pgr_bellmanFord - Experimental¶
pgr_bellmanFord — Shortest path(s) using Bellman-Ford algorithm.
Boost Graph Inside¶
Warning
Possible server crash
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Warning
Experimental functions
They are not officially of the current release.
They likely will not be officially be part of the next release:
The functions might not make use of ANY-INTEGER and ANY-NUMERICAL
Name might change.
Signature might change.
Functionality might change.
pgTap tests might be missing.
Might need c/c++ coding.
May lack documentation.
Documentation if any might need to be rewritten.
Documentation examples might need to be automatically generated.
Might need a lot of feedback from the comunity.
Might depend on a proposed function of pgRouting
Might depend on a deprecated function of pgRouting
Availability
Version 3.2.0
New experimental signature:
pgr_bellmanFord(Combinations)
Version 3.0.0
New experimental signatures:
pgr_bellmanFord(One to One)pgr_bellmanFord(One to Many)pgr_bellmanFord(Many to One)pgr_bellmanFord(Many to Many)
Description¶
Bellman-Ford’s algorithm, is named after Richard Bellman and Lester Ford, who
first published it in 1958 and 1956, respectively.It is a graph search algorithm
that computes shortest paths from a starting vertex (start_vid) to an ending
vertex (end_vid) in a graph where some of the edge weights may be negative.
Though it is more versatile, it is slower than Dijkstra’s algorithm.This
implementation can be used with a directed graph and an undirected graph.
- The main characteristics are:
Process is valid for edges with both positive and negative edge weights.
Values are returned when there is a path.
When the start vertex and the end vertex are the same, there is no path. The agg_cost would be \(0\).
When the start vertex and the end vertex are different, and there exists a path between them without having a negative cycle. The agg_cost would be some finite value denoting the shortest distance between them.
When the start vertex and the end vertex are different, and there exists a path between them, but it contains a negative cycle. In such case, agg_cost for those vertices keep on decreasing furthermore, Hence agg_cost can’t be defined for them.
When the start vertex and the end vertex are different, and there is no path. The agg_cost is \(\infty\).
For optimization purposes, any duplicated value in the start_vids or end_vids are ignored.
The returned values are ordered:
start_vid ascending
end_vid ascending
Running time: \(O(| start\_vids | * ( V * E))\)
Signatures¶
Summary
directed])directed])directed])directed])(seq, path_seq, [start_vid], [end_vid], node, edge, cost, agg_cost)One to One¶
directed])(seq, path_seq, node, edge, cost, agg_cost)- Example:
From vertex \(6\) to vertex \(10\) on a directed graph
SELECT * FROM pgr_bellmanFord(
'SELECT id, source, target, cost, reverse_cost FROM edges',
6, 10, true);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 | 1 | 6 | 4 | 1 | 0
2 | 2 | 7 | 8 | 1 | 1
3 | 3 | 11 | 9 | 1 | 2
4 | 4 | 16 | 16 | 1 | 3
5 | 5 | 15 | 3 | 1 | 4
6 | 6 | 10 | -1 | 0 | 5
(6 rows)
One to Many¶
directed])(seq, path_seq, end_vid, node, edge, cost, agg_cost)- Example:
From vertex \(6\) to vertices \(\{ 10, 17\}\) on a directed graph
SELECT * FROM pgr_bellmanFord(
'SELECT id, source, target, cost, reverse_cost FROM edges',
6, ARRAY[10, 17]);
seq | path_seq | end_vid | node | edge | cost | agg_cost
-----+----------+---------+------+------+------+----------
1 | 1 | 10 | 6 | 4 | 1 | 0
2 | 2 | 10 | 7 | 8 | 1 | 1
3 | 3 | 10 | 11 | 9 | 1 | 2
4 | 4 | 10 | 16 | 16 | 1 | 3
5 | 5 | 10 | 15 | 3 | 1 | 4
6 | 6 | 10 | 10 | -1 | 0 | 5
7 | 1 | 17 | 6 | 4 | 1 | 0
8 | 2 | 17 | 7 | 8 | 1 | 1
9 | 3 | 17 | 11 | 11 | 1 | 2
10 | 4 | 17 | 12 | 13 | 1 | 3
11 | 5 | 17 | 17 | -1 | 0 | 4
(11 rows)
Many to One¶
directed])(seq, path_seq, start_vid, node, edge, cost, agg_cost)- Example:
From vertices \(\{6, 1\}\) to vertex \(17\) on a directed graph
SELECT * FROM pgr_bellmanFord(
'SELECT id, source, target, cost, reverse_cost FROM edges',
ARRAY[6, 1], 17);
seq | path_seq | start_vid | node | edge | cost | agg_cost
-----+----------+-----------+------+------+------+----------
1 | 1 | 1 | 1 | 6 | 1 | 0
2 | 2 | 1 | 3 | 7 | 1 | 1
3 | 3 | 1 | 7 | 8 | 1 | 2
4 | 4 | 1 | 11 | 11 | 1 | 3
5 | 5 | 1 | 12 | 13 | 1 | 4
6 | 6 | 1 | 17 | -1 | 0 | 5
7 | 1 | 6 | 6 | 4 | 1 | 0
8 | 2 | 6 | 7 | 8 | 1 | 1
9 | 3 | 6 | 11 | 11 | 1 | 2
10 | 4 | 6 | 12 | 13 | 1 | 3
11 | 5 | 6 | 17 | -1 | 0 | 4
(11 rows)
Many to Many¶
directed])(seq, path_seq, start_vid, end_vid, node, edge, cost, agg_cost)- Example:
From vertices \(\{6, 1\}\) to vertices \(\{10, 17\}\) on an undirected graph
SELECT * FROM pgr_bellmanFord(
'SELECT id, source, target, cost, reverse_cost FROM edges',
ARRAY[6, 1], ARRAY[10, 17],
directed => false);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 | 1 | 1 | 10 | 1 | 6 | 1 | 0
2 | 2 | 1 | 10 | 3 | 7 | 1 | 1
3 | 3 | 1 | 10 | 7 | 4 | 1 | 2
4 | 4 | 1 | 10 | 6 | 2 | 1 | 3
5 | 5 | 1 | 10 | 10 | -1 | 0 | 4
6 | 1 | 1 | 17 | 1 | 6 | 1 | 0
7 | 2 | 1 | 17 | 3 | 7 | 1 | 1
8 | 3 | 1 | 17 | 7 | 8 | 1 | 2
9 | 4 | 1 | 17 | 11 | 11 | 1 | 3
10 | 5 | 1 | 17 | 12 | 13 | 1 | 4
11 | 6 | 1 | 17 | 17 | -1 | 0 | 5
12 | 1 | 6 | 10 | 6 | 2 | 1 | 0
13 | 2 | 6 | 10 | 10 | -1 | 0 | 1
14 | 1 | 6 | 17 | 6 | 4 | 1 | 0
15 | 2 | 6 | 17 | 7 | 8 | 1 | 1
16 | 3 | 6 | 17 | 11 | 11 | 1 | 2
17 | 4 | 6 | 17 | 12 | 13 | 1 | 3
18 | 5 | 6 | 17 | 17 | -1 | 0 | 4
(18 rows)
Combinations¶
(seq, path_seq, start_vid, end_vid, node, edge, cost, agg_cost)- Example:
Using a combinations table on an undirected graph.
The combinations table:
SELECT source, target FROM combinations;
source | target
--------+--------
5 | 6
5 | 10
6 | 5
6 | 15
6 | 14
(5 rows)
The query:
SELECT * FROM pgr_bellmanFord(
'SELECT id, source, target, cost, reverse_cost FROM edges',
'SELECT source, target FROM combinations',
false);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 | 1 | 5 | 6 | 5 | 1 | 1 | 0
2 | 2 | 5 | 6 | 6 | -1 | 0 | 1
3 | 1 | 5 | 10 | 5 | 1 | 1 | 0
4 | 2 | 5 | 10 | 6 | 2 | 1 | 1
5 | 3 | 5 | 10 | 10 | -1 | 0 | 2
6 | 1 | 6 | 5 | 6 | 1 | 1 | 0
7 | 2 | 6 | 5 | 5 | -1 | 0 | 1
8 | 1 | 6 | 15 | 6 | 2 | 1 | 0
9 | 2 | 6 | 15 | 10 | 3 | 1 | 1
10 | 3 | 6 | 15 | 15 | -1 | 0 | 2
(10 rows)
Parameters¶
Column |
Type |
Description |
|---|---|---|
|
Edges SQL as described below |
|
|
Combinations SQL as described below |
|
start vid |
|
Identifier of the starting vertex of the path. |
start vids |
|
Array of identifiers of starting vertices. |
end vid |
|
Identifier of the ending vertex of the path. |
end vids |
|
Array of identifiers of ending vertices. |
Optional parameters¶
Column |
Type |
Default |
Description |
|---|---|---|---|
|
|
|
|
Inner Queries¶
Edges SQL¶
Column |
Type |
Default |
Description |
|---|---|---|---|
|
ANY-INTEGER |
Identifier of the edge. |
|
|
ANY-INTEGER |
Identifier of the first end point vertex of the edge. |
|
|
ANY-INTEGER |
Identifier of the second end point vertex of the edge. |
|
|
ANY-NUMERICAL |
Weight of the edge ( |
|
|
ANY-NUMERICAL |
-1 |
Weight of the edge (
|
Where:
- ANY-INTEGER:
SMALLINT,INTEGER,BIGINT- ANY-NUMERICAL:
SMALLINT,INTEGER,BIGINT,REAL,FLOAT
Combinations SQL¶
Parameter |
Type |
Description |
|---|---|---|
|
ANY-INTEGER |
Identifier of the departure vertex. |
|
ANY-INTEGER |
Identifier of the arrival vertex. |
Where:
- ANY-INTEGER:
SMALLINT,INTEGER,BIGINT
Return columns¶
Returns set of (seq, path_seq [, start_vid] [, end_vid], node, edge, cost,
agg_cost)
Column |
Type |
Description |
|---|---|---|
|
|
Sequential value starting from 1. |
|
|
Relative position in the path. Has value 1 for the beginning of a path. |
|
|
Identifier of the starting vertex. Returned when multiple starting vetrices are in the query. |
|
|
Identifier of the ending vertex. Returned when multiple ending vertices are in the query. |
|
|
Identifier of the node in the path from |
|
|
Identifier of the edge used to go from |
|
|
Cost to traverse from |
|
|
Aggregate cost from |
Additional Examples¶
- Example 1:
Demonstration of repeated values are ignored, and result is sorted.
SELECT * FROM pgr_bellmanFord(
'SELECT id, source, target, cost, reverse_cost FROM edges',
ARRAY[7, 10, 15, 10, 10, 15], ARRAY[10, 7, 10, 15]);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 | 1 | 7 | 10 | 7 | 8 | 1 | 0
2 | 2 | 7 | 10 | 11 | 9 | 1 | 1
3 | 3 | 7 | 10 | 16 | 16 | 1 | 2
4 | 4 | 7 | 10 | 15 | 3 | 1 | 3
5 | 5 | 7 | 10 | 10 | -1 | 0 | 4
6 | 1 | 7 | 15 | 7 | 8 | 1 | 0
7 | 2 | 7 | 15 | 11 | 9 | 1 | 1
8 | 3 | 7 | 15 | 16 | 16 | 1 | 2
9 | 4 | 7 | 15 | 15 | -1 | 0 | 3
10 | 1 | 10 | 7 | 10 | 5 | 1 | 0
11 | 2 | 10 | 7 | 11 | 8 | 1 | 1
12 | 3 | 10 | 7 | 7 | -1 | 0 | 2
13 | 1 | 10 | 15 | 10 | 5 | 1 | 0
14 | 2 | 10 | 15 | 11 | 9 | 1 | 1
15 | 3 | 10 | 15 | 16 | 16 | 1 | 2
16 | 4 | 10 | 15 | 15 | -1 | 0 | 3
17 | 1 | 15 | 7 | 15 | 3 | 1 | 0
18 | 2 | 15 | 7 | 10 | 2 | 1 | 1
19 | 3 | 15 | 7 | 6 | 4 | 1 | 2
20 | 4 | 15 | 7 | 7 | -1 | 0 | 3
21 | 1 | 15 | 10 | 15 | 3 | 1 | 0
22 | 2 | 15 | 10 | 10 | -1 | 0 | 1
(22 rows)
- Example 2:
Making start vids the same as end vids.
SELECT * FROM pgr_bellmanFord(
'SELECT id, source, target, cost, reverse_cost FROM edges',
ARRAY[7, 10, 15], ARRAY[7, 10, 15]);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 | 1 | 7 | 10 | 7 | 8 | 1 | 0
2 | 2 | 7 | 10 | 11 | 9 | 1 | 1
3 | 3 | 7 | 10 | 16 | 16 | 1 | 2
4 | 4 | 7 | 10 | 15 | 3 | 1 | 3
5 | 5 | 7 | 10 | 10 | -1 | 0 | 4
6 | 1 | 7 | 15 | 7 | 8 | 1 | 0
7 | 2 | 7 | 15 | 11 | 9 | 1 | 1
8 | 3 | 7 | 15 | 16 | 16 | 1 | 2
9 | 4 | 7 | 15 | 15 | -1 | 0 | 3
10 | 1 | 10 | 7 | 10 | 5 | 1 | 0
11 | 2 | 10 | 7 | 11 | 8 | 1 | 1
12 | 3 | 10 | 7 | 7 | -1 | 0 | 2
13 | 1 | 10 | 15 | 10 | 5 | 1 | 0
14 | 2 | 10 | 15 | 11 | 9 | 1 | 1
15 | 3 | 10 | 15 | 16 | 16 | 1 | 2
16 | 4 | 10 | 15 | 15 | -1 | 0 | 3
17 | 1 | 15 | 7 | 15 | 3 | 1 | 0
18 | 2 | 15 | 7 | 10 | 2 | 1 | 1
19 | 3 | 15 | 7 | 6 | 4 | 1 | 2
20 | 4 | 15 | 7 | 7 | -1 | 0 | 3
21 | 1 | 15 | 10 | 15 | 3 | 1 | 0
22 | 2 | 15 | 10 | 10 | -1 | 0 | 1
(22 rows)
- Example 3:
Manually assigned vertex combinations.
SELECT * FROM pgr_bellmanFord(
'SELECT id, source, target, cost, reverse_cost FROM edges',
'SELECT * FROM (VALUES (6, 10), (6, 7), (12, 10)) AS combinations (source, target)');
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 | 1 | 6 | 7 | 6 | 4 | 1 | 0
2 | 2 | 6 | 7 | 7 | -1 | 0 | 1
3 | 1 | 6 | 10 | 6 | 4 | 1 | 0
4 | 2 | 6 | 10 | 7 | 8 | 1 | 1
5 | 3 | 6 | 10 | 11 | 9 | 1 | 2
6 | 4 | 6 | 10 | 16 | 16 | 1 | 3
7 | 5 | 6 | 10 | 15 | 3 | 1 | 4
8 | 6 | 6 | 10 | 10 | -1 | 0 | 5
9 | 1 | 12 | 10 | 12 | 13 | 1 | 0
10 | 2 | 12 | 10 | 17 | 15 | 1 | 1
11 | 3 | 12 | 10 | 16 | 16 | 1 | 2
12 | 4 | 12 | 10 | 15 | 3 | 1 | 3
13 | 5 | 12 | 10 | 10 | -1 | 0 | 4
(13 rows)
See Also¶
Indices and tables