# 4. Routing

#### 2024-12-06

Calculating shortest paths between pairs of nodes is a core task in network analysis. The `sfnetworks` package offers wrappers around the path calculation functions of `igraph`, making it easier to use them when working with spatial data and tidyverse packages. This vignette demonstrates their functionality.

In this regard it is important to remember that `sfnetworks` is a general-purpose package for spatial network analysis, not specifically optimized for a single task. If your only purpose is many-to-many routing in large networks, there might be other approaches that are faster and fit better to your needs. For example, the dodgr package was designed for many-to-many routing on large dual-weighted graphs, with its main focus on OpenStreetMap road network data. The cppRouting package contains functions to calculate shortest paths and isochrones/isodistances on weighted graphs. When working with OpenStreetMap data, it is also possible to use the interfaces to external routing engines such as graphhopper, osrm and opentripplanner. The stplanr package for sustainable transport planning lets you use many routing engines from a single interface, through `stplanr::route()`, including routing using local R objects. Of course, all these packages can be happily used alongside `sfnetworks`.

    library(sfnetworks)
    library(sf)
    library(tidygraph)
    library(dplyr)
    library(purrr)
    library(TSP)

## Calculating shortest paths

The function `st_network_paths()` is a wrapper around the igraph function `igraph::shortest_paths()`. There are two main differences:

- Besides node indices and node names, `st_network_paths()` gives the additional option to provide any (set of) geospatial point(s) as from and to location(s) of the shortest paths, either as sf or sfc object. Provided points that do not equal any node in the network will be snapped to their nearest node before calculating the paths.
- To allow smooth integration with the tidyverse, the output of `st_network_paths()` is a tibble, with one row per returned path. The column node_paths contains the ordered list of node indices in the path, and the column edge_paths contains the ordered list of edge indices in the path.

Just as `igraph::shortest_paths()`, the `st_network_paths()` function is meant for one-to-one and one-to-many routing. Hence, it is only possible to provide a single from location, while the to locations can be more than one.

Lets start with the most basic example of providing node indices as from and to locations. Remember that a node index in a sfnetwork refers to the position of the node in the nodes table of the network (i.e. its row number). There is also the possibility to use character encoded node names instead of numeric node indices. This requires the nodes table to have a column name with a unique name for each node.

We will use geographic edge lengths as the edge weights to be used for shortest paths calculation. In weighted networks, `igraph::shortest_paths()` applies the Dijkstra algorithm to find the shortest paths. In the case of unweighted networks, it uses breadth-first search instead.

    net = as_sfnetwork(roxel, directed = FALSE) %>%
      st_transform(3035) %>%
      activate("edges") %>%
      mutate(weight = edge_length())

    paths = st_network_paths(net, from = 495, to = c(458, 121), weights = "weight")
    paths

    #> # A tibble: 2 × 2
    #>   node_paths edge_paths
    #>   <list>     <list>
    #> 1 <int [29]> <int [28]>
    #> 2 <int [33]> <int [32]>

    paths %>%
      slice(1) %>%
      pull(node_paths) %>%
      unlist()
    #>  [1] 495 485 244 402 166  18 195 181  68 100 169 170 182 101 103 547 546 554 556
    #> [20] 555  86  61  88 531 543  90 229 527 458

    paths %>%
      slice(1) %>%
      pull(edge_paths) %>%
      unlist()
    #>  [1] 311 757 670 671  93 117 116 106  54  96  97 502 501 414 413 402 791 423 422
    #> [20] 788 392 393 394 779 395 540 539 357

    plot_path = function(node_path) {
      net %>%
        activate("nodes") %>%
        slice(node_path) %>%
        plot(cex = 1.5, lwd = 1.5, add = TRUE)
    }

    colors = sf.colors(3, categorical = TRUE)

    plot(net, col = "grey")
    paths %>%
      pull(node_paths) %>%
      walk(plot_path)
    net %>%
      activate("nodes") %>%
      st_as_sf() %>%
      slice(c(495, 121, 458)) %>%
      plot(col = colors, pch = 8, cex = 2, lwd = 2, add = TRUE)

Next we will create some geospatial points that do not intersect with any node in the network. Providing them to `st_network_paths()` will first find the nearest node to each of them, and then calculate the shortest paths accordingly.

    p1 = st_geometry(net, "nodes")[495] + st_sfc(st_point(c(50, -50)))
    st_crs(p1) = st_crs(net)
    p2 = st_geometry(net, "nodes")[458]
    p3 = st_geometry(net, "nodes")[121] + st_sfc(st_point(c(-10, 100)))
    st_crs(p3) = st_crs(net)

    paths = st_network_paths(net, from = p1, to = c(p2, p3), weights = "weight")

    plot(net, col = "grey")
    paths %>%
      pull(node_paths) %>%
      walk(plot_path)
    plot(c(p1, p2, p3), col = colors, pch = 8, cex = 2, lwd = 2, add = TRUE)

Simply finding the nearest node to given points is not always the best way to go. For example: when a provided point is close to a location on an edge linestring, but relatively far from any node, it gives better results when the routing does not start at its nearest node, but at that nearby location on the edge linestring. To accommodate this, you can first blend your input points into the network, and then proceed as normal. Blending is the name we gave to the combined process of snapping a point to its nearest point on its nearest edge, splitting the edge there, and adding the snapped point as node to the network. It is implement in the function `st_network_blend()`. See the vignette on spatial joins and filters for more details.

Another issue may arise wen your network consists of multiple components that are not connected to each other. In that case, it is possible that the nearest node to a provided point is located in a tiny component and only a few other nodes can be reached from it. In such cases it might be good to first reduce the network to its largest (or n largest) component(s) before calculating shortest paths. The tidygraph function `tidygraph::group_components()` can help with this. It assigns an integer to each node identifying the component it is in, with `1` being the largest component in the network, `2` the second largest, and so on.

    # Our network consists of several unconnected components.
    with_graph(net, graph_component_count())
    #> [1] 14

    connected_net = net %>%
      activate("nodes") %>%
      filter(group_components() == 1)

    plot(net, col = colors[2])
    plot(connected_net, cex = 1.1, lwd = 1.1, add = TRUE)

Another way to calculate shortest paths, which fits nicely in the tidygraph style of working, is by using the `to_spatial_shortest_paths()` morpher function. This will subset the original network to only contain those nodes and edges that appear in a shortest path between two nodes. See the vignette on Spatial morphers for details.

## Retrieving an OD cost matrix

The shortest paths calculation as described above is only supported for one-to-one and one-to-many routing. The alternative for many-to-many routing is the calculation of an origin-destination cost matrix. Instead of providing the individual paths, it returns a matrix in which entry \(i,j\) is the total cost (i.e. sum of weights) of the shortest path from node \(i\) to node \(j\). The origin-destination cost matrix is usually an important starting point for further analysis. For example, it can serve as input to route optimization algorithms, spatial clustering algorithms and the calculation of statistical measures based on spatial proximity.

The igraph function for this purpose is `igraph::distances()`, which in `sfnetworks` is wrapped by `st_network_cost()`, allowing again to provide sets of geospatial points as from and to locations. Note that the calculated costs refer to the paths between the nearest nodes of the input points. Their units are the same as the units of weights used in the calculation, in this case meters.

    st_network_cost(net, from = c(p1, p2, p3), to = c(p1, p2, p3), weights = "weight")
    #> Units: [m]
    #>          [,1]     [,2]     [,3]
    #> [1,]    0.000 1614.252 2094.555
    #> [2,] 1614.252    0.000 1315.965
    #> [3,] 2094.555 1315.965    0.000

If we don’t provide any from and to points, `st_network_cost()` will by default calculate the cost matrix for the entire network.

    # Our network has 701 nodes.
    with_graph(net, graph_order())
    #> [1] 701

    cost_matrix = st_network_cost(net, weights = "weight")
    dim(cost_matrix)
    #> [1] 701 701

In directed networks, `st_network_cost()` also gives you the possibility to define if you want to allow travel only over outbound edges from the from points (by setting `direction = "out"`, the default), only over inbound edges (by setting `direction = "in"`), or both (by setting `direction = "all"`, i.e. assuming an undirected network).

    net %>%
      convert(to_spatial_directed) %>%
      st_network_cost(
        from = c(p1, p2, p3),
        to = c(p1, p2, p3),
        direction = "in"
      )
    #> Units: [m]
    #>      [,1]     [,2] [,3]
    #> [1,]    0 1899.549  Inf
    #> [2,]  Inf    0.000  Inf
    #> [3,]  Inf 1816.688    0

    net %>%
      convert(to_spatial_directed) %>%
      st_network_cost(
        from = c(p1, p2, p3),
        to = c(p1, p2, p3),
        direction = "out"
      )
    #> Units: [m]
    #>          [,1] [,2]     [,3]
    #> [1,]    0.000  Inf      Inf
    #> [2,] 1899.549    0 1816.688
    #> [3,]      Inf  Inf    0.000

    net %>%
      convert(to_spatial_directed) %>%
      st_network_cost(
        from = c(p1, p2, p3),
        to = c(p1, p2, p3),
        direction = "all"
      )
    #> Units: [m]
    #>          [,1]     [,2]     [,3]
    #> [1,]    0.000 1614.252 2094.555
    #> [2,] 1614.252    0.000 1315.965
    #> [3,] 2094.555 1315.965    0.000

All `...` arguments are forwarded internally to `igraph::distances()`. Among other options, this means that you can define which algorithm is used for the paths calculation. By default, `igraph` will choose the most suitable algorithm based on characteristics of the request, such as the type of weights (e.g. only positive or also negative) and the number of given from locations. For details, see the igraph documentation.

## Applications

In this section, we will show a small set of applications of the routing related functions. It is definitely not meant to be an overview that covers everything! Also, remember again that `sfnetworks` is a general-purpose spatial network analysis package not optimized for a specific application. However, especially in combination with other packages it can address a wide variety of use-cases.

### Closest facility analysis

The purpose of closest facility analysis is, given a set of destination locations (also referred to as the facilities) and origin locations (also referred to as the sites), to find the closest n facilities to each site. For example, you might want to find the nearest transit hub for each address in a city, or the nearest hospital to high-risk road intersections.

To solve this problem, you can calculate the OD cost matrix with the sites as from points, and the facilities as to points. Then, for each row (i.e. each site) you find the column(s) with the lowest cost value.

    # Select a random set of sites and facilities.
    # We select random locations within the bounding box of the nodes.
    set.seed(128)
    hull = net %>%
      activate("nodes") %>%
      st_geometry() %>%
      st_combine() %>%
      st_convex_hull()

    sites = st_sample(hull, 50, type = "random")
    facilities = st_sample(hull, 5, type = "random")

    # Blend the sites and facilities into the network to get better results.
    # Also select only the largest connected component.
    new_net = net %>%
      activate("nodes") %>%
      filter(group_components() == 1) %>%
      st_network_blend(c(sites, facilities))
    #> Warning: st_network_blend assumes attributes are constant over geometries

    # Calculate the cost matrix.
    cost_matrix = st_network_cost(new_net, from = sites, to = facilities, weights = "weight")

    # Find for each site which facility is closest.
    closest = facilities[apply(cost_matrix, 1, function(x) which(x == min(x))[1])]

    # Create a line between each site and its closest facility, for visualization.
    draw_lines = function(sources, targets) {
      lines = mapply(
        function(a, b) st_sfc(st_cast(c(a, b), "LINESTRING"), crs = st_crs(net)),
        sources,
        targets,
        SIMPLIFY = FALSE
      )
      do.call("c", lines)
    }

    connections = draw_lines(sites, closest)

    # Plot the results.
    plot(new_net, col = "grey")
    plot(connections, col = colors[2], lwd = 2, add = TRUE)
    plot(facilities, pch = 8, cex = 2, lwd = 2, col = colors[3], add = TRUE)
    plot(sites, pch = 20, cex = 2, col = colors[2], add = TRUE)

### Route optimization

The traveling salesman problem aims to find the shortest tour that visits a set of locations exactly once and then returns to the starting location. In `sfnetworks`, there are no dedicated functions to solve this. However, we can bring in other packages here to assist us. Probably the best known R package for solving traveling salesman problems is TSP. To do so, it requires a matrix that specifies the travel cost between each pair of locations. As shown above, we can use `sfnetworks` to calculate such a cost matrix for our network.

Lets first generate a set of random points within the bounding box of the network. These will serve as the locations the traveling salesman has to visit.

    set.seed(403)
    rdm = net %>%
      st_bbox() %>%
      st_as_sfc() %>%
      st_sample(10, type = "random")

We can then compute the cost matrix using `st_network_cost()`. Internally, it will first snap the provided points to their nearest node in the network, and then use the weight column to calculate the travel costs between these nodes. Passing the matrix to `TSP::TSP()` will make it suitable for usage inside the `TSP` package.

    net = activate(net, "nodes")
    cost_matrix = st_network_cost(net, from = rdm, to = rdm, weights = "weight")

    # Use nearest node indices as row and column names.
    rdm_idxs = st_nearest_feature(rdm, net)
    row.names(cost_matrix) = rdm_idxs
    colnames(cost_matrix) = rdm_idxs

    round(cost_matrix, 0)
    #> Units: [m]
    #>      252  545  698   15  224  323   31  510   71   61
    #> 252    0  914 1335 1947 1010 1122  776 1052 1365 1204
    #> 545  914    0  421 1575  503  358  962  680  451  291
    #> 698 1335  421    0 1552  590  241 1383 1031  227  130
    #> 15  1947 1575 1552    0 2046 1408 1314 1015 1616 1620
    #> 224 1010  503  590 2046    0  767 1155 1151  551  459
    #> 323 1122  358  241 1408  767    0 1170  800  304  308
    #> 31   776  962 1383 1314 1155 1170    0  796 1413 1252
    #> 510 1052  680 1031 1015 1151  800  796    0 1094  970
    #> 71  1365  451  227 1616  551  304 1413 1094    0  161
    #> 61  1204  291  130 1620  459  308 1252  970  161    0

By passing the cost matrix to the solver `TSP::solve_TSP()` we obtain an object of class `TOUR`. This object contains a named vector specifying the indices of the provided locations in the optimal order of visit.

    tour = solve_TSP(TSP(units::drop_units(cost_matrix)))
    tour_idxs = as.numeric(names(tour))
    tour_idxs
    #>  [1]  71  61 545 224 252  31  15 510 323 698

    # Approximate length of the route.
    # In meters, since that was the unit of our cost values.
    round(tour_length(tour), 0)
    #> [1] 6337

Knowing the optimal order to visit all provided locations, we move back to `sfnetworks` to match this route to the network. We do so by computing the shortest paths between each location and its subsequent one, until we reach the starting point again.

For more details on solving travelling salesman problems in R, see the TSP package documentation.

    # Define the nodes to calculate the shortest paths from.
    # Define the nodes to calculate the shortest paths to.
    # All based on the calculated order of visit.
    from_idxs = tour_idxs
    to_idxs = c(tour_idxs[2:length(tour_idxs)], tour_idxs[1])

    # Calculate the specified paths.
    tsp_paths = mapply(st_network_paths,
        from = from_idxs,
        to = to_idxs,
        MoreArgs = list(x = net, weights = "weight")
      )["node_paths", ] %>%
      unlist(recursive = FALSE)

    # Plot the results.
    plot(net, col = "grey")
    plot(rdm, pch = 20, col = colors[2], add = TRUE)

    walk(tsp_paths, plot_path) # Reuse the plot_path function defined earlier.

    plot(
      st_as_sf(slice(net, rdm_idxs)),
      pch = 20, col = colors[3], add = TRUE
    )
    plot(
      st_as_sf(slice(net, tour_idxs[1])),
      pch = 8, cex = 2, lwd = 2, col = colors[3], add = TRUE
    )
    text(
      st_coordinates(st_as_sf(slice(net, tour_idxs[1]))) - c(200, 90),
      labels = "start/end\npoint"
    )

### Isochrones and isodistances

With respect to a given point \(p\) and a given travel time \(t\), an isochrone is the line for which it holds that the travel time from any point on the line to or from \(p\) is equal to \(t\). When using distances instead of time, it is called an isodistance.

In `sfnetworks` there are no dedicated, optimized functions for calculating isochrones or isodistances. However, we can roughly approximate them by using a combination of sf and tidygraph functions. Lets first calculate imaginary travel times for each edge, using randomly generated average walking speeds for each type of edge.

    # How many edge types are there?
    types = net %>%
      activate("edges") %>%
      pull(type) %>%
      unique()

    types
    #> [1] residential  secondary    path         footway      track
    #> [6] unclassified pedestrian   service      cycleway
    #> 9 Levels: cycleway footway path pedestrian residential secondary ... unclassified

    # Randomly define a walking speed in m/s for each type.
    # With values between 3 and 7 km/hr.
    set.seed(1)
    speeds = runif(length(types), 3 * 1000 / 60 / 60, 7 * 1000 / 60 / 60)

    # Assign a speed to each edge based on its type.
    # Calculate travel time for each edge based on that.
    net = net %>%
      activate("edges") %>%
      group_by(type) %>%
      mutate(speed = units::set_units(speeds[cur_group_id()], "m/s")) %>%
      mutate(time = weight / speed) %>%
      ungroup()

    net

    #> # A sfnetwork with 701 nodes and 851 edges
    #> #
    #> # CRS:  EPSG:3035
    #> #
    #> # An undirected multigraph with 14 components with spatially explicit edges
    #> #
    #> # Edge data: 851 × 8 (active)
    #>    from    to name            type                   geometry weight speed  time
    #>   <int> <int> <chr>           <fct>          <LINESTRING [m]>    [m] [m/s]   [s]
    #> 1     1     2 Havixbecker St… resi… (4151491 3207923, 415147…   28.9  1.06  27.3
    #> 2     3     4 Pienersallee    seco… (4151398 3207777, 415139…  108.   1.83  58.8
    #> 3     5     6 Schulte-Bernd-… resi… (4151408 3207539, 415141…   54.4  1.06  51.4
    #> 4     7     8 NA              path  (4151885 3206698, 415186…  155.   1.47 106.
    #> 5     9    10 Welsingheide    resi… (4151732 3207017, 415172…  209.   1.06 197.
    #> 6    11    12 NA              foot… (4152152 3206984, 415214…   63.0  1.25  50.6
    #> # ℹ 845 more rows
    #> #
    #> # Node data: 701 × 1
    #>            geometry
    #>         <POINT [m]>
    #> 1 (4151491 3207923)
    #> 2 (4151474 3207946)
    #> 3 (4151398 3207777)
    #> # ℹ 698 more rows

Now, we can calculate the total travel time for each shortest path between the (nearest node of the) origin point and all other nodes in the network, using the node measure function `tidygraph::node_distance_from()` with the values in the time column as weights. Then, we filter the nodes reachable within a given travel time from the origin. By drawing a convex hull around these selected nodes we roughly approximate the isochrone. If we wanted isochrones for travel times towards the central point, we could have used the node measure function `tidygraph::node_distance_to()` instead.

    net = activate(net, "nodes")

    p = net %>%
      st_geometry() %>%
      st_combine() %>%
      st_centroid()

    iso = net %>%
      filter(node_distance_from(st_nearest_feature(p, net), weights = time) <= 600)

    iso_poly = iso %>%
      st_geometry() %>%
      st_combine() %>%
      st_convex_hull()

    plot(net, col = "grey")
    plot(iso_poly, col = NA, border = "black", lwd = 3, add = TRUE)
    plot(iso, col = colors[2], add = TRUE)
    plot(p, col = colors[1], pch = 8, cex = 2, lwd = 2, add = TRUE)

The workflow presented above is generalized in a spatial morpher function `to_spatial_neighborhood()`, which can be used inside the `tidygraph::convert()` verb to filter only those nodes that can be reached within a given travel cost from the given origin node.

    # Define the threshold values (in seconds).
    # Define also the colors to plot the neighborhoods in.
    thresholds = rev(seq(60, 600, 60))
    palette = sf.colors(n = 10)

    # Plot the results.
    plot(net, col = "grey")

    for (i in c(1:10)) {
      nbh = convert(net, to_spatial_neighborhood, p, thresholds[i], weights = "time")
      plot(nbh, col = palette[i], add = TRUE)
    }

    plot(p, pch = 8, cex = 2, lwd = 2, add = TRUE)

### Custom routing

In many cases the shortest path based geographical distances or travel time is not necessarily the optimal path. The most appropriate route may depend on many factors, for example staying away from large and potentially unpleasant highways for motor traffic. All networks have different characteristics. In OpenStreetMap networks the ‘highway’ type is often a good (albeit crude) approximation of the network. The `roxel` demo dataset is derived from OpenStreetMap and represents a largely residential network:

    table(roxel$type)
    #>
    #>     cycleway      footway         path   pedestrian  residential    secondary
    #>           42           67           95            1          394           62
    #>      service        track unclassified
    #>          160            5           25

Building on the shortest paths calculated in a previous section we can try an alternative routing profile. Let’s take a look at these paths to see what type of ways they travel on:

    paths_sf = net %>%
      activate("edges") %>%
      slice(unlist(paths$edge_paths)) %>%
      st_as_sf()

    table(paths_sf$type)
    #>
    #>     cycleway      footway         path   pedestrian  residential    secondary
    #>            2            0            6            0           38            1
    #>      service        track unclassified
    #>            2            0            6

    plot(paths_sf["type"], lwd = 4, key.pos = 4, key.width = lcm(4))

As with the overall network, the paths we calculated are dominated by residential streets. For the purposes of illustration, lets imagine we’re routing for a vehicle that we want to keep away from residential roads, and which has a much lower cost per unit distance on secondary roads:

    weighting_profile = c(
      cycleway = Inf,
      footway = Inf,
      path = Inf,
      pedestrian = Inf,
      residential = 3,
      secondary = 1,
      service = 1,
      track = 10,
      unclassified = 1
    )

    weighted_net = net %>%
      activate("edges") %>%
      mutate(multiplier = weighting_profile[type]) %>%
      mutate(weight = edge_length() * multiplier)

We can now recalculate the routes. The result show routes that avoid residential networks.

    weighted_paths = st_network_paths(weighted_net, from = 495, to = c(458, 121), weights = "weight")

    weighted_paths_sf = weighted_net %>%
      activate("edges") %>%
      slice(unlist(weighted_paths$edge_paths)) %>%
      st_as_sf()

    table(weighted_paths_sf$type)
    #>
    #>     cycleway      footway         path   pedestrian  residential    secondary
    #>            0            0            0            0            9           43
    #>      service        track unclassified
    #>            3            0            0

    plot(st_as_sf(net, "edges")["type"], lwd = 4,
         key.pos = NULL, reset = FALSE, main = "Distance weights")
    plot(st_geometry(paths_sf), add = TRUE)
    plot(st_as_sf(net, "edges")["type"], lwd = 4,
         key.pos = NULL, reset = FALSE, main = "Custom weights")
    plot(st_geometry(weighted_paths_sf), add = TRUE)

Note that developing more sophisticated routing profiles is beyond the scope of this package. If you need complex mode-specific routing profiles, we recommend looking at routing profiles associated with open source routing engines, such as bike.lua in the OSRM project. Another direction of travel could be to extend on the approach illustrated here, but this work could be well-suited to a separate package that builds on `sfnetworks` (remembering that sophisticated routing profiles account for nodes and edges). If you’d like to work on such a project to improve mode-specific routing in R by building on this package, please let us know in the discussion room!