Contents

Graph::breadthFirstSearch

Makes a breadth first Search in a graph.

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

Graph::breadthFirstSearch(G, <StartVertex = v>)

Description

Graph::breadthFirstSearch traverses through a graph via breadth first search. The output shows the first time of identification and the predecessor of each vertex. If a vertex is a single vertex with no predecessor its predecessor is infinity.

Graph::breadthFirstSearch(G, StartVertex = v) traverses through a graph via breadth first search starting from vertex v. The output shows the first time of identification and the predecessor of each vertex. If a vertex is a single vertex with no predecessor its predecessor is infinity.

Examples

Example 1

A typical tree is created and drawn for a better understanding of the algorithm:

G := Graph([a, b, c, d, e, f, g, h, i, j, k, l],
           [[a, b], [a, c], [b, d], [b, e], [c, f], [c, g],
            [d, h], [e, i], [e, j], [f, k], [g, l]],
           Directed):
plot(
  Graph::plotGridGraph(G, VerticesPerLine = [12, 12, 12, 12], 
    VertexOrder = [
  None, None, None, None, None, None,
  a,    None, None, None, None, None,
  None, None, b,    None, None, None,
  None, None, None, c,    None, None,
  None, d,    None, None, e,    None,
  None, f,    None, None, g,    None,
  h,    None, None, i,    None, j,   
  None, None, k,    None, None, l
    ]
  )
)

Now we call breadthFirstSearch to find out the starting times and predecessors

Graph::breadthFirstSearch(G)

Vertex a is dicovered first, then vertex b and so on. The right table shows the predecessor of every vertex. The backtracking from a single vertex is therefore really simple. a as the first vertex discovered in its component can not be backtracked any further. The distance of each vertex in its component can be read in the middle table. Root-vertices always have the value 0 (they are the roots).

Example 2

What happens now, if there exist a vertex that has no connection to any other vertex. The upper example is taken and a single vertex is added without changing anything else. Then a breadth first search is invoked on the graph:

G := Graph([a, b, c, d, e, f, g, h, i, j, k, l],
           [[a, b], [a, c], [b, d], [b, e], [c, f], [c, g],
            [d, h], [e, i], [e, j], [f, k], [g, l]],
           Directed):
G2 := Graph::addVertices(G, [m]):
Graph::breadthFirstSearch(G2, StartVertex = [a])

The newly inserted vertex m has no predecessor. The predecessor therefore holds the value infinity.

Example 3

If we start somewhere in the graph without knowing the root of the DAG, the results are of course different:

G := Graph([a, b, c, d, e, f, g, h, i, j, k, l],
           [[a, b], [a, c], [b, d], [b, e], [c, f], [c, g],
            [d, h], [e, i], [e, j], [f, k], [g, l]],
           Directed):
Graph::breadthFirstSearch(G, StartVertex = [c])

The predecessor of c is c, but if we look at the graph it should be a. This is nevertheless not quite correct. Breadth first search takes the given vertex and uses this as the root of the graph (no in-vertices!). This explains also why the next call shows a infinity as predecessor to l.

Parameters

G

Graph

v

List containing one vertex.

Options

StartVertex

Defines a vertex from which to start the breadth first traversal.

Return Values

List containing three tables. The first table holds the timestamp of the discovery. The second the distance to the root-vertex. The last table holds the predecessor vertices.

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