Design techniques based on classical algorithms have proved useful for recent innovation on several large-scale problems, such as travel itineraries and routing challenges. For example, Dijkstra’s algorithm is often used to compute routes in graphs, but the size of the computation can increase quickly beyond the scale of a small town. The process of "partitioning" a road network, however, can greatly speed up algorithms by effectively shrinking how much of the graph is searched during computation.

In this post, we cover how we engineered a graph partitioning algorithm for road networks using ideas from classic algorithms, parts of which were presented in “Sketch-based Algorithms for Approximate Shortest Paths in Road Networks” at WWW 2021. Using random walks, a classical concept that is counterintuitively useful for computing shortest routes by decreasing the network size significantly, our algorithm can find a high quality partitioning of the whole road network of the North America continent nearly an order of magnitude faster¹ than other partitioning algorithms with similar output qualities.

Using Graphs to Model Road Networks
There is a well-known and useful correspondence between road networks and graphs, where intersections become nodes and roads become edges.

Image from Wikipedia

To understand how routing might benefit from partitioning, consider the most well-known solution for finding the fastest route: the Dijkstra algorithm, which works in a breadth-first search manner. The Dijkstra algorithm performs an exhaustive search starting from the source until it finds the destination. Because of this, as the distance between the source and the destination increases, the computation can become an order of magnitude slower. For example, it is faster to compute a route inside Seattle, WA than from Seattle, WA to San Francisco, CA. Moreover, even for intra-metro routes, the exhaustive volume of space explored by the Dijkstra algorithm during computation results in an impractical latency on the order of seconds. However, identifying regions that have more connections inside themselves, but fewer connections to the outside (such as Staten Island, NY) makes it possible to split the computation into multiple, smaller chunks.

Top: A routing problem around Staten Island, NY. Bottom: Corresponding partitioning as a graph. Blue nodes indicate the only entrances to/exits from Staten Island.

Consider driving from point A to point B in the above image. Once one decides where to enter Staten Island (Outerbridge or Goethals) and where to exit (Verrazzano), the problem can be broken into the three smaller pieces of driving: To the entrance, the exit, and then the destination using the best route available. That means a routing algorithm only needs to consider these special points (beacons) to navigate between points A and B and can thus find the shortest accurate path faster.

Note that beacons are only useful as long as there are not too many of them—the fewer beacons there are, the fewer shortcuts need to be added, the smaller the search space, and the faster the computation—so a good partitioning should have relatively fewer beacons for the number of components (i.e., a particular area of a road network).

As the example of Staten Island illustrates, real-life road networks have many beacons (special points such as bridges, tunnels, or mountain passes) that result in some areas being very well-connected (e.g., with large grids of streets) and others being poorly connected (e.g., an island only accessible via a couple of bridges). The question becomes how to efficiently define the components and identify the smallest number of beacons that connect the road network.

Our Partitioning Algorithm
Because each connection between two components is a potential beacon, the approach we take to ensure there are not too many beacons is to divide the road network in a way that minimizes the number of connections between components.

To do this, we start by dividing the network into two balanced (i.e., of similar size) components while also minimizing the number of roads that connect those two components, which results in an effectively small ratio of beacons to roads in each component. Then, the algorithm keeps dividing the network into two at a time until all the components reach the desired size, in terms of the number of roads inside, that yields a useful multi-component partition. There is a careful balance here: If the size is too small, we will get too many beacons; whereas if it is too large, then it will be useful only for long routes. Therefore the size is left as an input parameter and found through experimentation when the algorithm is being finalized.

While there are numerous partitioning schemes, such as METIS (for general networks), PUNCH and inertial-flow (both optimized for road-network likes), our solution is based on the inertial-flow algorithm, augmented to run as efficiently on whole continents as it does on cities.

Balanced Partitioning for Road Networks
How does one divide a road network represented as a graph into two balanced components, as mentioned above? A first step is to make a graph smaller by grouping closely connected nodes together, which allows us to speed up the following two-way partitioning phase. This is where a random walk is useful.

Random walks enjoy many useful theoretical properties—which is why they have been used to study a range of topics from the motion of mosquitoes in a forest to heat diffusion—and that most relevant for our application is that they tend to get “trapped” in regions that are well connected inside but poorly connected outside. Consider a random walk on the streets of Staten Island for a fixed number of steps: because relatively few roads exit the island, most of the steps happen inside the island, and the probability of stepping outside the island is low.

Illustration of a random walk. Suppose the blue graph is a hypothetical road network corresponding to Staten Island. 50 random walks are performed, all starting at the middle point. Each random walk continues for 10 steps or until it steps out of the island. The numbers at each node depict how many times they were visited by a random walk. By the end, any node inside the island is visited much more frequently than the nodes outside.

After finding these small components, which will be highly connected nodes grouped together (such as Staten Island in the above example), the algorithm contracts each group into a new, single node.

Reducing the size of the original graph (left) by finding groups of nodes (middle) and coalescing each group into a single “super” node (right). Example here chosen manually to better illustrate the rest of the algorithm.

The final steps of the algorithm are to partition this much smaller graph into two parts and then refine the partitioning on this small graph to one on the original graph of the road network. We then use the inertial flow algorithm to find the cut on the smaller graph that minimizes the ratio of beacons (i.e., edges being cut) to nodes.

The algorithm evaluates different directions. For each direction, we find the division that minimizes the number of edges cut (e.g., beacons) between the first and last 10% of the nodes

Having found a cut on the small graph, the algorithm performs a refinement step to project the cut back to the original graph of the road network.

Conclusion
This work shows how classical algorithms offer many useful tools for solving problems at large scale. Graph partitioning can be used to break down a large scale graph problem into smaller subproblems to be solved independently and in parallel—which is particularly relevant in Google maps, where this partitioning algorithm is used to efficiently compute routes.

Acknowledgements
We thank our collaborators Lisa Fawcett, Sreenivas Gollapudi, Kostas Kollias, Ravi Kumar, Andrew Tomkins, Ameya Velingker from Google Research and Pablo Beltran, Geoff Hulten, Steve Jackson, Du Nguyen from Google Maps.

¹This technique can also be used for any network structure, such as that for brain neurons. ^↩

Mapping algorithms used for navigation often rely on Dijkstra’s algorithm, a fundamental textbook solution for finding shortest paths in graphs. Dijkstra’s algorithm is simple and elegant -- rather than considering all possible routes (an exponential number) it iteratively improves an initial solution, and works in polynomial time. The original algorithm and practical extensions of it (such as the A* algorithm) are used millions of times per day for routing vehicles on the global road network. However, due to the fact that most vehicles are gas-powered, these algorithms ignore refueling considerations because a) gas stations are usually available everywhere at the cost of a small detour, and b) the time needed to refuel is typically only a few minutes and is negligible compared to the total travel time.

This situation is different for electric vehicles (EVs). First, EV charging stations are not as commonly available as gas stations, which can cause range anxiety, the fear that the car will run out of power before reaching a charging station. This concern is common enough that it is considered one of the barriers to the widespread adoption of EVs. Second, charging an EV’s battery is a more decision-demanding task, because the charging time can be a significant fraction of the total travel time and can vary widely by station, vehicle model, and battery level. In addition, the charging time is non-linear — e.g., it takes longer to charge a battery from 90% to 100% than from 20% to 30%.

The EV can only travel a distance up to the illustrated range before needing to recharge. Different roads and different stations have different time costs. The goal is to optimize for the total trip time.

Today, we present a new approach for routing of EVs integrated into the latest release of Google Maps built into your car for participating EVs that reduces range anxiety by integrating recharging stations into the navigational route. Based on the battery level and the destination, Maps will recommend the charging stops and the corresponding charging levels that will minimize the total duration of the trip. To accomplish this we engineered a highly scalable solution for recommending efficient routes through charging stations, which optimizes the sum of the driving time and the charging time together.

The fastest route from Berlin to Paris for a gas fueled car is shown in the top figure. The middle figure shows the optimal route for a 400 km range EV (travel time indicated - charging time excluded), where the larger white circles along the route indicate charging stops. The bottom figure shows the optimal route for a 200 km range EV.

Routing Through Charging Stations
A fundamental constraint on route selection is that the distance between recharging stops cannot be higher than what the vehicle can reach on a full charge. Consequently, the route selection model emphasizes the graph of charging stations, as opposed to the graph of road segments of the road network, where each charging station is a node and each trip between charging stations is an edge. Taking into consideration the various characteristics of each EV (such as the weight, maximum battery level, plug type, etc.) the algorithm identifies which of the edges are feasible for the EV under consideration and which are not. Once the routing request comes in, Maps EV routing augments the feasible graph with two new nodes, the origin and the destination, and with multiple new (feasible) edges that outline the potential trips from the origin to its nearby charging stations and to the destination from each of its nearby charging stations.

Routing using Dijkstra’s algorithm or A* on this graph is sufficient to give a feasible solution that optimizes for the travel time for drivers that do not care at all about the charging time, (i.e., drivers who always fully charge their batteries at each charging station). However, such algorithms are not sufficient to account for charging times. In this case, the algorithm constructs a new graph by replicating each charging station node multiple times. Half of the copies correspond to entering the station with a partially charged battery, with a charge, x, ranging from 0%-100%. The other half correspond to exiting the station with a fractional charge, y (again from 0%-100%). We add an edge from the entry node at the charge x to the exit node at charge y (constrained by y > x), with a corresponding charging time to get from x to y. When the trip from Station A to Station B spends some fraction (z) of the battery charge, we introduce an edge between every exit node of Station A to the corresponding entry node of Station B (at charge x-z). After performing this transformation, using Dijkstra or A* recovers the solution.

An example of our node/edge replication. In this instance the algorithm opts to pass through the first station without charging and charges at the second station from 20% to 80% battery.

Graph Sparsification
To perform the above operations while addressing range anxiety with confidence, the algorithm must compute the battery consumption of each trip between stations with good precision. For this reason, Maps maintains detailed information about the road characteristics along the trip between any two stations (e.g., the length, elevation, and slope, for each segment of the trip), taking into consideration the properties of each type of EV.

Due to the volume of information required for each segment, maintaining a large number of edges can become a memory intensive task. While this is not a problem for areas where EV charging stations are sparse, there exist locations in the world (such as Northern Europe) where the density of stations is very high. In such locations, adding an edge for every pair of stations between which an EV can travel quickly grows to billions of possible edges.

The figure on the left illustrates the high density of charging stations in Northern Europe. Different colors correspond to different plug types. The figure on the right illustrates why the routing graph scales up very quickly in size as the density of stations increases. When there are many stations within range of each other, the induced routing graph is a complete graph that stores detailed information for each edge.

However, this high density implies that a trip between two stations that are relatively far apart will undoubtedly pass through multiple other stations. In this case, maintaining information about the long edge is redundant, making it possible to simply add the smaller edges (spanners) in the graph, resulting in sparser, more computationally feasible, graphs.

The spanner construction algorithm is a direct generalization of the greedy geometric spanner. The trips between charging stations are sorted from fastest to slowest and are processed in that order. For each trip between points a and b, the algorithm examines whether smaller subtrips already included in the spanner subsume the direct trip. To do so it compares the trip time and battery consumption that can be achieved using subtrips already in the spanner, against the same quantities for the direct a-b route. If they are found to be within a tiny error threshold, the direct trip from a to b is not added to the spanner, otherwise it is. Applying this sparsification algorithm has a notable impact and allows the graph to be served efficiently in responding to users’ routing requests.

On the left is the original road network (EV stations in light red). The station graph in the middle has edges for all feasible trips between stations. The sparse graph on the right maintains the distances with much fewer edges.

Summary
In this work we engineer a scalable solution for routing EVs on long trips to include access to charging stations through the use of graph sparsification and novel framing of standard routing algorithms. We are excited to put algorithmic ideas and techniques in the hands of Maps users and look forward to serving stress-free routes for EV drivers across the globe!

Acknowledgements
We thank our collaborators Dixie Wang, Xin Wei Chow, Navin Gunatillaka, Stephen Broadfoot, Alex Donaldson, and Ivan Kuznetsov.