# pgRouting Manual (3.6)

## pgr_kruskalBFS

# `pgr_kruskalBFS`

`pgr_kruskalBFS` — Kruskal’s algorithm for Minimum Spanning Tree with breadth First Search ordering.

Boost Graph Inside

Availability

- Version 3.0.0

  - New Official function

## Description

Visits and extracts the nodes information in Breath First Search ordering of the Minimum Spanning Tree created using Kruskal’s algorithm.

The main Characteristics are:

- It’s implementation is only on undirected graph.

- Process is done only on edges with positive costs.

- When the graph is connected

  - The resulting edges make up a tree

- When the graph is not connected,

  - Finds a minimum spanning tree for each connected component.

  - The resulting edges make up a forest.

- The total weight of all the edges in the tree or forest is minimized.

- Kruskal’s running time: \(O(E * log E)\)

- Returned tree nodes from a root vertex are on Breath First Search order

- Breath First Search Running time: \(O(E + V)\)

## Signatures

pgr_kruskalBFS(Edges SQL, root vid, [`max_depth`])
pgr_kruskalBFS(Edges SQL, root vids, [`max_depth`])
Returns set of `(seq, depth, start_vid, node, edge, cost, agg_cost)`

### Single vertex

pgr_kruskalBFS(Edges SQL, root vid, [`max_depth`])
Returns set of `(seq, depth, start_vid, node, edge, cost, agg_cost)`

Example:

  The Minimum Spanning Tree having as root vertex \(6\)

    SELECT * FROM pgr_kruskalBFS(
      'SELECT id, source, target, cost, reverse_cost FROM edges ORDER BY id',
      6);
     seq | depth | start_vid | node | edge | cost | agg_cost
    -----+-------+-----------+------+------+------+----------
       1 |     0 |         6 |    6 |   -1 |    0 |        0
       2 |     1 |         6 |    5 |    1 |    1 |        1
       3 |     1 |         6 |   10 |    2 |    1 |        1
       4 |     2 |         6 |   15 |    3 |    1 |        2
       5 |     3 |         6 |   16 |   16 |    1 |        3
       6 |     4 |         6 |   17 |   15 |    1 |        4
       7 |     5 |         6 |   12 |   13 |    1 |        5
       8 |     6 |         6 |   11 |   11 |    1 |        6
       9 |     6 |         6 |    8 |   12 |    1 |        6
      10 |     7 |         6 |    7 |   10 |    1 |        7
      11 |     7 |         6 |    9 |   14 |    1 |        7
      12 |     8 |         6 |    3 |    7 |    1 |        8
      13 |     9 |         6 |    1 |    6 |    1 |        9
    (13 rows)

### Multiple vertices

pgr_kruskalBFS(Edges SQL, root vids, [`max_depth`])
Returns set of `(seq, depth, start_vid, node, edge, cost, agg_cost)`

Example:

  The Minimum Spanning Tree starting on vertices \(\{9, 6\}\) with \(depth \leq 3\)

    SELECT * FROM pgr_kruskalBFS(
      'SELECT id, source, target, cost, reverse_cost FROM edges ORDER BY id',
      ARRAY[9, 6], max_depth => 3);
     seq | depth | start_vid | node | edge | cost | agg_cost
    -----+-------+-----------+------+------+------+----------
       1 |     0 |         6 |    6 |   -1 |    0 |        0
       2 |     1 |         6 |    5 |    1 |    1 |        1
       3 |     1 |         6 |   10 |    2 |    1 |        1
       4 |     2 |         6 |   15 |    3 |    1 |        2
       5 |     3 |         6 |   16 |   16 |    1 |        3
       6 |     0 |         9 |    9 |   -1 |    0 |        0
       7 |     1 |         9 |    8 |   14 |    1 |        1
       8 |     2 |         9 |    7 |   10 |    1 |        2
       9 |     2 |         9 |   12 |   12 |    1 |        2
      10 |     3 |         9 |    3 |    7 |    1 |        3
      11 |     3 |         9 |   11 |   11 |    1 |        3
      12 |     3 |         9 |   17 |   13 |    1 |        3
    (12 rows)

## Parameters

Parameter | Type | Description
--- | --- | ---
Edges SQL | `TEXT` | Edges SQL as described below.

root vid

`BIGINT`

Identifier of the root vertex of the tree.

- When value is \(0\) then gets the spanning forest starting in aleatory nodes for each tree in the forest.

root vids

`ARRAY` [ ANY-INTEGER ]

Array of identifiers of the root vertices.

> - \(0\) values are ignored

> - For optimization purposes, any duplicated value is ignored.

Where:

ANY-INTEGER:

  SMALLINT, INTEGER, BIGINT

ANY-NUMERIC:

  SMALLINT, INTEGER, BIGINT, REAL, FLOAT, NUMERIC

### BFS optional parameters

Parameter | Type | Default | Description
--- | --- | --- | ---

`max_depth`

`BIGINT`

\(9223372036854775807\)

Upper limit of the depth of the tree.

- When negative throws an error.

## Inner Queries

### Edges SQL

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`)

- 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`

## Result columns

Returns set of `(seq, depth, start_vid, node, edge, cost, agg_cost)`

Parameter | Type | Description
--- | --- | ---
`seq` | `BIGINT` | Sequential value starting from \(1\).

`depth`

`BIGINT`

Depth of the `node`.

- \(0\) when `node` = `start_vid`.

`start_vid` | `BIGINT` | Identifier of the root vertex.
`node` | `BIGINT` | Identifier of `node` reached using `edge`.

`edge`

`BIGINT`

Identifier of the `edge` used to arrive to `node`.

- \(-1\) when `node` = `start_vid`.

`cost` | `FLOAT` | Cost to traverse `edge`.
`agg_cost` | `FLOAT` | Aggregate cost from `start_vid` to `node`.

Where:

ANY-INTEGER:

  SMALLINT, INTEGER, BIGINT

ANY-NUMERIC:

  SMALLINT, INTEGER, BIGINT, REAL, FLOAT, NUMERIC

## See Also

- Spanning Tree - Category

- Kruskal - Family of functions

- Sample Data

- Boost: Kruskal’s algorithm

- Wikipedia: Kruskal’s algorithm

Indices and tables

- Index

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