distance_table
Shortest (directed or undirected) paths between vertices
distances
calculates the length of all the shortest paths from
or to the vertices in the network. shortest_paths
calculates one
shortest path (the path itself, and not just its length) from or to the
given vertex.
- Keywords
- graphs
Usage
distance_table(graph, directed = TRUE)mean_distance(graph, directed = TRUE, unconnected = TRUE)
distances(
graph,
v = V(graph),
to = V(graph),
mode = c("all", "out", "in"),
weights = NULL,
algorithm = c("automatic", "unweighted", "dijkstra", "bellman-ford", "johnson")
)
shortest_paths(
graph,
from,
to = V(graph),
mode = c("out", "all", "in"),
weights = NULL,
output = c("vpath", "epath", "both"),
predecessors = FALSE,
inbound.edges = FALSE
)
all_shortest_paths(
graph,
from,
to = V(graph),
mode = c("out", "all", "in"),
weights = NULL
)
Arguments
- graph
The graph to work on.
- directed
Whether to consider directed paths in directed graphs, this argument is ignored for undirected graphs.
- unconnected
What to do if the graph is unconnected (not strongly connected if directed paths are considered). If TRUE only the lengths of the existing paths are considered and averaged; if FALSE the length of the missing paths are counted having length
vcount(graph)
, one longer than the longest possible geodesic in the network.- v
Numeric vector, the vertices from which the shortest paths will be calculated.
- to
Numeric vector, the vertices to which the shortest paths will be calculated. By default it includes all vertices. Note that for
distances
every vertex must be included here at most once. (This is not required forshortest_paths
.- mode
Character constant, gives whether the shortest paths to or from the given vertices should be calculated for directed graphs. If
out
then the shortest paths from the vertex, ifin
then to it will be considered. Ifall
, the default, then the corresponding undirected graph will be used, ie. not directed paths are searched. This argument is ignored for undirected graphs.- weights
Possibly a numeric vector giving edge weights. If this is
NULL
and the graph has aweight
edge attribute, then the attribute is used. If this isNA
then no weights are used (even if the graph has aweight
attribute).- algorithm
Which algorithm to use for the calculation. By default igraph tries to select the fastest suitable algorithm. If there are no weights, then an unweighted breadth-first search is used, otherwise if all weights are positive, then Dijkstra's algorithm is used. If there are negative weights and we do the calculation for more than 100 sources, then Johnson's algorithm is used. Otherwise the Bellman-Ford algorithm is used. You can override igraph's choice by explicitly giving this parameter. Note that the igraph C core might still override your choice in obvious cases, i.e. if there are no edge weights, then the unweighted algorithm will be used, regardless of this argument.
- from
Numeric constant, the vertex from or to the shortest paths will be calculated. Note that right now this is not a vector of vertex ids, but only a single vertex.
- output
Character scalar, defines how to report the shortest paths. “vpath” means that the vertices along the paths are reported, this form was used prior to igraph version 0.6. “epath” means that the edges along the paths are reported. “both” means that both forms are returned, in a named list with components “vpath” and “epath”.
- predecessors
Logical scalar, whether to return the predecessor vertex for each vertex. The predecessor of vertex
i
in the tree is the vertex from which vertexi
was reached. The predecessor of the start vertex (in thefrom
argument) is itself by definition. If the predecessor is zero, it means that the given vertex was not reached from the source during the search. Note that the search terminates if all the vertices into
are reached.- inbound.edges
Logical scalar, whether to return the inbound edge for each vertex. The inbound edge of vertex
i
in the tree is the edge via which vertexi
was reached. The start vertex and vertices that were not reached during the search will have zero in the corresponding entry of the vector. Note that the search terminates if all the vertices into
are reached.
Details
The shortest path, or geodesic between two pair of vertices is a path with the minimal number of vertices. The functions documented in this manual page all calculate shortest paths between vertex pairs.
distances
calculates the lengths of pairwise shortest paths from
a set of vertices (from
) to another set of vertices (to
). It
uses different algorithms, depending on the algorithm
argument and
the weight
edge attribute of the graph. The implemented algorithms
are breadth-first search (‘unweighted
’), this only works for
unweighted graphs; the Dijkstra algorithm (‘dijkstra
’), this
works for graphs with non-negative edge weights; the Bellman-Ford algorithm
(‘bellman-ford
’), and Johnson's algorithm
(‘"johnson"
’). The latter two algorithms work with arbitrary
edge weights, but (naturally) only for graphs that don't have a negative
cycle.
igraph can choose automatically between algorithms, and chooses the most
efficient one that is appropriate for the supplied weights (if any). For
automatic algorithm selection, supply ‘automatic
’ as the
algorithm
argument. (This is also the default.)
shortest_paths
calculates a single shortest path (i.e. the path
itself, not just its length) between the source vertex given in from
,
to the target vertices given in to
. shortest_paths
uses
breadth-first search for unweighted graphs and Dijkstra's algorithm for
weighted graphs. The latter only works if the edge weights are non-negative.
all_shortest_paths
calculates all shortest paths between
pairs of vertices. More precisely, between the from
vertex to the
vertices given in to
. It uses a breadth-first search for unweighted
graphs and Dijkstra's algorithm for weighted ones. The latter only supports
non-negative edge weights.
mean_distance
calculates the average path length in a graph, by
calculating the shortest paths between all pairs of vertices (both ways for
directed graphs). This function does not consider edge weights currently and
uses a breadth-first search.
distance_table
calculates a histogram, by calculating the shortest
path length between each pair of vertices. For directed graphs both
directions are considered, so every pair of vertices appears twice in the
histogram.
Value
For distances
a numeric matrix with length(to)
columns and length(v)
rows. The shortest path length from a vertex to
itself is always zero. For unreachable vertices Inf
is included.
For shortest_paths
a named list with four entries is returned:
This itself is a list, of length length(to)
; list
element i
contains the vertex ids on the path from vertex from
to vertex to[i]
(or the other way for directed graphs depending on
the mode
argument). The vector also contains from
and i
as the first and last elements. If from
is the same as i
then
it is only included once. If there is no path between two vertices then a
numeric vector of length zero is returned as the list element. If this
output is not requested in the output
argument, then it will be
NULL
.
This is a list similar to vpath
, but the
vectors of the list contain the edge ids along the shortest paths, instead
of the vertex ids. This entry is set to NULL
if it is not requested
in the output
argument.
Numeric vector, the
predecessor of each vertex in the to
argument, or NULL
if it
was not requested.
Numeric vector, the inbound edge
for each vertex, or NULL
, if it was not requested.
For all_shortest_paths a list is returned, each list element contains a shortest path from from to a vertex in to. The shortest paths to the same vertex are collected into consecutive elements of the list.
For mean_distance a single number is returned.
distance_table returns a named list with two entries: res is a numeric vector, the histogram of distances, unconnected is a numeric scalar, the number of pairs for which the first vertex is not reachable from the second. The sum of the two entries is always n(n-1) for directed graphs and n(n-1)/2 for undirected graphs.
References
West, D.B. (1996). Introduction to Graph Theory. Upper Saddle River, N.J.: Prentice Hall.
Examples
# NOT RUN {
g <- make_ring(10)
distances(g)
shortest_paths(g, 5)
all_shortest_paths(g, 1, 6:8)
mean_distance(g)
## Weighted shortest paths
el <- matrix(nc=3, byrow=TRUE,
c(1,2,0, 1,3,2, 1,4,1, 2,3,0, 2,5,5, 2,6,2, 3,2,1, 3,4,1,
3,7,1, 4,3,0, 4,7,2, 5,6,2, 5,8,8, 6,3,2, 6,7,1, 6,9,1,
6,10,3, 8,6,1, 8,9,1, 9,10,4) )
g2 <- add_edges(make_empty_graph(10), t(el[,1:2]), weight=el[,3])
distances(g2, mode="out")
# }