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 <title>MADlib: Single Source Shortest Path</title>
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 <div class="header">
   <div class="headertitle">
 <div class="title">Single Source Shortest Path<div class="ingroups"><a class="el" href="group__grp__graph.html">Graph</a></div></div>  </div>
 </div><!--header-->
 <div class="contents">
 <div class="toc"><b>Contents</b> <ul>
 <li>
 <a href="#sssp">SSSP</a> </li>
 <li>
 <a href="#examples">Examples</a> </li>
 <li>
 <a href="#notes">Notes</a> </li>
 <li>
 <a href="#literature">Literature</a> </li>
 </ul>
 </div><p>Given a graph and a source vertex, the single source shortest path (SSSP) algorithm finds a path from the source vertex to every other vertex in the graph, such that the sum of the weights of the path edges is minimized.</p>
 <p><a class="anchor" id="sssp"></a></p><dl class="section user"><dt>SSSP</dt><dd><pre class="syntax">
 graph_sssp( vertex_table,
             vertex_id,
             edge_table,
             edge_args,
             source_vertex,
             out_table,
             grouping_cols
           )
 </pre></dd></dl>
 <p><b>Arguments</b> </p><dl class="arglist">
 <dt>vertex_table </dt>
 <dd><p class="startdd">TEXT. Name of the table containing the vertex data for the graph. Must contain the column specified in the 'vertex_id' parameter below.</p>
 <p class="enddd"></p>
 </dd>
 <dt>vertex_id </dt>
 <dd><p class="startdd">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.</p>
 <p class="enddd"></p>
 </dd>
 <dt>edge_table </dt>
 <dd><p class="startdd">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.</p>
 <p class="enddd"></p>
 </dd>
 <dt>edge_args </dt>
 <dd><p class="startdd">TEXT. A comma-delimited string containing multiple named arguments of the form "name=value". The following parameters are supported for this string argument:</p><ul>
 <li>src (INTEGER or BIGINT): Name of the column containing the source vertex ids in the edge table. Default column name is 'src'.</li>
 <li>dest (INTEGER or BIGINT): Name of the column containing the destination vertex ids in the edge table. Default column name is 'dest'.</li>
 <li>weight (FLOAT8): Name of the column containing the edge weights in the edge table. Default column name is 'weight'.</li>
 </ul>
 <p class="enddd"></p>
 </dd>
 <dt>source_vertex </dt>
 <dd><p class="startdd">INTEGER or BIGINT. The source vertex id for the algorithm to start. This vertex id must exist in the 'vertex_id' column of 'vertex_table'.</p>
 <p class="enddd"></p>
 </dd>
 <dt>out_table </dt>
 <dd><p class="startdd">TEXT. Name of the table to store the result of SSSP. It contains a row for every vertex of every group and have the following columns (in addition to the grouping columns):</p><ul>
 <li>vertex_id : The id for the destination. Will use the input parameter 'vertex_id' for column naming.</li>
 <li>weight : The total weight of the shortest path from the source vertex to this particular vertex. Will use the input parameter 'weight' for column naming.</li>
 <li>parent : The parent of this vertex in the shortest path from source. Will use 'parent' for column naming.</li>
 </ul>
 <p>A summary table named &lt;out_table&gt;_summary is also created. This is an internal table that keeps a record of the input parameters and is used by the path function described below. </p>
 <p class="enddd"></p>
 </dd>
 <dt>grouping_cols (optional) </dt>
 <dd>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 SSSP result is generated.  </dd>
 </dl>
 <dl class="section user"><dt>Path Retrieval</dt><dd></dd></dl>
 <p>The path retrieval function returns the shortest path from the source vertex to a specified desination vertex.</p>
 <pre class="syntax">
 graph_sssp_get_path( sssp_table,
                      dest_vertex,
                      path_table
                     )
 </pre><p><b>Arguments</b> </p><dl class="arglist">
 <dt>sssp_table </dt>
 <dd><p class="startdd">TEXT. Name of the table that contains the SSSP output.</p>
 <p class="enddd"></p>
 </dd>
 <dt>dest_vertex </dt>
 <dd><p class="startdd">INTEGER or BIGINT. The vertex that will be the destination of the desired path.</p>
 <p class="enddd"></p>
 </dd>
 <dt>path_table </dt>
 <dd><p class="startdd">TEXT. Name of the output table that contains the path. It contains a row for every group and has the following columns:</p><ul>
 <li>grouping_cols : The grouping columns given in the creation of the SSSP table. If there are no grouping columns, these columns will not exist and the table will have a single row.</li>
 <li>path (ARRAY) : The shortest path from the source vertex (as specified in the SSSP execution) to the destination vertex. </li>
 </ul>
 <p class="enddd"></p>
 </dd>
 </dl>
 <p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
 <ol type="1">
 <li>Create vertex and edge tables to represent the graph: <pre class="syntax">
 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);
 </pre></li>
 <li>Calculate the shortest paths from vertex 0: <pre class="syntax">
 DROP TABLE IF EXISTS out, out_summary;
 SELECT madlib.graph_sssp(
                          'vertex',      -- Vertex table
                          NULL,          -- Vertix id column (NULL means use default naming)
                          'edge',        -- Edge table
                          NULL,          -- Edge arguments (NULL means use default naming)
                          0,             -- Source vertex for path calculation
                          'out');        -- Output table of shortest paths
 SELECT * FROM out ORDER BY id;
 </pre> <pre class="result">
  id | weight | parent
 ----+--------+--------
   0 |      0 |      0
   1 |      1 |      0
   2 |      1 |      0
   3 |      2 |      2
   4 |     10 |      0
   5 |      2 |      2
   6 |      3 |      5
   7 |      4 |      6
 (8 rows)
 </pre></li>
 <li>Get the shortest path to vertex 5: <pre class="syntax">
 DROP TABLE IF EXISTS out_path;
 SELECT madlib.graph_sssp_get_path('out',5,'out_path');
 SELECT * FROM out_path;
 </pre> <pre class="result">
   path
 ---------
  {0,2,5}
 </pre></li>
 <li>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: <pre class="syntax">
 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;
 </pre></li>
 <li>Get the shortest path from vertex 1: <pre class="syntax">
 DROP TABLE IF EXISTS out_alt, out_alt_summary;
 SELECT madlib.graph_sssp(
                          'vertex_alt',                  -- Vertex table
                          'v_id',                        -- Vertex id column (NULL means use default naming)
                          'edge_alt',                    -- Edge table
                          'src=e_src, weight=e_weight',  -- Edge arguments (NULL means use default naming)
                          1,                             -- Source vertex for path calculation
                          'out_alt');                    -- Output table of shortest paths
 SELECT * FROM out_alt ORDER BY v_id;
 </pre> <pre class="result">
  v_id | e_weight | parent
 ------+----------+--------
     0 |        4 |      3
     1 |        0 |      1
     2 |        2 |      1
     3 |        3 |      2
     4 |       14 |      0
     5 |        3 |      2
     6 |        4 |      5
     7 |        5 |      6
 (8 rows)
 </pre></li>
 <li>Create a graph with 2 groups: <pre class="syntax">
 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 &lt; 6 AND dest &lt; 6
 );
 INSERT INTO edge_gr VALUES
 (4,5,-20,1);
 </pre></li>
 <li>Find SSSP for all groups <pre class="syntax">
 DROP TABLE IF EXISTS out_gr, out_gr_summary;
 SELECT madlib.graph_sssp(
                          'vertex',      -- Vertex table
                          NULL,          -- Vertex id column (NULL means use default naming)
                          'edge_gr',     -- Edge table
                          NULL,          -- Edge arguments (NULL means use default naming)
                          0,             -- Source vertex for path calculation
                          'out_gr',      -- Output table of shortest paths
                          'grp'          -- Grouping columns
 );
 SELECT * FROM out_gr ORDER BY grp,id;
 </pre> <pre class="result">
  grp | id | weight | parent
 -----+----+--------+--------
    0 |  0 |      0 |      0
    0 |  1 |      1 |      0
    0 |  2 |      1 |      0
    0 |  3 |      2 |      2
    0 |  4 |     10 |      0
    0 |  5 |      2 |      2
    0 |  6 |      3 |      5
    0 |  7 |      4 |      6
    1 |  0 |      0 |      0
    1 |  1 |      1 |      0
    1 |  2 |      1 |      0
    1 |  3 |      2 |      2
    1 |  4 |     10 |      0
    1 |  5 |    -10 |      4
 </pre></li>
 <li>Find the path to vertex 5 in every group <pre class="syntax">
 DROP TABLE IF EXISTS out_gr_path;
 SELECT madlib.graph_sssp_get_path('out_gr',5,'out_gr_path');
 SELECT * FROM out_gr_path ORDER BY grp;
 </pre> <pre class="result">
  grp |  path
 -----+---------
    0 | {0,2,5}
    1 | {0,4,5}
 </pre></li>
 </ol>
 <p><a class="anchor" id="notes"></a></p><dl class="section user"><dt>Notes</dt><dd></dd></dl>
 <ol type="1">
 <li>On a Greenplum cluster, the edge table should be distributed by the source vertex id column for better performance.</li>
 <li>The Bellman-Ford algorithm [1] is used to implement SSSP. This algorithm allows negative edges but not negative cycles. In the case of graphs with negative cycles, an error will be given and no output table will be generated.</li>
 <li>Also see the Grail project [2] for more background on graph analytics processing in relational databases.</li>
 </ol>
 <p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
 <p>[1] Bellman–Ford algorithm. <a href="https://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithm">https://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithm</a></p>
 <p>[2] The case against specialized graph analytics engines, J. Fan, G. Soosai Raj, and J. M. Patel. CIDR 2015. <a href="http://cidrdb.org/cidr2015/Papers/CIDR15_Paper20.pdf">http://cidrdb.org/cidr2015/Papers/CIDR15_Paper20.pdf</a> </p>
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	<div class="header">
	<div class="headertitle">
	<div class="title">Single Source Shortest Path<div class="ingroups"><a class="el" href="group__grp__graph.html">Graph</a></div></div> </div>
	</div><!--header-->
	<div class="contents">
	<div class="toc"><b>Contents</b> <ul>
	<li>
	<a href="#sssp">SSSP</a> </li>
	<li>
	<a href="#examples">Examples</a> </li>
	<li>
	<a href="#notes">Notes</a> </li>
	<li>
	<a href="#literature">Literature</a> </li>
	</ul>
	</div><p>Given a graph and a source vertex, the single source shortest path (SSSP) algorithm finds a path from the source vertex to every other vertex in the graph, such that the sum of the weights of the path edges is minimized.</p>
	<p><a class="anchor" id="sssp"></a></p><dl class="section user"><dt>SSSP</dt><dd><pre class="syntax">
	graph_sssp( vertex_table,
	vertex_id,
	edge_table,
	edge_args,
	source_vertex,
	out_table,
	grouping_cols
	)
	</pre></dd></dl>
	<p><b>Arguments</b> </p><dl class="arglist">
	<dt>vertex_table </dt>
	<dd><p class="startdd">TEXT. Name of the table containing the vertex data for the graph. Must contain the column specified in the 'vertex_id' parameter below.</p>
	<p class="enddd"></p>
	</dd>
	<dt>vertex_id </dt>
	<dd><p class="startdd">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.</p>
	<p class="enddd"></p>
	</dd>
	<dt>edge_table </dt>
	<dd><p class="startdd">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.</p>
	<p class="enddd"></p>
	</dd>
	<dt>edge_args </dt>
	<dd><p class="startdd">TEXT. A comma-delimited string containing multiple named arguments of the form "name=value". The following parameters are supported for this string argument:</p><ul>
	<li>src (INTEGER or BIGINT): Name of the column containing the source vertex ids in the edge table. Default column name is 'src'.</li>
	<li>dest (INTEGER or BIGINT): Name of the column containing the destination vertex ids in the edge table. Default column name is 'dest'.</li>
	<li>weight (FLOAT8): Name of the column containing the edge weights in the edge table. Default column name is 'weight'.</li>
	</ul>
	<p class="enddd"></p>
	</dd>
	<dt>source_vertex </dt>
	<dd><p class="startdd">INTEGER or BIGINT. The source vertex id for the algorithm to start. This vertex id must exist in the 'vertex_id' column of 'vertex_table'.</p>
	<p class="enddd"></p>
	</dd>
	<dt>out_table </dt>
	<dd><p class="startdd">TEXT. Name of the table to store the result of SSSP. It contains a row for every vertex of every group and have the following columns (in addition to the grouping columns):</p><ul>
	<li>vertex_id : The id for the destination. Will use the input parameter 'vertex_id' for column naming.</li>
	<li>weight : The total weight of the shortest path from the source vertex to this particular vertex. Will use the input parameter 'weight' for column naming.</li>
	<li>parent : The parent of this vertex in the shortest path from source. Will use 'parent' for column naming.</li>
	</ul>
	<p>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 function described below. </p>
	<p class="enddd"></p>
	</dd>
	<dt>grouping_cols (optional) </dt>
	<dd>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 SSSP result is generated. </dd>
	</dl>
	<dl class="section user"><dt>Path Retrieval</dt><dd></dd></dl>
	<p>The path retrieval function returns the shortest path from the source vertex to a specified desination vertex.</p>
	<pre class="syntax">
	graph_sssp_get_path( sssp_table,
	dest_vertex,
	path_table
	)
	</pre><p><b>Arguments</b> </p><dl class="arglist">
	<dt>sssp_table </dt>
	<dd><p class="startdd">TEXT. Name of the table that contains the SSSP output.</p>
	<p class="enddd"></p>
	</dd>
	<dt>dest_vertex </dt>
	<dd><p class="startdd">INTEGER or BIGINT. The vertex that will be the destination of the desired path.</p>
	<p class="enddd"></p>
	</dd>
	<dt>path_table </dt>
	<dd><p class="startdd">TEXT. Name of the output table that contains the path. It contains a row for every group and has the following columns:</p><ul>
	<li>grouping_cols : The grouping columns given in the creation of the SSSP table. If there are no grouping columns, these columns will not exist and the table will have a single row.</li>
	<li>path (ARRAY) : The shortest path from the source vertex (as specified in the SSSP execution) to the destination vertex. </li>
	</ul>
	<p class="enddd"></p>
	</dd>
	</dl>
	<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
	<ol type="1">
	<li>Create vertex and edge tables to represent the graph: <pre class="syntax">
	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);
	</pre></li>
	<li>Calculate the shortest paths from vertex 0: <pre class="syntax">
	DROP TABLE IF EXISTS out, out_summary;
	SELECT madlib.graph_sssp(
	'vertex', -- Vertex table
	NULL, -- Vertix id column (NULL means use default naming)
	'edge', -- Edge table
	NULL, -- Edge arguments (NULL means use default naming)
	0, -- Source vertex for path calculation
	'out'); -- Output table of shortest paths
	SELECT * FROM out ORDER BY id;
	</pre> <pre class="result">
	id \| weight \| parent
	----+--------+--------
	0 \| 0 \| 0
	1 \| 1 \| 0
	2 \| 1 \| 0
	3 \| 2 \| 2
	4 \| 10 \| 0
	5 \| 2 \| 2
	6 \| 3 \| 5
	7 \| 4 \| 6
	(8 rows)
	</pre></li>
	<li>Get the shortest path to vertex 5: <pre class="syntax">
	DROP TABLE IF EXISTS out_path;
	SELECT madlib.graph_sssp_get_path('out',5,'out_path');
	SELECT * FROM out_path;
	</pre> <pre class="result">
	path
	---------
	{0,2,5}
	</pre></li>
	<li>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: <pre class="syntax">
	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;
	</pre></li>
	<li>Get the shortest path from vertex 1: <pre class="syntax">
	DROP TABLE IF EXISTS out_alt, out_alt_summary;
	SELECT madlib.graph_sssp(
	'vertex_alt', -- Vertex table
	'v_id', -- Vertex id column (NULL means use default naming)
	'edge_alt', -- Edge table
	'src=e_src, weight=e_weight', -- Edge arguments (NULL means use default naming)
	1, -- Source vertex for path calculation
	'out_alt'); -- Output table of shortest paths
	SELECT * FROM out_alt ORDER BY v_id;
	</pre> <pre class="result">
	v_id \| e_weight \| parent
	------+----------+--------
	0 \| 4 \| 3
	1 \| 0 \| 1
	2 \| 2 \| 1
	3 \| 3 \| 2
	4 \| 14 \| 0
	5 \| 3 \| 2
	6 \| 4 \| 5
	7 \| 5 \| 6
	(8 rows)
	</pre></li>
	<li>Create a graph with 2 groups: <pre class="syntax">
	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);
	</pre></li>
	<li>Find SSSP for all groups <pre class="syntax">
	DROP TABLE IF EXISTS out_gr, out_gr_summary;
	SELECT madlib.graph_sssp(
	'vertex', -- Vertex table
	NULL, -- Vertex id column (NULL means use default naming)
	'edge_gr', -- Edge table
	NULL, -- Edge arguments (NULL means use default naming)
	0, -- Source vertex for path calculation
	'out_gr', -- Output table of shortest paths
	'grp' -- Grouping columns
	);
	SELECT * FROM out_gr ORDER BY grp,id;
	</pre> <pre class="result">
	grp \| id \| weight \| parent
	-----+----+--------+--------
	0 \| 0 \| 0 \| 0
	0 \| 1 \| 1 \| 0
	0 \| 2 \| 1 \| 0
	0 \| 3 \| 2 \| 2
	0 \| 4 \| 10 \| 0
	0 \| 5 \| 2 \| 2
	0 \| 6 \| 3 \| 5
	0 \| 7 \| 4 \| 6
	1 \| 0 \| 0 \| 0
	1 \| 1 \| 1 \| 0
	1 \| 2 \| 1 \| 0
	1 \| 3 \| 2 \| 2
	1 \| 4 \| 10 \| 0
	1 \| 5 \| -10 \| 4
	</pre></li>
	<li>Find the path to vertex 5 in every group <pre class="syntax">
	DROP TABLE IF EXISTS out_gr_path;
	SELECT madlib.graph_sssp_get_path('out_gr',5,'out_gr_path');
	SELECT * FROM out_gr_path ORDER BY grp;
	</pre> <pre class="result">
	grp \| path
	-----+---------
	0 \| {0,2,5}
	1 \| {0,4,5}
	</pre></li>
	</ol>
	<p><a class="anchor" id="notes"></a></p><dl class="section user"><dt>Notes</dt><dd></dd></dl>
	<ol type="1">
	<li>On a Greenplum cluster, the edge table should be distributed by the source vertex id column for better performance.</li>
	<li>The Bellman-Ford algorithm [1] is used to implement SSSP. This algorithm allows negative edges but not negative cycles. In the case of graphs with negative cycles, an error will be given and no output table will be generated.</li>
	<li>Also see the Grail project [2] for more background on graph analytics processing in relational databases.</li>
	</ol>
	<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
	<p>[1] Bellman–Ford algorithm. <a href="https://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithm">https://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithm</a></p>
	<p>[2] The case against specialized graph analytics engines, J. Fan, G. Soosai Raj, and J. M. Patel. CIDR 2015. <a href="http://cidrdb.org/cidr2015/Papers/CIDR15_Paper20.pdf">http://cidrdb.org/cidr2015/Papers/CIDR15_Paper20.pdf</a> </p>
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