pgr_binaryBreadthFirstSearch
— Returns the shortest path(s) in a binary graph.
Any graph whose edge-weights belongs to the set {0,X}, where ‘X’ is any non-negative real integer, is termed as a ‘binary graph’.
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It is well-known that the shortest paths between a single source and all other vertices can be found using Breadth First Search in \(O(|E|)\) in an unweighted graph, i.e. the distance is the minimal number of edges that you need to traverse from the source to another vertex. We can interpret such a graph also as a weighted graph, where every edge has the weight 1. If not all edges in graph have the same weight, that we need a more general algorithm, like Dijkstra’s Algorithm which runs in \(O(|E|log|V|)\) time.
However if the weights are more constrained, we can use a faster algorithm. This algorithm, termed as ‘Binary Breadth First Search’ as well as ‘0-1 BFS’, is a variation of the standard Breadth First Search problem to solve the SSSP (single-source shortest path) problem in \(O(|E|)\), if the weights of each edge belongs to the set {0,X}, where ‘X’ is any non-negative real integer.
The main Characteristics are:
- Process is done only on ‘binary graphs’. (‘Binary Graph’: Any graph whose edge-weights belongs to the set {0,X}, where ‘X’ is any non-negative real integer.)
- 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: \(O(| start\_vids | * |E|)\)
pgr_binaryBreadthFirstSearch(edges_sql, start_vid, end_vid [, directed])
pgr_binaryBreadthFirstSearch(edges_sql, start_vid, end_vids [, directed])
pgr_binaryBreadthFirstSearch(edges_sql, start_vids, end_vid [, directed])
pgr_binaryBreadthFirstSearch(edges_sql, start_vids, end_vids [, directed])
RETURNS SET OF (seq, path_seq [, start_vid] [, end_vid], node, edge, cost, agg_cost)
OR EMPTY SET
pgr_binaryBreadthFirstSearch(TEXT edges_sql, BIGINT start_vid, BIGINT end_vid)
RETURNS SET OF (seq, path_seq, node, edge, cost, agg_cost) or EMPTY SET
Example: | From vertex \(2\) to vertex \(3\) on a directed binary graph |
---|
SELECT * FROM pgr_binaryBreadthFirstSearch(
'SELECT id, source, target, road_work as cost, reverse_road_work as reverse_cost FROM roadworks',
2, 3
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 | 1 | 2 | 4 | 0 | 0
2 | 2 | 5 | 8 | 1 | 0
3 | 3 | 6 | 9 | 1 | 1
4 | 4 | 9 | 16 | 0 | 2
5 | 5 | 4 | 3 | 0 | 2
6 | 6 | 3 | -1 | 0 | 2
(6 rows)
pgr_binaryBreadthFirstSearch(TEXT edges_sql, BIGINT start_vid, BIGINT end_vid,
BOOLEAN directed:=true);
RETURNS SET OF (seq, path_seq, node, edge, cost, agg_cost)
OR EMPTY SET
Example: | From vertex \(2\) to vertex \(3\) on an undirected binary graph |
---|
SELECT * FROM pgr_binaryBreadthFirstSearch(
'SELECT id, source, target, road_work as cost, reverse_road_work as reverse_cost FROM roadworks',
2, 3,
FALSE
);
seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
1 | 1 | 2 | 2 | 1 | 0
2 | 2 | 3 | -1 | 0 | 1
(2 rows)
pgr_binaryBreadthFirstSearch(TEXT edges_sql, BIGINT start_vid, ARRAY[ANY_INTEGER] end_vids,
BOOLEAN directed:=true);
RETURNS SET OF (seq, path_seq, end_vid, node, edge, cost, agg_cost)
OR EMPTY SET
Example: | From vertex \(2\) to vertices \(\{3, 5\}\) on an undirected binary graph |
---|
SELECT * FROM pgr_binaryBreadthFirstSearch(
'SELECT id, source, target, road_work as cost FROM roadworks',
2, ARRAY[3,5],
FALSE
);
seq | path_seq | end_vid | node | edge | cost | agg_cost
-----+----------+---------+------+------+------+----------
1 | 1 | 3 | 2 | 4 | 0 | 0
2 | 2 | 3 | 5 | 8 | 1 | 0
3 | 3 | 3 | 6 | 5 | 1 | 1
4 | 4 | 3 | 3 | -1 | 0 | 2
5 | 1 | 5 | 2 | 4 | 0 | 0
6 | 2 | 5 | 5 | -1 | 0 | 0
(6 rows)
pgr_binaryBreadthFirstSearch(TEXT edges_sql, ARRAY[ANY_INTEGER] start_vids, BIGINT end_vid,
BOOLEAN directed:=true);
RETURNS SET OF (seq, path_seq, start_vid, node, edge, cost, agg_cost)
OR EMPTY SET
Example: | From vertices \(\{2, 11\}\) to vertex \(5\) on a directed binary graph |
---|
SELECT * FROM pgr_binaryBreadthFirstSearch(
'SELECT id, source, target, road_work as cost, reverse_road_work as reverse_cost FROM roadworks',
ARRAY[2,11], 5
);
seq | path_seq | start_vid | node | edge | cost | agg_cost
-----+----------+-----------+------+------+------+----------
1 | 1 | 2 | 2 | 4 | 0 | 0
2 | 2 | 2 | 5 | -1 | 0 | 0
3 | 1 | 11 | 11 | 13 | 1 | 0
4 | 2 | 11 | 12 | 15 | 0 | 1
5 | 3 | 11 | 9 | 16 | 0 | 1
6 | 4 | 11 | 4 | 3 | 0 | 1
7 | 5 | 11 | 3 | 2 | 1 | 1
8 | 6 | 11 | 2 | 4 | 0 | 2
9 | 7 | 11 | 5 | -1 | 0 | 2
(9 rows)
pgr_binaryBreadthFirstSearch(TEXT edges_sql, ARRAY[ANY_INTEGER] start_vids, ARRAY[ANY_INTEGER] end_vids,
BOOLEAN directed:=true);
RETURNS SET OF (seq, path_seq, start_vid, end_vid, node, edge, cost, agg_cost)
OR EMPTY SET
Example: | From vertices \(\{2, 11\}\) to vertices \(\{3, 5\}\) on an undirected binary graph |
---|
SELECT * FROM pgr_binaryBreadthFirstSearch(
'SELECT id, source, target, road_work as cost, reverse_road_work as reverse_cost FROM roadworks',
ARRAY[2,11], ARRAY[3,5],
FALSE
);
seq | path_seq | start_vid | end_vid | node | edge | cost | agg_cost
-----+----------+-----------+---------+------+------+------+----------
1 | 1 | 2 | 3 | 2 | 2 | 1 | 0
2 | 2 | 2 | 3 | 3 | -1 | 0 | 1
3 | 1 | 2 | 5 | 2 | 4 | 0 | 0
4 | 2 | 2 | 5 | 5 | -1 | 0 | 0
5 | 1 | 11 | 3 | 11 | 13 | 1 | 0
6 | 2 | 11 | 3 | 12 | 15 | 0 | 1
7 | 3 | 11 | 3 | 9 | 16 | 0 | 1
8 | 4 | 11 | 3 | 4 | 3 | 0 | 1
9 | 5 | 11 | 3 | 3 | -1 | 0 | 1
10 | 1 | 11 | 5 | 11 | 12 | 0 | 0
11 | 2 | 11 | 5 | 10 | 10 | 1 | 0
12 | 3 | 11 | 5 | 5 | -1 | 0 | 1
(12 rows)
Parameter | Type | Default | Description |
---|---|---|---|
edges_sql | TEXT |
Inner SQL query as described below. | |
start_vid | BIGINT |
Identifier of the starting vertex of the path. | |
start_vids | ARRAY[BIGINT] |
Array of identifiers of starting vertices. | |
end_vid | BIGINT |
Identifier of the ending vertex of the path. | |
end_vids | ARRAY[BIGINT] |
Array of identifiers of ending vertices. | |
directed | BOOLEAN |
true |
|
Column | Type | Default | Description |
---|---|---|---|
id | ANY-INTEGER |
Identifier of the edge. | |
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)
|
|
reverse_cost | ANY-NUMERICAL |
-1 | Weight of the edge (target, source),
|
Where:
ANY-INTEGER: | SMALLINT, INTEGER, BIGINT |
---|---|
ANY-NUMERICAL: | SMALLINT, INTEGER, BIGINT, REAL, FLOAT |
Returns set of (seq, path_id, path_seq [, start_vid] [, end_vid], node, edge, cost, agg_cost)
Column | Type | Description |
---|---|---|
seq | INT |
Sequential value starting from 1. |
path_id | INT |
Path identifier. Has value 1 for the first of a path. Used when there are multiple paths for the same start_vid to end_vid combination. |
path_seq | INT |
Relative position in the path. Has value 1 for the beginning of a path. |
start_vid | BIGINT |
Identifier of the starting vertex. Returned when multiple starting vetrices are in the query. |
end_vid | BIGINT |
Identifier of the ending vertex. Returned when multiple ending vertices are in the query. |
node | BIGINT |
Identifier of the node in the path from start_vid to end_vid . |
edge | BIGINT |
Identifier of the edge used to go from node to the next node in the path sequence. -1 for the last node of the path. |
cost | FLOAT |
Cost to traverse from node using edge to the next node in the path sequence. |
agg_cost | FLOAT |
Aggregate cost from start_v to node . |
This type of data is used on the examples of this page.
Edwards-Moore Algorithm is best applied when trying to answer queries such as the following: “Find the path with the minimum number from Source to Destination” Here: * Source = Source Vertex, Destination = Any arbitrary destination vertex * X is an event/property * Each edge in the graph is either “X” or “Not X” .
Example: “Find the path with the minimum number of road works from Source to Destination”
Here, a road under work(aka road works) means that part of the road is occupied for construction work/maintenance.
Here: * Edge ( u , v ) has weight = 0 if no road work is ongoing on the road from u to v. * Edge ( u, v) has weight = 1 if road work is ongoing on the road from u to v.
Then, upon running the algorithm, we obtain the path with the minimum number of road works from the given source and destination.
Thus, the queries used in the previous section can be interpreted in this manner.
The queries in the previous sections use the table ‘roadworks’. The data of the table:
DROP TABLE IF EXISTS roadworks CASCADE;
NOTICE: table "roadworks" does not exist, skipping
DROP TABLE
CREATE table roadworks (
id BIGINT not null primary key,
source BIGINT,
target BIGINT,
road_work FLOAT,
reverse_road_work FLOAT
);
CREATE TABLE
INSERT INTO roadworks(
id, source, target, road_work, reverse_road_work) VALUES
(1, 1, 2, 0, 0),
(2, 2, 3, -1, 1),
(3, 3, 4, -1, 0),
(4, 2, 5, 0, 0),
(5, 3, 6, 1, -1),
(6, 7, 8, 1, 1),
(7, 8, 5, 0, 0),
(8, 5, 6, 1, 1),
(9, 6, 9, 1, 1),
(10, 5, 10, 1, 1),
(11, 6, 11, 1, -1),
(12, 10, 11, 0, -1),
(13, 11, 12, 1, -1),
(14, 10, 13, 1, 1),
(15, 9, 12, 0, 0),
(16, 4, 9, 0, 0),
(17, 14, 15, 0, 0),
(18, 16, 17, 0, 0);
INSERT 0 18