 # pgr_johnson¶

## Synopsis¶

pgr_johnson - Returns the sum of the costs of the shortest path for each pair of nodes in the graph using Floyd-Warshall algorithm.

The Johnson algorithm, is a good choice to calculate the sum of the costs of the shortest path for each pair of nodes in the graph, for sparse graphs. It usees the Boost’s implementation which runs in $$O(V E \log V)$$ time,

## Characteristics¶

The main Characteristics are:
• It does not return a path.
• Returns the sum of the costs of the shortest path for each pair of nodes in the graph.
• Process is done only on edges with positive costs.
• Boost returns a $$V \times V$$ matrix, where the infinity values. Represent the distance between vertices for which there is no path.
• We return only the non infinity values in form of a set of (start_vid, end_vid, agg_cost).
• Let be the case the values returned are stored in a table, so the unique index would be the pair: (start_vid, end_vid).
• For the undirected graph, the results are symmetric.
• The agg_cost of (u, v) is the same as for (v, u).
• When start_vid = end_vid, the agg_cost = 0.

## Signature Summary¶

pgr_johnson(edges_sql)
pgr johnson(edges_sql, directed)
RETURNS SET OF (start_vid, end_vid,  agg_cost) or EMPTY SET


## Signatures¶

### Minimal Signature¶

pgr_johnson(edges_sql)
RETURNS SET OF (start_vid, end_vid,  agg_cost) or EMPTY SET

Example 1: On a directed graph.
SELECT * FROM pgr_johnson(
'SELECT source, target, cost FROM edge_table WHERE id < 5
ORDER BY id'
);
start_vid | end_vid | agg_cost
-----------+---------+----------
1 |       2 |        1
1 |       5 |        2
2 |       5 |        1
(3 rows)


### Complete Signature¶

pgr_johnson(edges_sql, directed)
RETURNS SET OF (start_vid, end_vid,  agg_cost) or EMPTY SET

Example 2: On an undirected graph.
SELECT * FROM pgr_johnson(
'SELECT source, target, cost FROM edge_table WHERE id < 5
ORDER BY id',
false
);
start_vid | end_vid | agg_cost
-----------+---------+----------
1 |       2 |        1
1 |       5 |        2
2 |       1 |        1
2 |       5 |        1
5 |       1 |        2
5 |       2 |        1
(6 rows)


## Description of the Signatures¶

### Description of the edges_sql query¶

edges_sql: an SQL query, which should return a set of rows with the following columns:
Column Type Default Description
source ANY-INTEGER   Identifier of the first end point vertex of the edge.
target ANY-INTEGER   Identifier of the second end point vertex of the edge.
cost ANY-NUMERICAL
Weight of the edge (source, target)
• When negative: edge (source, target) does not exist, therefore it’s not part of the graph.
reverse_cost ANY-NUMERICAL -1
Weight of the edge (target, source),
• When negative: edge (target, source) does not exist, therefore it’s not part of the graph.

Where:

ANY-INTEGER: SMALLINT, INTEGER, BIGINT SMALLINT, INTEGER, BIGINT, REAL, FLOAT

### Description of the parameters of the signatures¶

Parameter Type Description
edges_sql TEXT SQL query as described above.
directed BOOLEAN (optional) Default is true (is directed). When set to false the graph is considered as Undirected

### Description of the return values¶

Returns set of (start_vid, end_vid, agg_cost)

Column Type Description
start_vid BIGINT Identifier of the starting vertex.
end_vid BIGINT Identifier of the ending vertex.
agg_cost FLOAT Total cost from start_vid to end_vid.

History

• Re-design of pgr_apspJohnson in Version 2.2.0