# pgr_kruskalDFS¶

pgr_kruskalDFS — Kruskal’s algorithm for Minimum Spanning Tree with Depth First Search ordering.

Availability

Version 3.7.0:

• Standarizing output columns to (seq, depth, start_vid, pred, node, edge, cost, agg_cost)

• Added pred result columns.

Version 3.0.0:
• New Official function

## Description¶

Visits and extracts the nodes information in Depth 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 Depth First Search order

• Depth First Search Running time: $$O(E + V)$$

## Signatures¶

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

### Single vertex¶

pgr_kruskalDFS(Edges SQL, root vid, [max_depth])
Returns set of (seq, depth, start_vid, pred, node, edge, cost, agg_cost)
Example:

The Minimum Spanning Tree having as root vertex $$6$$

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



### Multiple vertices¶

pgr_kruskalDFS(Edges SQL, root vids, [max_depth])
Returns set of (seq, depth, start_vid, pred, node, edge, cost, agg_cost)
Example:

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

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

Root vids

ARRAY[ANY-INTEGER]

Array of identifiers of the root vertices.

• $$0$$ values are ignored

• For optimization purposes, any duplicated value is ignored.

distance

FLOAT

Upper limit for the inclusion of a node in the result.

Where:

ANY-NUMERIC:

SMALLINT, INTEGER, BIGINT, REAL, FLOAT

### DFS 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, pred, 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.

• $$depth-1$$ is the depth of pred

start_vid

BIGINT

Identifier of the root vertex.

pred

BIGINT

Predecessor of node.

• When node = start_vid then has the value node.

node

BIGINT

Identifier of node reached using edge.

edge

BIGINT

Identifier of the edge used to arrive from pred to node.

• $$-1$$ when node = start_vid.

cost

FLOAT

Cost to traverse edge.

agg_cost

FLOAT

Aggregate cost from start_vid to node.