dodgr
Mark Padgham
2017-09-27
1 Background: Directed Graphs
dodgr is an R package for calculating Distances On Directed Graphs. It does so very efficiently, and is able to process larger graphs than many other comparable R packages. Skip straight to the Intro if you know what directed graphs are (but maybe make a brief stop-in to Dual-Weighted Directed Graphs below.) Directed graphs are ones in which the “distance” (or some equivalent measure) from A to B is not necessarily equal to that from B to A. In Fig. 1, for example, the weights between the graph vertices (A, B, C, and D) differ depending on the direction of travel, and it is only possible to traverse the entire graph in an anti-clockwise direction.
Figure 1: A weighted, directed graph
Graphs in dodgr are represented by simple flat data.frame objects, so the graph of Fig. 1, presuming the edge weights to take values of 1, 2, and 3, would be,
## from to d
## 1 A B 1
## 2 B A 2
## 3 B C 1
## 4 B D 3
## 5 C B 2
## 6 C D 1
## 7 D C 2
## 8 D A 1The primary function of dodgr is dodgr_dists, which calculates pair-wise shortest distances between all vertices of a graph.
dodgr_dists (graph)## A B C D
## A 0 1 2 3
## B 2 0 1 2
## C 2 2 0 1
## D 1 2 2 0dodgr_dists (graph, from = c ("A", "C"), to = c ("B", "C", "D"))## B C D
## A 1 2 3
## C 2 0 11.1 Dual-Weighted Directed Graphs
Shortest-path distances on weighted graphs can be calculated using a number of other R packages, such as igraph or e1071. dodgr comes into its own through its ability to trace paths through dual-weighted directed graphs, illustrated in Fig. 2.
Figure 2: A dual-weighted, directed graph in which the weights (represented by thicknesses) of the grey arrows differ from those of the black arrows.
Dual-weighted directed graphs are common in many areas, a foremost example being routing through street networks. Routes through street networks depends on mode of transport: the route a pedestrian might take will generally differ markedly from the route the same person might take if behind the wheel of an automobile. Routing through street networks thus generally requires each edge to be specified with two weights or distances: one quantifying the physical distance, and a second weighted version reflecting the mode of transport (or some other preferential weighting).
dodgr calculates shortest paths using one set of weights (called “weights” or anything else starting with “w”), but returns the actual lengths of them using a second set of weights (called “distances”, or anything else starting with “d”). If no weights are specified, distances alone are used both for routing and final distance calculations. Consider that the weights and distances of Fig. 2 are the black and grey lines, respectively, with the latter all equal to one. In this case, the graph and associated shortest distances are,
## from to w d
## 1 A B 1 1
## 2 B A 2 1
## 3 B C 1 1
## 4 B D 3 1
## 5 C B 2 1
## 6 C D 1 1
## 7 D C 2 1
## 8 D A 1 1## A B C D
## A 0 1 2 3
## B 1 0 1 2
## C 2 1 0 1
## D 1 2 1 0Note that even though the shortest “distance” from A to D is actually A\(\to\)B\(\to\)D with a distance of only 2, that path has a weighted distance of 1 + 3 = 4. The shortest weighted path is A\(\to\)B\(\to\)C\(\to\)D, with a distance both weighted and unweighted of 1 + 1 + 1 = 3. Thus d(A,D) = 3 and not 2.
2 Introduction to dodgr
Although the package has been intentionally developed to be adaptable to any kinds of networks, most of the applications illustrated here concern street networks, and also illustrate several helper functions the package offers for working with street networks. The basic graph object of dodgr is nevertheless arbitrary, and need only minimally contain three or four columns as demonstrated in the simple examples at the outset.
The package may be used to calculate a matrix of distances between a given set of geographic coordinates. We can start by simply generating some random coordinates, in this case within the bounding box defining the city of York in the U.K.
bb <- osmdata::getbb ("york uk")
npts <- 1000
xy <- apply (bb, 1, function (i) min (i) + runif (npts) * diff (i))
bb; head (xy)## min max
## x -1.241536 -0.9215361
## y 53.799056 54.1190555## x y
## [1,] -1.1713502 53.89409
## [2,] -1.2216108 54.01065
## [3,] -1.0457199 53.83613
## [4,] -0.9384666 53.93545
## [5,] -0.9445541 53.89436
## [6,] -1.1207099 54.01262Those points can simply be passed to dodgr_dists():
system.time(
d <- dodgr_dists (from = xy, wt_profile = "foot", quiet = FALSE)
)## No graph submitted to dodgr_dists; downloading street network ... done
## Converting network to dodgr graph ... done
## Calculating shortest paths ... done## user system elapsed
## 26.620 0.132 28.567dim (d); range (d, na.rm = TRUE)## [1] 1000 1000## [1] 0.00000 46.60609The result is a matrix of 1000-by-1000 distances of up to 47km long, measured along routes weighted for optimal pedestrian travel. In this case, the single call to dodgr_dists() automatically downloaded the entire street network of York and calculated one million shortest-path distances, all in under 30 seconds.
3 Graphs and Street Networks
Although the above code is short and fast, most users will probably want more control over their graphs and routing possibilities. Each of the steps indicated above (through the quiet = FALSE option) can be implemented separately. To illustrate, the remainder of this vignette analyses the much smaller street network of Hampi, Karnataka, India, included in the dodgr package as the dataset hampi. This data set may be re-created with the following single line:
hampi <- dodgr_streetnet ("hampi india")Or with the equivalent version bundled with the package:
class (hampi)## [1] "sf" "data.frame"class (hampi$geometry)## [1] "sfc_LINESTRING" "sfc"dim (hampi)## [1] 189 4The streetnet is an sf (Simple Features) object containing 182 LINESTRING geometries. In other words, it’s got an sf representation of 182 street segments. The R package osmplotr can be used to visualise this street network (with the help of magrittr pipe operator, %>%):
library (osmplotr)
library (magrittr)
map <- osm_basemap (hampi, bg = "gray95") %>%
add_osm_objects (hampi, col = "gray5") %>%
add_axes () %>%
print_osm_map ()
(Note that this requires current dev version of osmplotr, which can be installed with devtools::install_github ("ropensci/osmplotr").)
The sf class data representing the street network of Hampi can then be converted into a flat data.frame object by
graph <- weight_streetnet (hampi, wt_profile = "foot")
dim (graph)## [1] 6096 11head (graph)## edge_id from_id from_lon from_lat to_id to_lon to_lat
## 1 1 339318500 76.47489 15.34169 339318502 76.47612 15.34173
## 2 2 339318502 76.47612 15.34173 339318500 76.47489 15.34169
## 3 3 339318502 76.47612 15.34173 2398958028 76.47620 15.34174
## 4 4 2398958028 76.47620 15.34174 339318502 76.47612 15.34173
## 5 5 2398958028 76.47620 15.34174 1427116077 76.47627 15.34179
## 6 6 1427116077 76.47627 15.34179 2398958028 76.47620 15.34174
## d d_weighted highway component
## 1 0.075941518 0.084379472 service 1
## 2 0.075941518 0.084379472 service 1
## 3 0.005191200 0.005768001 service 1
## 4 0.005191200 0.005768001 service 1
## 5 0.005337372 0.005930413 service 1
## 6 0.005337372 0.005930413 service 1Note that the actual graph contains around 30 times as many edges as there are streets, indicating that each street is composed on average of around 30 individual segments. The individual points points or vertices from those segments can be extracted with,
vt <- dodgr_vertices (graph)
head(vt)## id x y n
## 1 339318500 76.47489 15.34169 0
## 2 339318502 76.47612 15.34173 1
## 4 2398958028 76.47620 15.34174 2
## 6 1427116077 76.47627 15.34179 3
## 8 339318503 76.47641 15.34190 4
## 10 2398958034 76.47650 15.34199 5dim (vt)## [1] 2987 4From which we see that the OpenStreetMap representation of the streets of Hampi has 182 line segments with 2,814 unique points and 5,742 edges between those points. The number of edges per vertex in the entire network is thus,
nrow (graph) / nrow (vt)## [1] 2.040844A simple straight line has two edges between all intermediate nodes, and this thus indicates that the network in it’s entirety is quite simple. The data.frame resulting from weight_streetnet() is what dodgr uses to calculate shortest path routes, as will be described below, following a brief description of weighting street networks.
3.1 Graph Components
The foregoing graph object returned from weight_streetnet() also includes a $component column enumerating all of the distinct inter-connected components of the graph.
table (graph$component)##
## 1 2 3
## 3934 2144 18Components are numbered in order of decreasing size, with $component = 1 always denoting the largest component. In this case, that component contains 3,596 edges, representing 63% of the graph. There are clearly only three distinct components, but this number may be much larger for larger graphs, and may be obtained from,
length (unique (graph$component))## [1] 3Component numbers can be determined for any types of graph with the dodgr_components() function. For example, the following lines reduce the previous graph to a minmial (non-spatial) structure of four columns, and then (re-)calculate a fifth column of $components:
cols <- c ("edge_id", "from_id", "to_id", "d")
graph_min <- graph [, which (names (graph) %in% cols)]
graph_min <- dodgr_components (graph_min)
head (graph_min)## edge_id from_id to_id d component
## 1 1 339318500 339318502 0.075941518 1
## 2 2 339318502 339318500 0.075941518 1
## 3 3 339318502 2398958028 0.005191200 1
## 4 4 2398958028 339318502 0.005191200 1
## 5 5 2398958028 1427116077 0.005337372 1
## 6 6 1427116077 2398958028 0.005337372 1The component column column can be used to select or filter any component in a graph. It is particularly useful to ensure routing calculations consider only connected vertices through simply removing all minor components:
graph_connected <- graph [graph$component == 1, ]This is explored further below (under Distance Matrices).
3.2 Weighting Profiles
Dual-weights for street networks are generally obtained by multiplying the distance of each segment by a weighting factor reflecting the type of highway. As demonstrated above, this can be done easily within dodgr with the weight_streetnet() function, which using the named weighting profiles included with the dodgr package. Each profile assigns weights to each distinct type of highway.
names (weighting_profiles)## [1] "name" "way" "value"class (weighting_profiles)## [1] "data.frame"unique (weighting_profiles$name)## [1] "foot" "horse" "wheelchair" "bicycle" "moped"
## [6] "motorcycle" "motorcar" "goods" "hgv" "psv"weighting_profiles [weighting_profiles$name == "foot", ]## name way value
## 1 foot motorway 0
## 2 foot trunk 40
## 3 foot primary 50
## 4 foot secondary 60
## 5 foot tertiary 70
## 6 foot unclassified 80
## 7 foot residential 90
## 8 foot service 90
## 9 foot track 95
## 10 foot cycleway 95
## 11 foot path 100
## 12 foot steps 80
## 13 foot ferry 20Each profile is defined by a series of percentage weights quantifying highway-type preferences for a particular mode of travel. The distinct types of highways within the Hampi graph obtained above can be tabulated with:
table (graph$highway)##
## footway living_street path pedestrian primary
## 424 116 2200 142 1194
## residential service steps tertiary track
## 94 176 128 396 912
## unclassified
## 314Hampi is unlike most other human settlements on the planet in being a Unesco World Heritage area in which automobiles are generally prohibited. Accordingly, numbers of "footway", "path", and "pedestrian" ways far exceed typical categories denoting automobile traffic ("primary", "residential", "tertiary")
3.3 Random Sub-Graphs
The dodgr packages includes a function to select a random connected portion of graph including a specified number of vertices. This function is used in the compare_heaps() function described below, but is also useful for general statistical analyses of large graphs which may otherwise take too long to compute.
graph_sub <- dodgr_sample (graph, nverts = 100)
nrow (graph_sub)## [1] 210The random sample has around twice as many edges as vertices, in accordance with the statistics calculated above.
nrow (dodgr_vertices (graph_sub))## [1] 1024 Distance Matrices: dodgr_dists()
As demonstrated at the outset, an entire network can simply be submitted to dodgr_dists(), in which case a square matrix will be returned containing pair-wise distances between all vertices. Doing that for the graph of York will return a square matrix of around 90,000-times-90,000 (or 8 billion) distances. It might be possible to do that on some computers, but is possibly neither recommended nor desirable. The dodgr_dists() function accepts additional arguments of from and to defining points from and to which distances are to be calculated. If only from is provided, a square matrix is returned of pair-wise distances between all listed points.
4.1 Aligning Routing Points to Graphs
For spatial graphs—that is, those containing columns of latitudes and longitudes (or “x” and “y”)—routing points can be represented by a simple matrix of arbitrary latitudes and longitudes (or, again, “x” and “y”). dodgr_dists() will map these points to the closest network points, and return corresponding shortest-path distances. This may be illustrated by generating random points within the bounding box of the above map of Hampi. As demonstrated above, the coordinates of all vertices may be extracted with the dodgr_vertices() function, enabling random points to be generated with the following lines:
vt <- dodgr_vertices (graph)
n <- 100 # number of points to generate
xy <- data.frame (x = min (vt$x) + runif (n) * diff (range (vt$x)),
y = min (vt$y) + runif (n) * diff (range (vt$y)))Submitting these to dodgr_dists() as points from which to route will generate a distance matrix from each of these 100 points to every other point in the graph:
d <- dodgr_dists (graph, from = xy)
dim (d); range (d, na.rm = TRUE)## [1] 100 2987## [1] 0.00000 16.54437If the to argument is also specified, the matrix returned will have rows matching from and columns matching to
d <- dodgr_dists (graph, from = xy, to = xy [1:10, ])
dim (d)## [1] 100 10Some of the resultant distances in the above cases are NA because the points were sampled from the entire bounding box, and the street network near the boundaries may be cut off from the rest. As demonstrated above, the weight_streetnet() function returns a component vector, and such disconnected edges will have graph$component > 1, because graph$component == 1 always denotes the largest connected component. This means that the graph can always be reduced to the single largest component with the following single line:
graph_connected <- graph [graph$component == 1, ]A distance matrix obtained from running dodgr_dists on graph_connected should generally contain no NA values, although some points may still be effectively unreachable due to one-way connections (or streets). Thus, routing on the largest connected component of a directed graph ought to be expected to yield the minimal number of NA values, which may sometimes be more than zero. Note further that spatial routing points (expressed as from and/or to arguments) will in this case be mapped to the nearest vertices of graph_connected, rather than the potentially closer nearest points of the full graph. This may make the spatial mapping of routing points less accurate than results obtained by repeating extraction of the street network using an expanded bounding box. For automatic extraction of street networks with dodgr_dists(), the extent by which the bounding box exceeds the range of routing points (from and to arguments) is determined by an extra parameter expand, quantifying the relative extent to which the bounding box should exceed the spatial range of the routing points. This is illustrated in the following code which calculates distances between 100 random points:
bb <- osmdata::getbb ("york uk")
npts <- 100
xy <- apply (bb, 1, function (i) min (i) + runif (npts) * diff (i))
routed_points <- function (expand = 0, pts)
{
gr0 <- dodgr_streetnet (pts = pts, expand = expand) %>%
weight_streetnet ()
d0 <- dodgr_dists (gr0, from = pts)
length (which (is.na (d0))) / length (d0)
}vapply (c (0, 0.05, 0.1), function (i) routed_points (i, pts = xy),
numeric (1))## [1] 0.0780 0.0586 0.0000With a street network that precisely encompasses the submitted routing points (expand = 0), 7.8% of pairwise distances are unable to be calculated; with a bounding box expanded to 5% larger than the submitted points, this is reduced to 5.9%, and with expansion to 10%, all points can be connected.
For non-spatial graphs, from and to must match precisely on to vertices named in the graph itself. In the graph considered above, these vertex names were contained in the columns, from_id and to_id. The minimum that a dodgr graph requires is,
head (graph [, names (graph) %in% c ("from_id", "to_id", "d")])## from_id to_id d
## 1 339318500 339318502 0.075941518
## 2 339318502 339318500 0.075941518
## 3 339318502 2398958028 0.005191200
## 4 2398958028 339318502 0.005191200
## 5 2398958028 1427116077 0.005337372
## 6 1427116077 2398958028 0.005337372in which case the from values submitted to dodgr_dists() (and to, if given) must directly name the vertices in the from_id and to_id columns of the graph. This is illustrated in the following code:
graph_min <- graph [, names (graph) %in% c ("from_id", "to_id", "d")]
fr <- sample (graph_min$from_id, size = 10) # 10 random points
to <- sample (graph_min$to_id, size = 20)
d <- dodgr_dists (graph_min, from = fr, to = to)
dim (d)## [1] 10 20The result is a 10-by-20 matrix of distances between these named graph vertices.
4.2 Shortest Path Calculations: Priority Queues
dodgr uses an internal library (S. Saunders and Takaoka 2003, Shane Saunders (2004)) for the calculation of shortest paths using a variety of priority queues (see Miller 1960 for an overview). In the context of shortest paths, priority queues determine the order in which a graph is traversed (R. Tarjan 1983), and the choice of priority queue can have a considerable effect on computational efficiency for different kinds of graphs (Johnson 1977). In contrast to dodgr, most other R packages for shortest path calculations do not use priority queues, and so may often be less efficient. Shortest path distances can be calculated in dodgr with priority queues that use the following heaps:
- Binary heaps;
- Fibonacci heaps (Fredman and Tarjan 1987);
- Trinomial and extended trinomial heaps (T. Takaoka 2000); and
- 2-3 heaps (T. Takaoka 1999).
Differences in how these heaps operate are often largely extraneous to direct application of routing algorithms, even though heap choice may strongly affect performance. To avoid users needing to know anything about algorithmic details, dodgr provides a function compare_heaps to which a particular graph may be submitted in order to determine the optimal kind of heap.
The comparisons are actually made on a randomly selected sub-component of the graph containing a defined number of vertices (with a default of 1,000, or the entire graph if it contains fewer than 1,000 vertices).
compare_heaps (graph, nverts = 100, replications = 1)## test
## 6 d <- dodgr_dists(graph_contracted, from = from_id, to = to_id, heap = "BHeap")
## 7 d <- dodgr_dists(graph_contracted, from = from_id, to = to_id, heap = "FHeap")
## 8 d <- dodgr_dists(graph_contracted, from = from_id, to = to_id, heap = "TriHeap")
## 9 d <- dodgr_dists(graph_contracted, from = from_id, to = to_id, heap = "TriHeapExt")
## 10 d <- dodgr_dists(graph_contracted, from = from_id, to = to_id, heap = "Heap23")
## 1 d <- dodgr_dists(graph, from = from_id, to = to_id, heap = "BHeap")
## 2 d <- dodgr_dists(graph, from = from_id, to = to_id, heap = "FHeap")
## 4 d <- dodgr_dists(graph, from = from_id, to = to_id, heap = "TriHeapExt")
## 3 d <- dodgr_dists(graph, from = from_id, to = to_id, heap = "TriHeap")
## 5 d <- dodgr_dists(graph, from = from_id, to = to_id, heap = "Heap23")
## 11 d <- igraph::distances(igr, v = from_id, to = to_id, mode = "out")
## replications elapsed relative user.self sys.self user.child sys.child
## 6 10 0.008 1.000 0.008 0.000 0 0
## 7 10 0.008 1.000 0.008 0.000 0 0
## 8 10 0.008 1.000 0.008 0.000 0 0
## 9 10 0.008 1.000 0.008 0.000 0 0
## 10 10 0.008 1.000 0.008 0.000 0 0
## 1 10 0.011 1.375 0.008 0.004 0 0
## 2 10 0.012 1.500 0.016 0.000 0 0
## 4 10 0.012 1.500 0.012 0.000 0 0
## 3 10 0.013 1.625 0.012 0.000 0 0
## 5 10 0.013 1.625 0.012 0.000 0 0
## 11 10 0.067 8.375 0.064 0.000 0 0The key column of that data.frame is relative, which quantifies the relative performance of each test in relation to the best which is given a score of 1. dodgr using the default heap = "BHeap", which is a binary heap priority queue, performs faster than igraph (Csardi and Nepusz 2006) for these graphs. Different kind of graphs will perform differently with different priority queue structures, and this function enables users to empirically discern the optimal heap for their kind of graph.
Note, however, that this is not an entirely fair comparison, because dodgr calculates dual-weighted distances, whereas igraph—and indeed all other R packages—only directly calculate distances based on a single set of weights. Implementing dual-weighted routing in these cases requires explicitly re-tracing all paths and summing the second set of weights along each path. A time comparison in that case would be very strongly in favour of dodgr. Moreover, dodgr can convert graphs to contracted form through removing redundant vertices, as detailed in the following section. Doing so greatly improves performance with respect to igraph.
For those wishing to do explicit comparisons themselves, the following code generates the igraph equivalent of dodgr_dists(), although of course for single-weighted graphs only:
edges <- cbind (graph$from_id, graph$to_id)
pts <- sample (unique (as.vector (edges)), 100) # set of random routing points
edges <- as.vector (t (edges))
igr <- igraph::make_directed_graph (edges)
igraph::E (igr)$weight <- graph$d
d <- igraph::distances (igr, v = pts, to = pts, mode = "out")5 Graph Contraction
A further unique feature of dodgr is the ability to remove redundant vertices from graphs (see Fig. 3), thereby speeding up routing calculations.
Figure 3: (A) Vertex#2 is redundant for routing calculations from anywhere other than that point itself. (B) Equivalent graph to (A) contracted through removal of vertex#2. (C) Vertex#2 is not redundant.
In Fig. 3(A), the only way to get from vertex 1 to 3, 4 or 5 is through C. The intermediate vertex B is redundant for routing purposes (and than to or from that precise point) and may simply be removed, with directional edges inserted directly between vertices 1 and 3. This yields the equivalent contracted graph of Fig. 3(B), in which, for example, the distance (or weight) between 1 and 3 is the sum of previous distances (or weights) between 1 \(\to\) 2 and 2 \(\to\) 3. Note that if one of the two edges between, say, 3 and 2 were removed, vertex 2 would no longer be redundant (Fig. 3(C)).
Different kinds of graphs have different degrees of redundancy, and even street networks differ, through for example dense inner-urban networks generally being less redundant than less dense extra-urban or rural networks. The contracted version of a graph can be obtained with the function dodgr_compact_graph(), illustrated here with the York example from above.
grc <- dodgr_contract_graph (graph)The function dodgr_compact_graph() returns a list with both the original and compact graphs. Relative sizes are
nrow (graph); nrow (grc); nrow (grc) / nrow (graph)## [1] 6096## [1] 662## [1] 0.1085958equivalent to the removal of just under half of all edges. In order to fairly compre differences in routing efficiency, these graphs can first be converted to a standardised minimal form used for routing with the dodgr_convert_graph() function:
graph <- dodgr_convert_graph (graph)$graph
grc <- dodgr_convert_graph (grc)$graph
names (grc)## [1] "edge_id" "from" "to" "d" "w" "component"The dodgr_convert_graph() function also returns a matrix of all vertices ($xy), with the graph again in a list item named $graph. The difference in routing efficiency can then be seen with the following code
from <- sample (grc$from, size = 100)
to <- sample (grc$to, size = 100)
rbenchmark::benchmark (
d2 <- dodgr_dists (grc, from = from, to = to),
d2 <- dodgr_dists (graph, from = from, to = to),
replications = 2)## test replications elapsed
## 2 d2 <- dodgr_dists(graph, from = from, to = to) 2 0.045
## 1 d2 <- dodgr_dists(grc, from = from, to = to) 2 0.007
## relative user.self sys.self user.child sys.child
## 2 6.429 0.048 0 0 0
## 1 1.000 0.008 0 0 0And reducing numbers of graph edges by half enables pairwise routing between 100 points to be conducted around 4 times faster. All routing algorithms scale non-linearly with size, and relative improvements in efficiency will be even greater for larger graphs.
5.1 Routing on Contracted Graphs
Routing is often desired between defined points, and these points may inadvertedly be removed in graph contraction. The dodgr_contract_graph() function accepts an additional argument specifying vertices to keep within the contracted graph. This list of vertices must directly match the vertex ID values in the graph.
The following code illustrates how to retain specific vertices within contracted graphs:
grc <- dodgr_contract_graph (graph)
nrow (grc)## [1] 662verts <- sample (dodgr_vertices (graph)$id, size = 100)
head (verts) # a character vector## [1] "313796415" "286632888" "1148815136" "4337842055" "2398957885"
## [6] "977473005"grc <- dodgr_contract_graph (graph, verts)
nrow (grc)## [1] 846Retaining the nominated vertices yields a graph with considerably more edges than the fully contracted graph excluding these vertices. The dodgr_dists() function can be applied to the latter graph to obtain accurate distances precisely routed between these points, yet using the speed advantages of graph contraction.
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