A Cloud-Based Time-Dependent Routing Service

Abstract

We present a cloud-based time-dependent routing service, which is a core component in providing efficient personalized renewable mobility services in smart cities. We describe the architecture of the time-dependent routing engine, consisting of a core routing module along with the so-called urban-traffic knowledge base, which creates, maintains and stores historic traffic data, as well as live traffic updates such as road blockages or unforeseen congestion. We also provide the crucial algorithmic details for providing the sought efficient time-dependent routing service. Our cloud-based time-dependent routing service exhibits an excellent practical behavior on a diversity (w.r.t. to scale and type) of real-world road networks.

A Hueristics for Improving Performance

We describe some heuristic algorithmic improvements, as well as some implementation details that we apply during the creation and maintenance of traffic metadata in the UTKB, in order to obtain even better performance, both wrt the required preprocessing space and wrt the efficiency of the query phase.

Approximately constant functions. The TRAP approximation method introduces one intermediate breakpoint per interval that satisfies the required approximation guarantee. To keep our algorithm space-efficient, the first thing that we do, when dealing with every subinterval of the time period, is that we check for approximately constant functions. More precisely, we perform an additional sampling at the middle point $t_m$ of each $[t_s, t_f)$ and we consider the upper-approximation $\overline{\varDelta }[\ell ,v][t_s, t_f)$ to be “almost constant”, if the following condition holds: $D[\ell ,v](t_s) = D[\ell ,v](t_f) = D[\ell ,v](t_m)$. For those approximately constant functions, the insertion of the additional breakpoint at $t_m$ is unnecessary.

Piecewise composition. Many shortest paths are likely to contain at least one arc with piecewise travel time, making the shortest-path function also piecewise. In our case, keeping the predecessor vertex of v in every sample of $\overline{\varDelta }[\ell ,v](t)$, allows us to analyze any $\ell$-v approximate shortest path into two sub-paths $\ell$-p-v. Starting from the destination v, we travel back to a predecessor p, as long as all vertices u up to p have constantly the same predecessor kept in the corresponding samples of $\overline{\varDelta }[\ell ,u](t)$, and all arcs involved in the p-v subpath have a constant travel-time function. Therefore, there is no need to keep any samples for the approximate function $\overline{\varDelta }[\ell ,v](t)$; instead, we store the necessary information in the form: (constant-travel-time, predecessor-p, approximate-path-predecessor). In this way, the approximate travel-time function $\overline{\varDelta }[\ell ,v](t)$ is given by taking into account $\overline{\varDelta }[p,v](t)$ and the (approximately) constant travel-time from v to p, i.e., $\overline{\varDelta }[\ell ,v](t) = \overline{\varDelta }[\ell ,p](t) + \overline{\varDelta }[p,v](t)$. In our experiments, this method leads to around 40–50% reduction of the space requirements.

Delay shifts. In cases that such a predecessor p does not exist, we have to store the approximate travel-time function as a sequence of breakpoints. However, the samples collected for $\overline{\varDelta }[\ell ,v](t)$ may have small delay variation. Based on this fact, the required space can be further reduced. We store the minimum travel-time value and for each leg we only need to store the small shift from this value. This conversion leads to around 5–10% reduction of the required preprocessing space.

Fixed range. For a one-day time period, departure-times have a bounded value range. The same holds for travel-times which are at most one-day for any query within a country or city area. When the considered precision of the traffic data is within seconds, we handle time-values as integers in the range [0, 86399], rather than real values, sacrificing precision for space reduction. In particular, we convert all floating-point time-values $t_f$ to integers $t_i$ with fewer bytes and a given unit of measure. For a unit of measure s, the resulting integer is $t_i = \lceil t_f/s \rceil$ and needs size $\lceil \log _2 (t_f/s)/8\rceil$ bytes. Converting $t_f$ to $t_i$ results to an absolute error of at most 2 s. Therefore, for storing the time-values of approximate travel-time summaries, we can consider different resolutions, depending on the scale factor s, to achieve further reduction of the preprocessing space.

Compression. Since there is no need for all landmarks to be concurrently active, we can compress their data blocks. This method leads to significant reduction of the space requirements, especially for large-scale networks.

Contraction. The space of the preprocessed travel-time summaries can be further reduced if we consider a subset of vertices in the network as inactive. More precisely, we can conduct a preprocessing of the instance that contracts all vertices which are not junctions, i.e., they form paths with no intersections. Each such path can be represented by a single shortcut arc, which is added at the endpoints of the chain and equipped with an arc-traversal time function equal to the corresponding (exact) path-travel-time function. The arcs involved in the contracted paths are also considered as inactive. All contracted nodes are ignored and therefore the number of possible destinations from a landmark is smaller. At the query phase, these paths can be easily retrieved, by exploiting the appropriate information kept on all inserted shortcuts and all contracted nodes for this purpose.

Parallelism. We can speed up the preprocessing time for computing the one-to-all approximate travel-time functions, from properly selected landmarks towards all reachable destinations, as well as the real-time responsiveness to live-traffic reports, i.e., the re-computation of the travel-time summaries for the subset of affected landmarks, by exploiting the inherent parallelism of the entire process.