# dodgr In dodgr: Distances on Directed Graphs

library (dodgr)


# 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. 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,

graph <- data.frame (from = c ("A", "B", "B", "B", "C", "C", "D", "D"),
to = c ("B", "A", "C", "D", "B", "D", "C", "A"),
d = c (1, 2, 1, 3, 2, 1, 2, 1))
graph


The primary function of dodgr is dodgr_dists, which calculates pair-wise shortest distances between all vertices of a graph.

dodgr_dists (graph)
dodgr_dists (graph, from = c ("A", "C"), to = c ("B", "C", "D"))


## 1.1 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. 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,

graph <- data.frame (from = c ("A", "B", "B", "B", "C", "C", "D", "D"),
to = c ("B", "A", "C", "D", "B", "D", "C", "A"),
w = c (1, 2, 1, 3, 2, 1, 2, 1),
d = c (1, 1, 1, 1, 1, 1, 1, 1))
graph
dodgr_dists (graph)


Note 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 <- rbind (c (-1.241536, -0.9215361), c (53.799056, 54.1190555))
rownames (bb) <- c ("x", "y")
colnames (bb) <- c ("min", "max")
bb
x <- c (-1.1713502, -1.2216108, -1.0457199, -0.9384666, -0.9445541, -1.1207099)
y <- c (53.89409, 54.01065, 53.83613, 53.93545, 53.89436, 54.01262)
cbind (x, y)


The following lines download the street network within that bounding box, weight it for pedestrian travel, and use the weighted network to calculate the pairwise distances between all of the xypoints.

net <- dodgr_streetnet (bb)
net <- weight_streetnet (net, wt_profile = "foot")
system.time (
d <- dodgr_dists (net, from = xy, to = xy)
)

c (user = 38.828, system = 0.036, elapsed = 5.424)

dim (d); range (d, na.rm = TRUE)

c (1000, 1000)
c (0.00000, 57021.18)


The result is a matrix of 1000-by-1000 distances of up to 57km long, measured along routes weighted for optimal pedestrian travel. In this case, the single call to dodgr_distances() 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. 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)

class (hampi$geometry)  dim (hampi)  The streetnet is an sf (Simple Features) object containing 189 LINESTRING geometries. In other words, it's got an sf representation of 189 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 ()  library (magrittr)  map <- osm_basemap (hampi, bg = "gray95") %>% add_osm_objects (hampi, col = "gray5") %>% add_axes () %>% print_osm_map (filename = "hampi.png", width = 2000, units = "px") 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)  head (graph)  Note 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 or vertices from those segments can be extracted with, vt <- dodgr_vertices (graph) head(vt)  dim (vt)  From which we see that the OpenStreetMap representation of the streets of Hampi has 189 line segments with 2,987 unique points and 6,096 edges between those points. The number of edges per vertex in the entire network is thus, nrow (graph) / nrow (vt)  A 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)  Components are numbered in order of decreasing size, with $component = 1 always denoting the largest component. In this case, that component contains 3,934 edges, representing 65% 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))  Component 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 minimal (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)


The 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$component == 1, ]  This is explored further below (under Distance Matrices). ## 3.2 Weighting Profiles Dual-weights for street networks are generally obtained by multiplying the distance of each segment by a weighting factor reflecting the type of highway. As demonstrated above, this can be done easily within dodgr with the weight_streetnet() function, which applies the named weighting profiles included with the dodgr package to OpenStreetMap networks extracted with the osmdata package. This function uses the internal data dodgr::weighting_profiles, which is a list of three items: 1. weighting_profiles; 2. surface_speeds; and 3. penalties Most of these data are used to calculate routing times with the dodgr_times function, as detailed in an additional vignette. The only aspects relevant for distances are the profiles themselves, which assign preferential weights to each distinct type of highway. wp <- weighting_profiles$weighting_profiles
names (wp)
class (wp)
unique (wp$name) wp [wp$name == "foot", ]


Each 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)  Hampi 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") It is also possible to use other types of (non-OpenStreetMap) street networks, an example of which is the os_roads_bristol data provided with the package. "OS" is the U.K. Ordnance Survey, and these data are provided as a Simple Features (sf) data.frame with a decidedly different structure to osmdata data.frame objects: names (hampi) # many fields manually removed to reduce size of this object names (os_roads_bristol)  The latter may be converted to a dodgr network by first specifying a weighting profile, here based on the formOfWay column: colnm <- "formOfWay" table (os_roads_bristol [[colnm]]) wts <- data.frame (name = "custom", way = unique (os_roads_bristol [[colnm]]), value = c (0.1, 0.2, 0.8, 1)) net <- weight_streetnet (os_roads_bristol, wt_profile = wts, type_col = colnm, id_col = "identifier")  The resultant net object contains the street network of os_roads_bristol weighted by the specified profile, and in a format suitable for submission to any dodgr routine. ## 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)  The random sample has around twice as many edges as vertices, in accordance with the statistics calculated above. nrow (dodgr_vertices (graph_sub))  # 4 Distance Matrices: dodgr_dists() As demonstrated at the outset, an entire network can simply be submitted to dodgr_distances(), 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_distances() 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_distances() 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$y)))  Submitting these to dodgr_distances() as points from which to route will generate a distance matrix from each of these 100 points to every other point in the graph: d <- dodgr_dists (graph, from = xy) dim (d); range (d, na.rm = TRUE)  If the to argument is also specified, the matrix returned will have rows matching from and columns matching to d <- dodgr_dists (graph, from = xy, to = xy [1:10, ]) dim (d)  Some of the resultant distances in the above cases are NA because the points were sampled from the entire bounding box, and the street network near the boundaries may be cut off from the rest. As demonstrated above, the weight_streetnet() function returns a component vector, and such disconnected edges will have graph$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_distances 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_distances(), 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, 0.2), function (i) routed_points (i, pts = xy),
numeric (1))

c (0.04007477, 0.02326452, 0.02131992, 0)


With a street network that precisely encompasses the submitted routing points (expand = 0), 4% of pairwise distances are unable to be calculated; with a bounding box expanded to 5% larger than the submitted points, this is reduced to 2.3%, and with expansion to 20%, 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")])


in 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)


The 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 [@Saunders2003, @Saunders2004] for the calculation of shortest paths using a variety of priority queues [see @Miller1960 for an overview]. In the context of shortest paths, priority queues determine the order in which a graph is traversed [@Tarjan1983], and the choice of priority queue can have a considerable effect on computational efficiency for different kinds of graphs [@Johnson1977]. 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:

1. Binary heaps;
2. Fibonacci heaps [@Fredman1987];
3. Trinomial and extended trinomial heaps [@Takaoka2000]; and
4. 2-3 heaps [@Takaoka1999].

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 = 1000, replications = 1)


The 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 [@Csardi2006] 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_distances(), although of course for single-weighted graphs only:

v <- dodgr_vertices (graph)
pts <- sample (v$id, 1000) igr <- dodgr_to_igraph (graph) d <- igraph::distances (igr, v = pts, to = pts, mode = "out")  # 5 Graph Contraction A further unique feature of dodgr is the ability to remove redundant vertices from graphs (see Fig. 3), thereby speeding up routing calculations. In Fig. 3(A), the only way to get from vertex 1 to 3, 4 or 5 is through C. The intermediate vertex B is redundant for routing purposes (and than to or from that precise point) and may simply be removed, with directional edges inserted directly between vertices 1 and 3. This yields the equivalent contracted graph of Fig. 3(B), in which, for example, the distance (or weight) between 1 and 3 is the sum of previous distances (or weights) between 1$\to$2 and 2$\to$3. Note that if one of the two edges between, say, 3 and 2 were removed, vertex 2 would no longer be redundant (Fig. 3(C)). Different kinds of graphs have different degrees of redundancy, and even street networks differ, through for example dense inner-urban networks generally being less redundant than less dense extra-urban or rural networks. The contracted version of a graph can be obtained with the function dodgr_contract_graph(), illustrated here with the York example from above. grc <- dodgr_contract_graph (graph)  The function dodgr_contract_graph() returns the contracted version of the original graph, containing the same number of columns, but with each row representing an edge between two junction vertices (or between the submitted verts, which may or may not be junctions). Relative sizes are nrow (graph); nrow (grc); nrow (grc) / nrow (graph)  equivalent to the removal of around 90% of all edges. The difference in routing efficiency can then be seen with the following code from <- sample (grc$from_id, size = 100)
to <- sample (grc$to_id, size = 100) rbenchmark::benchmark ( d2 <- dodgr_dists (grc, from = from, to = to), d2 <- dodgr_dists (graph, from = from, to = to), replications = 2)  And contracting the graph has a similar effect of speeding up pairwise routing between these 100 points. All routing algorithms scale non-linearly with size, and relative improvements in efficiency will be even greater for larger graphs. ## 5.1 Routing on Contracted Graphs Routing is often desired between defined points, and these points may inadvertently be removed in graph contraction. The dodgr_contract_graph() function accepts an additional argument specifying vertices to keep within the contracted graph. This list of vertices must directly match the vertex ID values in the graph. The following code illustrates how to retain specific vertices within contracted graphs: grc <- dodgr_contract_graph (graph) nrow (grc) verts <- sample (dodgr_vertices (graph)$id, size = 100)
head (verts) # a character vector
grc <- dodgr_contract_graph (graph, verts)
nrow (grc)


Retaining the nominated vertices yields a graph with considerably more edges than the fully contracted graph excluding these vertices. The dodgr_distances() 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.

# 6 Shortest Paths

Shortest paths can also be extracted with the dodgr_paths() function. For given vectors of from and to points, this returns a nested list so that if,

dp <- dodgr_paths (graph, from = from, to = to)


then dp [[i]] [[j]] will contain the path from from [i] to to [j]. The paths are represented as sequences of vertex names. Consider the following example,

graph <- weight_streetnet (hampi, wt_profile = "foot")


The columns of from_id and to_id contain the names of the vertices. To extract shortest paths between some of these, first take some small samples of from and to points, and submit them to dodgr_paths():

from <- sample (graph$from_id, size = 10) to <- sample (graph$to_id, size = 5)
dp <- dodgr_paths (graph, from = from, to = to)
length (dp)


The result (dp) is a list of 10 items, each of which contains 5 vectors. An example is,

dp [] []

# make sure there are some paths:
maxlen <- max (unlist (lapply (dp, function (i)
max (unlist (lapply (i, length))))))
if (maxlen > 0)
{
n <- 0
i <- 0
while (all (n == 0))
{
i <- i + 1
n <- which (lapply (dp [[i]], length) > 0)
}
j <- n 
dp [[i]] [[j]]
}


For spatial graphs, the coordinates of these paths can be obtained by extracting the vertices with dodgr_vertices() and matching the vertex IDs:

verts <- dodgr_vertices (graph)
path1 <- verts [match (dp [] [], verts$id), ] head (path1)  verts <- dodgr_vertices (graph) path1 <- verts [match (dp [[i]] [[j]], verts$id), ]