in addition to the grouping columns):

• source_vertex: The id for the source vertex. Will use the input edge column 'src' for column naming.
• dest_vertex: The id for the destination vertex. Will use the input edge column 'dest' for column naming.
• weight: The total weight of the shortest path from the source vertex to the destination vertex. Will use the input parameter 'weight' for column naming.
• parent: The parent of the destination vertex in the shortest path from source. Parent will equal dest_vertex if there are no intermediate vertices. Will use 'parent' for column naming.

A summary table named <out_table>_summary is also created. This is an internal table that keeps a record of the input parameters and is used by the path retrieval function described below.

grouping_cols (optional)
TEXT, default = NULL. List of columns used to group the input into discrete subgraphs. These columns must exist in the edge table. When this value is null, no grouping is used and a single APSP result is generated.
Path Retrieval

The path retrieval function returns the shortest path from the source vertex to a specified desination vertex.

graph_apsp_get_path( apsp_table,
source_vertex,
dest_vertex,
path_table
)


Arguments

apsp_table

TEXT. Name of the table that contains the APSP output.

source_vertex

INTEGER or BIGINT. The vertex that will be the source of the desired path.

dest_vertex

INTEGER or BIGINT. The vertex that will be the destination of the desired path.

path_table

TEXT. Name of the output table that contains the path. It contains a row for every group and has the following columns:

• grouping_cols: The grouping columns given in the creation of the APSP table. If there are no grouping columns, these columns will not exist and the table will have a single row.
• path (ARRAY): The shortest path from the source vertex to the destination vertex.

Examples
1. Create vertex and edge tables to represent the graph:
DROP TABLE IF EXISTS vertex, edge;
CREATE TABLE vertex(
id INTEGER
);
CREATE TABLE edge(
src INTEGER,
dest INTEGER,
weight FLOAT8
);
INSERT INTO vertex VALUES
(0),
(1),
(2),
(3),
(4),
(5),
(6),
(7);
INSERT INTO edge VALUES
(0, 1, 1.0),
(0, 2, 1.0),
(0, 4, 10.0),
(1, 2, 2.0),
(1, 3, 10.0),
(2, 3, 1.0),
(2, 5, 1.0),
(2, 6, 3.0),
(3, 0, 1.0),
(4, 0, -2.0),
(5, 6, 1.0),
(6, 7, 1.0);

2. Calculate the shortest paths:
DROP TABLE IF EXISTS out, out_summary;
'vertex',      -- Vertex table
NULL,          -- Vertix id column (NULL means use default naming)
'edge',        -- Edge table
NULL,          -- Edge arguments (NULL means use default naming)
'out');        -- Output table of shortest paths
SELECT * FROM out ORDER BY src,dest;

 src | dest |  weight  | parent
-----+------+----------+--------
0 |    0 |        0 |      0
0 |    1 |        1 |      1
0 |    2 |        1 |      2
0 |    3 |        2 |      2
0 |    4 |       10 |      4
0 |    5 |        2 |      2
0 |    6 |        3 |      5
0 |    7 |        4 |      6
1 |    0 |        4 |      3
1 |    1 |        0 |      1
1 |    2 |        2 |      2
1 |    3 |        3 |      2
1 |    4 |       14 |      0
1 |    5 |        3 |      2
1 |    6 |        4 |      5
1 |    7 |        5 |      6
(showing only 16 of 64 rows)

3. Get the shortest path from vertex 0 to vertex 5:
DROP TABLE IF EXISTS out_path;
SELECT * FROM out_path;

  path
---------
{0,2,5}

4. Now let's do a similar example except using different column names in the tables (i.e., not the defaults). Create the vertex and edge tables:
DROP TABLE IF EXISTS vertex_alt, edge_alt;
CREATE TABLE vertex_alt AS SELECT id AS v_id FROM vertex;
CREATE TABLE edge_alt AS SELECT src AS e_src, dest, weight AS e_weight FROM edge;

5. Calculate the shortest paths:
DROP TABLE IF EXISTS out_alt, out_alt_summary;
'vertex_alt',                  -- Vertex table
'v_id',                        -- Vertex id column
'edge_alt',                    -- Edge table
'src=e_src, weight=e_weight',  -- Edge arguments
'out_alt');                    -- Output table of shortest paths
SELECT * FROM out_alt ORDER BY e_src, dest;

 e_src | dest | e_weight | parent
-------+------+----------+--------
0 |    0 |        0 |      0
0 |    1 |        1 |      1
0 |    2 |        1 |      2
0 |    3 |        2 |      2
0 |    4 |       10 |      4
0 |    5 |        2 |      2
0 |    6 |        3 |      5
0 |    7 |        4 |      6
1 |    0 |        4 |      3
1 |    1 |        0 |      1
1 |    2 |        2 |      2
1 |    3 |        3 |      2
1 |    4 |       14 |      0
1 |    5 |        3 |      2
1 |    6 |        4 |      5
1 |    7 |        5 |      6
(showing only 16 of 64 rows)

6. Create a graph with 2 groups and find APSP for each group:
DROP TABLE IF EXISTS edge_gr;
CREATE TABLE edge_gr AS
(
SELECT *, 0 AS grp FROM edge
UNION
SELECT *, 1 AS grp FROM edge WHERE src < 6 AND dest < 6
);
INSERT INTO edge_gr VALUES
(4,5,-20,1);

7. Find APSP for all groups:
DROP TABLE IF EXISTS out_gr, out_gr_summary;
'vertex',      -- Vertex table
NULL,          -- Vertex id column (NULL means use default naming)
'edge_gr',     -- Edge table
NULL,          -- Edge arguments (NULL means use default naming)
'out_gr',      -- Output table of shortest paths
'grp'          -- Grouping columns
);
SELECT * FROM out_gr WHERE src < 2 ORDER BY grp,src,dest;

 grp | src | dest | weight | parent
-----+-----+------+--------+--------
0 |   0 |    0 |      0 |      0
0 |   0 |    1 |      1 |      1
0 |   0 |    2 |      1 |      2
0 |   0 |    3 |      2 |      2
0 |   0 |    4 |     10 |      4
0 |   0 |    5 |      2 |      2
0 |   0 |    6 |      4 |      2
0 |   0 |    7 |      5 |      6
0 |   1 |    0 |      4 |      3
0 |   1 |    1 |      0 |      1
0 |   1 |    2 |      2 |      2
0 |   1 |    3 |      3 |      2
0 |   1 |    4 |     14 |      0
0 |   1 |    5 |      3 |      2
0 |   1 |    6 |      4 |      5
0 |   1 |    7 |      5 |      6
1 |   0 |    0 |      0 |      0
1 |   0 |    1 |      1 |      1
1 |   0 |    2 |      1 |      2
1 |   0 |    3 |      2 |      2
1 |   0 |    4 |     10 |      4
1 |   0 |    5 |    -10 |      4
1 |   1 |    0 |      4 |      3
1 |   1 |    1 |      0 |      1
1 |   1 |    2 |      2 |      2
1 |   1 |    3 |      3 |      2
1 |   1 |    4 |     14 |      0
1 |   1 |    5 |     -6 |      4
(28 rows)

8. Find the path from vertex 0 to vertex 5 in every group
DROP TABLE IF EXISTS out_gr_path;
SELECT * FROM out_gr_path ORDER BY grp;

 grp |  path
-----+---------
0 | {0,2,5}
1 | {0,4,5}


Notes
1. Graphs with negative edges are supported but graphs with negative cycles are not.
2. The implementation for APSP is analogous to a matrix multiplication operation. Please refer to the MADlib design document and references [1] and [2] for more details.
3. Also see the Grail project [3] for more background on graph analytics processing in relational databases.

Literature

[3] The case against specialized graph analytics engines, J. Fan, G. Soosai Raj, and J. M. Patel. CIDR 2015. http://cidrdb.org/cidr2015/Papers/CIDR15_Paper20.pdf