Contraction - Family of functions

Introduction

In large graphs, like the road graphs, or electric networks, graph contraction can be used to speed up some graph algorithms. Contraction reduces the size of the graph by removing some of the vertices and edges and, for example, might add edges that represent a sequence of original edges decreasing the total time and space used in graph algorithms.

This implementation gives a flexible framework for adding contraction algorithms in the future, currently, it supports two algorithms:

  1. Dead end contraction
  2. Linear contraction

Allowing the user to:

  • Forbid contraction on a set of nodes.
  • Decide the order of the contraction algorithms and set the maximum number of times they are to be executed.

Dead end contraction

In the algorithm, dead end contraction is represented by 1.

Dead end

In case of an undirected graph, a node is considered a dead end node when

In case of a directed graph, a node is considered a dead end node when

When the conditions are true then the Operation: Dead End Contraction can be done.

The number of adjacent vertices is 1.

  • The green nodes are dead end nodes
  • The blue nodes have an unlimited number of incoming and outgoing edges.

Directed graph

digraph G { u, v [shape=circle;style=filled;width=.4;color=deepskyblue]; w, z [style=filled; color=green]; G [shape=tripleoctagon;width=1.5;style=filled;color=deepskyblue;label = "Rest of the Graph"]; rankdir=LR; G -> {u, v} [dir=none, weight=1, penwidth=3]; u -> w -> u; v -> z; }

Undirected graph

graph G { u, v [shape=circle;style=filled;width=.4;color=deepskyblue]; w, z [style=filled; color=green]; G [shape=tripleoctagon;width=1.5;style=filled;color=deepskyblue;label = "Rest of the Graph"]; rankdir=LR; G -- {u, v} [dir=none, weight=1, penwidth=3]; u -- w [color=black]; u -- w [color=darkgray]; v -- z; }

There are no outgoing edges and has at least one incoming edge.

  • The green nodes are dead end nodes
  • The blue nodes have an unlimited number of incoming and outgoing edges.

Directed graph

digraph G { u, v [shape=circle;style=filled;width=.4;color=deepskyblue]; w, z [style=filled; color=green]; G [shape=tripleoctagon;width=1.5;style=filled;color=deepskyblue;label = "Rest of the Graph"]; rankdir=LR; G -> {u, v} [dir=none, weight=1, penwidth=3]; u -> w; v -> w; v -> z; }

There are no incoming edges and has at least one outgoing edge.

  • The green nodes are dead end nodes
  • The blue nodes have an unlimited number of incoming and outgoing edges.
  • Considering that the nodes are dead starts nodes

Directed graph

digraph G { u, v [shape=circle;style=filled;width=.4;color=deepskyblue]; w, z [style=filled; color=green]; G [shape=tripleoctagon;width=1.5;style=filled;color=deepskyblue;label = "Rest of the Graph"]; rankdir=LR; {u, v} -> G [dir=none, weight=1, penwidth=3]; w -> u; w -> v; z -> v; }

Operation: Dead End Contraction

The dead end contraction will stop until there are no more dead end nodes. For example from the following graph where w is the dead end node:

digraph G { u, v [shape=circle;style=filled;width=.4;color=deepskyblue]; w [style=filled; color=green]; "G" [shape=tripleoctagon;style=filled;color=deepskyblue; label = "Rest of the Graph"]; rankdir=LR; G -> u [dir=none, weight=1, penwidth=3]; u -> v -> w; }

After contracting w, node v is now a dead end node and is contracted:

digraph G { u [shape=circle;style=filled;width=.4;color=deepskyblue]; v [style=filled; color=green, label="v{w}"]; "G" [shape=tripleoctagon;style=filled;color=deepskyblue; label = "Rest of the Graph"]; rankdir=LR; G -> u [dir=none, weight=1, penwidth=3]; u -> v; }

After contracting v, stop. Node u has the information of nodes that were contrcted.

digraph G { u [style=filled; color=green, label="u{v,w}"]; "G" [shape=tripleoctagon;style=filled;color=deepskyblue; label = "Rest of the Graph"]; rankdir=LR; G -> u [dir=none, weight=1, penwidth=3]; }

Node u has the information of nodes that were contracted.

Linear contraction

In the algorithm, linear contraction is represented by 2.

Linear

In case of an undirected graph, a node is considered a linear node when

In case of a directed graph, a node is considered a linear node when

The number of adjacent vertices is 2.

  • The green nodes are linear nodes
  • The blue nodes have an unlimited number of incoming and outgoing edges.

Directed

digraph G { u, c, a, w [shape=circle;style=filled;width=.4;color=deepskyblue]; v, b [style=filled; color=green]; G [shape=tripleoctagon;width=1.5;style=filled;color=deepskyblue;label = "Rest of the Graph"]; rankdir=LR; {w, c} -> G -> {u, a} [dir=none, weight=1, penwidth=3]; u -> v -> w; a -> b -> c; c -> b -> a[color=darkgray]; }

Undirected

graph G { u, w [shape=circle;style=filled;width=.4;color=deepskyblue]; v [style=filled; color=green]; G [shape=tripleoctagon;width=1.5;style=filled;color=deepskyblue;label = "Rest of the Graph"]; rankdir=LR; w -- G -- u [dir=none, weight=1, penwidth=3]; u -- v -- w; }

Linearity is symmetrical

Using a contra example, vertex v is not linear because it’s not possible to go from w to u via v.

digraph G { u, w, v [shape=circle;style=filled;width=.4;color=deepskyblue]; G [shape=tripleoctagon;width=1.5;style=filled;color=deepskyblue;label = "Rest of the Graph"]; rankdir=LR; {w} -> G -> {u} [dir=none, weight=1, penwidth=3]; u -> v -> w -> v; }

Operation: Linear Contraction

The linear contraction will stop until there are no more linear nodes. For example from the following graph where v and w are linear nodes:

digraph G { u, z [shape=circle;style=filled;color=deepskyblue]; v, w [style=filled; color=green]; "G" [shape=tripleoctagon; style=filled;color=deepskyblue;label = "Rest of the Graph"]; rankdir=LR; G -> {u, z} [dir=none, weight=1, penwidth=3]; u -> v -> w -> z; }

After contracting w,

  • The vertex w is removed from the graph
    • The edges \(v \rightarrow w\) and \(w \rightarrow z\) are removed from the graph.
  • A new edge \(v \rightarrow z\) is inserted represented with red color.
digraph G { u, z [shape=circle;style=filled;color=deepskyblue]; v [style=filled; color=green]; "G" [shape=tripleoctagon; style=filled;color=deepskyblue;label = "Rest of the Graph"]; rankdir=LR; G -> {u, z} [dir=none, weight=1, penwidth=3]; u -> v; v -> z [label="{w}";color=red] }

Contracting v:

  • The vertex v is removed from the graph
    • The edges \(u \rightarrow v\) and \(v \rightarrow z\) are removed from the graph.
  • A new edge \(u \rightarrow z\) is inserted represented with red color.
digraph G { u, z [shape=circle;style=filled;color=deepskyblue]; "G" [shape=tripleoctagon; style=filled;color=deepskyblue;label = "Rest of the Graph"]; rankdir=LR; G -> {u, z} [dir=none, weight=1, penwidth=3]; u -> z [label="{v, w}";color=red] }

Edge \(u \rightarrow z\) has the information of nodes that were contracted.

The cycle

Contracting a graph, can be done with more than one operation. The order of the operations affect the resulting contracted graph, after applying one operation, the set of vertices that can be contracted by another operation changes.

This implementation, cycles max_cycles times through operations_order .

<input>
do max_cycles times {
    for (operation in operations_order)
     { do operation }
}
<output>

Contracting Sample Data

In this section, building and using a contracted graph will be shown by example.

  • The Sample Data for an undirected graph is used
  • a dead end operation first followed by a linear operation.

Construction of the graph in the database

Original Data

The following query shows the original data involved in the contraction operation.

SELECT id, source, target, cost, reverse_cost FROM edge_table;
 id | source | target | cost | reverse_cost
----+--------+--------+------+--------------
  1 |      1 |      2 |    1 |            1
  2 |      2 |      3 |   -1 |            1
  3 |      3 |      4 |   -1 |            1
  4 |      2 |      5 |    1 |            1
  5 |      3 |      6 |    1 |           -1
  6 |      7 |      8 |    1 |            1
  7 |      8 |      5 |    1 |            1
  8 |      5 |      6 |    1 |            1
  9 |      6 |      9 |    1 |            1
 10 |      5 |     10 |    1 |            1
 11 |      6 |     11 |    1 |           -1
 12 |     10 |     11 |    1 |           -1
 13 |     11 |     12 |    1 |           -1
 14 |     10 |     13 |    1 |            1
 15 |      9 |     12 |    1 |            1
 16 |      4 |      9 |    1 |            1
 17 |     14 |     15 |    1 |            1
 18 |     16 |     17 |    1 |            1
(18 rows)

The original graph:

_images/undirected_sampledata_a.png

Contraction Results

The results do not represent the contracted graph. They represent the changes done to the graph after applying the contraction algorithm.

Observe that vertices, for example, \(6\) do not appear in the results because it was not affected by the contraction algorithm.

SELECT * FROM pgr_contraction(
    'SELECT id, source, target, cost, reverse_cost FROM edge_table',
    array[1,2], directed:=false);
 type | id | contracted_vertices | source | target | cost
------+----+---------------------+--------+--------+------
 v    |  5 | {7,8}               |     -1 |     -1 |   -1
 v    | 15 | {14}                |     -1 |     -1 |   -1
 v    | 17 | {16}                |     -1 |     -1 |   -1
 e    | -1 | {1,2}               |      3 |      5 |    2
 e    | -2 | {4}                 |      3 |      9 |    2
 e    | -3 | {10,13}             |      5 |     11 |    2
 e    | -4 | {12}                |      9 |     11 |    2
(7 rows)

After doing the dead end contraction operation:

_images/undirected_sampledata_b.png

After doing the linear contraction operation to the graph above:

_images/undirected_sampledata_c.png

The process to create the contraction graph on the database:

Add additional columns

Adding extra columns to the edge_table and edge_table_vertices_pgr tables, where:

Column Description
contracted_vertices The vertices set belonging to the vertex/edge
is_contracted

On the vertex table

  • when true the vertex is contracted, its not part of the contracted graph.
  • when false the vertex is not contracted, its part of the contracted graph.
is_new

On the edge table:

  • when true the edge was generated by the contraction algorithm. its part of the contracted graph.
  • when false the edge is an original edge, might be or not part of the contracted graph.
 
ALTER TABLE edge_table_vertices_pgr ADD is_contracted BOOLEAN DEFAULT false;
ALTER TABLE
ALTER TABLE edge_table_vertices_pgr ADD contracted_vertices BIGINT[];
ALTER TABLE
ALTER TABLE edge_table ADD is_new BOOLEAN DEFAULT false;
ALTER TABLE
ALTER TABLE edge_table ADD contracted_vertices BIGINT[];
ALTER TABLE

Store contraction information

Store the contraction results in a table

SELECT * INTO contraction_results
FROM pgr_contraction(
    'SELECT id, source, target, cost, reverse_cost FROM edge_table',
    array[1,2], directed:=false);
SELECT 7

Update the vertices and edge tables

Update the vertex table using the contraction information

Use edge_table_vertices_pgr.is_contracted to indicate the vertices that are contracted.

UPDATE edge_table_vertices_pgr
SET is_contracted = true
WHERE id IN (SELECT  unnest(contracted_vertices) FROM  contraction_results);
UPDATE 10

Add to edge_table_vertices_pgr.contracted_vertices the contracted vertices belonging to the vertices.

UPDATE edge_table_vertices_pgr
SET contracted_vertices = contraction_results.contracted_vertices
FROM contraction_results
WHERE type = 'v' AND edge_table_vertices_pgr.id = contraction_results.id;
UPDATE 3

The modified edge_table_vertices_pgr.

SELECT id, contracted_vertices, is_contracted
FROM edge_table_vertices_pgr
ORDER BY id;
 id | contracted_vertices | is_contracted
----+---------------------+---------------
  1 |                     | t
  2 |                     | t
  3 |                     | f
  4 |                     | t
  5 | {7,8}               | f
  6 |                     | f
  7 |                     | t
  8 |                     | t
  9 |                     | f
 10 |                     | t
 11 |                     | f
 12 |                     | t
 13 |                     | t
 14 |                     | t
 15 | {14}                | f
 16 |                     | t
 17 | {16}                | f
(17 rows)

Update the edge table using the contraction information

Insert the new edges generated by pgr_contraction.

INSERT INTO edge_table(source, target, cost, reverse_cost, contracted_vertices, is_new)
SELECT source, target, cost, -1, contracted_vertices, true
FROM contraction_results
WHERE type = 'e';
INSERT 0 4

The modified edge_table.

SELECT id, source, target, cost, reverse_cost, contracted_vertices, is_new
FROM edge_table
ORDER BY id;
 id | source | target | cost | reverse_cost | contracted_vertices | is_new
----+--------+--------+------+--------------+---------------------+--------
  1 |      1 |      2 |    1 |            1 |                     | f
  2 |      2 |      3 |   -1 |            1 |                     | f
  3 |      3 |      4 |   -1 |            1 |                     | f
  4 |      2 |      5 |    1 |            1 |                     | f
  5 |      3 |      6 |    1 |           -1 |                     | f
  6 |      7 |      8 |    1 |            1 |                     | f
  7 |      8 |      5 |    1 |            1 |                     | f
  8 |      5 |      6 |    1 |            1 |                     | f
  9 |      6 |      9 |    1 |            1 |                     | f
 10 |      5 |     10 |    1 |            1 |                     | f
 11 |      6 |     11 |    1 |           -1 |                     | f
 12 |     10 |     11 |    1 |           -1 |                     | f
 13 |     11 |     12 |    1 |           -1 |                     | f
 14 |     10 |     13 |    1 |            1 |                     | f
 15 |      9 |     12 |    1 |            1 |                     | f
 16 |      4 |      9 |    1 |            1 |                     | f
 17 |     14 |     15 |    1 |            1 |                     | f
 18 |     16 |     17 |    1 |            1 |                     | f
 19 |      3 |      5 |    2 |           -1 | {1,2}               | t
 20 |      3 |      9 |    2 |           -1 | {4}                 | t
 21 |      5 |     11 |    2 |           -1 | {10,13}             | t
 22 |      9 |     11 |    2 |           -1 | {12}                | t
(22 rows)

The contracted graph

Vertices that belong to the contracted graph.

SELECT id
FROM edge_table_vertices_pgr
WHERE is_contracted = false
ORDER BY id;
 id
----
  3
  5
  6
  9
 11
 15
 17
(7 rows)

Edges that belong to the contracted graph.

WITH
vertices_in_graph AS (
    SELECT id
    FROM edge_table_vertices_pgr
    WHERE is_contracted = false
)
SELECT id, source, target, cost, reverse_cost, contracted_vertices
FROM edge_table
WHERE source IN (SELECT * FROM vertices_in_graph)
AND target IN (SELECT * FROM vertices_in_graph)
ORDER BY id;
 id | source | target | cost | reverse_cost | contracted_vertices
----+--------+--------+------+--------------+---------------------
  5 |      3 |      6 |    1 |           -1 |
  8 |      5 |      6 |    1 |            1 |
  9 |      6 |      9 |    1 |            1 |
 11 |      6 |     11 |    1 |           -1 |
 19 |      3 |      5 |    2 |           -1 | {1,2}
 20 |      3 |      9 |    2 |           -1 | {4}
 21 |      5 |     11 |    2 |           -1 | {10,13}
 22 |      9 |     11 |    2 |           -1 | {12}
(8 rows)
_images/undirected_sampledata_c.png

Using the contracted graph

Using the contracted graph with pgr_dijkstra

There are three cases when calculating the shortest path between a given source and target in a contracted graph:

  • Case 1: Both source and target belong to the contracted graph.
  • Case 2: Source and/or target belong to an edge subgraph.
  • Case 3: Source and/or target belong to a vertex.

Case 1: Both source and target belong to the contracted graph.

Using the Edges that belong to the contracted graph. on lines 10 to 19.

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CREATE OR REPLACE FUNCTION my_dijkstra(
    departure BIGINT, destination BIGINT,
    OUT seq INTEGER, OUT path_seq INTEGER,
    OUT node BIGINT, OUT edge BIGINT,
    OUT cost FLOAT, OUT agg_cost FLOAT)
RETURNS SETOF RECORD AS
$BODY$
SELECT * FROM pgr_dijkstra(
    $$
    WITH
    vertices_in_graph AS (
        SELECT id
        FROM edge_table_vertices_pgr
        WHERE is_contracted = false
    )
    SELECT id, source, target, cost, reverse_cost
    FROM edge_table
    WHERE source IN (SELECT * FROM vertices_in_graph)
    AND target IN (SELECT * FROM vertices_in_graph)
    $$,
    departure, destination, false);
$BODY$
LANGUAGE SQL VOLATILE;
CREATE FUNCTION

Case 1

When both source and target belong to the contracted graph, a path is found.

SELECT * FROM my_dijkstra(3, 11);
 seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
   1 |        1 |    3 |    5 |    1 |        0
   2 |        2 |    6 |   11 |    1 |        1
   3 |        3 |   11 |   -1 |    0 |        2
(3 rows)

Case 2

When source and/or target belong to an edge subgraph then a path is not found.

In this case, the contracted graph do not have an edge connecting with node \(4\).

SELECT * FROM my_dijkstra(4, 11);
 seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
(0 rows)

Case 3

When source and/or target belong to a vertex then a path is not found.

In this case, the contracted graph do not have an edge connecting with node \(7\) and of node \(4\) of the second case.

SELECT * FROM my_dijkstra(4, 7);
 seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
(0 rows)

Case 2: Source and/or target belong to an edge subgraph.

Refining the above function to include nodes that belong to an edge.

  • The vertices that need to be expanded are calculated on lines 10 to 16.
  • Adding to the contracted graph that additional section on lines 25 to 27.
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CREATE OR REPLACE FUNCTION my_dijkstra(
    departure BIGINT, destination BIGINT,
    OUT seq INTEGER, OUT path_seq INTEGER,
    OUT node BIGINT, OUT edge BIGINT,
    OUT cost FLOAT, OUT agg_cost FLOAT)
RETURNS SETOF RECORD AS
$BODY$
SELECT * FROM pgr_dijkstra(
    $$
    WITH
    edges_to_expand AS (
        SELECT id
        FROM edge_table
        WHERE ARRAY[$$ || departure || $$]::BIGINT[] <@ contracted_vertices
           OR ARRAY[$$ || destination || $$]::BIGINT[] <@ contracted_vertices
    ),

    vertices_in_graph AS (
        SELECT id
        FROM edge_table_vertices_pgr
        WHERE is_contracted = false

        UNION

        SELECT unnest(contracted_vertices)
        FROM edge_table
        WHERE id IN (SELECT id FROM edges_to_expand)
    )

    SELECT id, source, target, cost, reverse_cost
    FROM edge_table
    WHERE source IN (SELECT * FROM vertices_in_graph)
    AND target IN (SELECT * FROM vertices_in_graph)
    $$,
    departure, destination, false);
$BODY$
LANGUAGE SQL VOLATILE;
CREATE FUNCTION

Case 1

When both source and target belong to the contracted graph, a path is found.

SELECT * FROM my_dijkstra(3, 11);
 seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
   1 |        1 |    3 |    5 |    1 |        0
   2 |        2 |    6 |   11 |    1 |        1
   3 |        3 |   11 |   -1 |    0 |        2
(3 rows)

Case 2

When source and/or target belong to an edge subgraph, now, a path is found.

The routing graph now has an edge connecting with node \(4\).

SELECT * FROM my_dijkstra(4, 11);
 seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
   1 |        1 |    4 |   16 |    1 |        0
   2 |        2 |    9 |   22 |    2 |        1
   3 |        3 |   11 |   -1 |    0 |        3
(3 rows)

Case 3

When source and/or target belong to a vertex then a path is not found.

In this case, the contracted graph do not have an edge connecting with node \(7\).

SELECT * FROM my_dijkstra(4, 7);
 seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
(0 rows)

Case 3: Source and/or target belong to a vertex.

Refining the above function to include nodes that belong to an edge.

  • The vertices that need to be expanded are calculated on lines 18 to 23.
  • Adding to the contracted graph that additional section on lines 38 to 40.
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CREATE OR REPLACE FUNCTION my_dijkstra(
    departure BIGINT, destination BIGINT,
    OUT seq INTEGER, OUT path_seq INTEGER,
    OUT node BIGINT, OUT edge BIGINT,
    OUT cost FLOAT, OUT agg_cost FLOAT)
RETURNS SETOF RECORD AS
$BODY$
SELECT * FROM pgr_dijkstra(
    $$
    WITH
    edges_to_expand AS (
        SELECT id
        FROM edge_table
        WHERE ARRAY[$$ || departure || $$]::BIGINT[] <@ contracted_vertices
           OR ARRAY[$$ || destination || $$]::BIGINT[] <@ contracted_vertices
    ),

    vertices_to_expand AS (
        SELECT id
        FROM edge_table_vertices_pgr
        WHERE ARRAY[$$ || departure || $$]::BIGINT[] <@ contracted_vertices
           OR ARRAY[$$ || destination || $$]::BIGINT[] <@ contracted_vertices
    ),

    vertices_in_graph AS (
        SELECT id
        FROM edge_table_vertices_pgr
        WHERE is_contracted = false

        UNION

        SELECT unnest(contracted_vertices)
        FROM edge_table
        WHERE id IN (SELECT id FROM edges_to_expand)

        UNION

        SELECT unnest(contracted_vertices)
        FROM edge_table_vertices_pgr
        WHERE id IN (SELECT id FROM vertices_to_expand)
    )

    SELECT id, source, target, cost, reverse_cost
    FROM edge_table
    WHERE source IN (SELECT * FROM vertices_in_graph)
    AND target IN (SELECT * FROM vertices_in_graph)
    $$,
    departure, destination, false);
$BODY$
LANGUAGE SQL VOLATILE;
CREATE FUNCTION

Case 1

When both source and target belong to the contracted graph, a path is found.

SELECT * FROM my_dijkstra(3, 11);
 seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
   1 |        1 |    3 |    5 |    1 |        0
   2 |        2 |    6 |   11 |    1 |        1
   3 |        3 |   11 |   -1 |    0 |        2
(3 rows)

Case 2

The code change do not affect this case so when source and/or target belong to an edge subgraph, a path is still found.

SELECT * FROM my_dijkstra(4, 11);
 seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
   1 |        1 |    4 |   16 |    1 |        0
   2 |        2 |    9 |   22 |    2 |        1
   3 |        3 |   11 |   -1 |    0 |        3
(3 rows)

Case 3

When source and/or target belong to a vertex, now, a path is found.

Now, the routing graph has an edge connecting with node \(7\).

SELECT * FROM my_dijkstra(4, 7);
 seq | path_seq | node | edge | cost | agg_cost
-----+----------+------+------+------+----------
   1 |        1 |    4 |    3 |    1 |        0
   2 |        2 |    3 |   19 |    2 |        1
   3 |        3 |    5 |    7 |    1 |        3
   4 |        4 |    8 |    6 |    1 |        4
   5 |        5 |    7 |   -1 |    0 |        5
(5 rows)