pgr_binaryBreadthFirstSearch - Experimental

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’.

Boost Graph Inside

Warning

Possible server crash

• These functions might create a server crash

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 function:

    • pgr_binaryBreadthFirstSearch(Combinations)

• Version 3.0.0

  • New experimental function

Description

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: \(O(| start\_vids | * |E|)\)

Signatures

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])
pgr_binaryBreadthFirstSearch(Edges SQL, Combinations SQL [, directed]) -- Proposed on v3.2
RETURNS SET OF (seq, path_seq [, start_vid] [, end_vid], node, edge, cost, agg_cost)
OR EMPTY SET

pgr_binaryBreadthFirstSearch(Edges SQL, start_vid, 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)

One to One

pgr_binaryBreadthFirstSearch(Edges SQL, start_vid, end_vid [, directed]);
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)

One to many

pgr_binaryBreadthFirstSearch(Edges SQL, start_vid, end_vids [, directed]);
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)

Many to One

pgr_binaryBreadthFirstSearch(Edges SQL, start_vids, end_vid [, directed]);
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)

Many to Many

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

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)

Combinations

pgr_binaryBreadthFirstSearch(Edges SQL, Combinations SQL [, directed]);
RETURNS SET OF (seq, path_seq, start_vid, end_vid, node, edge, cost, agg_cost)
OR EMPTY SET

Example

Using a combinations table 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',
    'SELECT * FROM ( VALUES (2, 3), (11, 5) ) AS t(source, target)',
    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 |        11 |       5 |   11 |   12 |    0 |        0
   4 |        2 |        11 |       5 |   10 |   10 |    1 |        0
   5 |        3 |        11 |       5 |    5 |   -1 |    0 |        1
(5 rows)

Parameters

Parameter	Type	Default	Description
Edges SQL	TEXT		Edges query as described below.
Combinations SQL	TEXT		Combinations 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	When true Graph is considered Directed When false the graph is considered as Undirected.

Inner queries

Edges query

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) 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

ANY-NUMERICAL

SMALLINT, INTEGER, BIGINT, REAL, FLOAT

Combinations query

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.

Where:

ANY-INTEGER

SMALLINT, INTEGER, BIGINT

Return Columns

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. Many to One Many to Many
end_vid	BIGINT	Identifier of the ending vertex. Returned when multiple ending vertices are in the query. One to Many Many to Many
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.

Example Data

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.

Table Data

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

See Also

• https://cp-algorithms.com/graph/01_bfs.html

• https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Specialized_variants

Indices and tables

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