# The ε-Approximation of the Time-Dependent Shortest Path Problem Solution for All Departure Times

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## Abstract

**:**

## 1. Introduction

## 2. Definitions and Preliminaries

#### 2.1. Road Network

- There are two consecutive edges $(s,u)$ and $(u,d)$ with AFs ${f}_{su}$, ${f}_{ud}$. The operation combination${f}_{ud}\ast {f}_{su}:t\mapsto {f}_{ud}\left({f}_{su}\left(t\right)\right)$ represents AF from s to d. In Figure 1b there are AFs as results of the combination along the paths $(s,u,d)$ (solid red line) and $(s,v,d)$ (solid blue line).
- There are two parallel paths ${p}_{1}$, ${p}_{2}$ from s to d with AFs ${f}_{sd}^{1}$, ${f}_{sd}^{2}$. The operation minimum$min({f}_{sd}^{1},{f}_{sd}^{2}):t\mapsto min\{{f}_{sd}^{1},{f}_{sd}^{2}\}$ represents the earliest AF from s to d. In Figure 1a ${p}_{1}=(s,u,d)$ and ${p}_{2}=(s,v,d)$. In Figure 1c you can see this earliest AF as a result of the operation minimum (green line).

#### 2.2. Problem Definition

#### 2.3. Approximation

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

#### 2.4. Related Work

**V**points represent points that are created as images of the breakpoints that lie on the edge arrival functions $\left\{{f}_{uv}\right|(u,v)\in E\}$. The

**X**points are created as an intersection of two AF in the $minimum$ operation. It can be proven that the AF between two consecutive

**V**points is concave or a line segment [1] (see example in Figure 1c). The algorithms described in [1,4] use this concavity. First, the

**V**points are computed using one backward probe and two forward probes (more in [3]), and then the approximation of AF between the

**V**points is determined. The main problem of this approach is that the computation of

**V**points requires $3{\sum}_{(u,v)\in E}\left|{f}_{uv}\right|$ probes [3].

- The node labels are AFs from s.
- The key of the priority queue is the minimum of AF ($minf$).
- The relaxation of the edge $(u,v)$ is performed using ${f}_{v}=min({f}_{v},{f}_{uv}\ast {f}_{u})$.

Algorithm 1: LCA in the exact form. |

## 3. Proposed Algorithms

#### 3.1. $\epsilon $-LCA Algorithm

#### 3.2. $\epsilon $-LCA-BS Algorithm

Algorithm 2:$\epsilon $-LCA-BS. |

Algorithm 3: backSearch. |

#### 3.3. Heuristic Improvement

- Split the origin interval into equal subintervals.
- Split the origin interval into inhomogeneous subintervals (e.g., longer at night and shorter by day).

## 4. Experiments

^{(R)}Core

^{(TM)}i5-8250U CPU with 1.60GHz and 16 GB RAM.

#### 4.1. The $\u03f5$-LCA-BS Testing

**V**breakpoints was implemented. This computation is the first step of the all algorithms presented in [1,3,4,16]. As mentioned above, the step needs $3{\sum}_{(u,v)\in E}\left|{f}_{uv}\right|$ static shortest path computation. The

**V**breakpoints determination was performed on G1 and takes 21 min.

#### 4.2. Splitting Tests

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Example of calculation of the arrival function from node s to d (${t}_{d}$—departure time, ${t}_{a}$—arrival time).

**Figure 6.**The relative number of breakpoints and the relative time related to exact LCA (II—Imai and Iri, DP—Douglas–Peucker).

**Table 1.**Results of testing $\u03f5$-LCA-BS (${t}_{r}$—the relative time related to the exact version of LCA, $bps$—the relative number of breakpoints, $\epsilon $—the maximum allowed relative error).

Dataset | # Edges | $\mathit{\epsilon}$ | Imai and Iri | Douglas Peucker | ||
---|---|---|---|---|---|---|

${\mathit{t}}_{\mathit{r}}$ [%] | $\mathit{bps}$ [%] | ${\mathit{t}}_{\mathit{r}}$ [%] | $\mathit{bps}$ [%] | |||

G 1 | 10,798 | ${10}^{-2}$ | 19.7 | 0.8 | 13.4 | 1.3 |

${10}^{-3}$ | 20.6 | 2.3 | 43.3 | 4.6 | ||

${10}^{-4}$ | 46.0 | 7.2 | 123.6 | 13.7 | ||

${10}^{-5}$ | 101.7 | 19.0 | 288.5 | 32.3 | ||

G2 | 33,354 | ${10}^{-2}$ | 13.3 | 0.8 | 11.6 | 1.3 |

${10}^{-3}$ | 15.0 | 2.1 | 34.9 | 4.1 | ||

${10}^{-4}$ | 33.4 | 6.4 | 107.3 | 12.4 | ||

${10}^{-5}$ | 76.8 | 17.0 | 262.7 | 30.1 | ||

G3 | 107,476 | ${10}^{-2}$ | 9.4 | 0.8 | 9.0 | 1.3 |

${10}^{-3}$ | 12.8 | 2.1 | 26.5 | 4.0 | ||

${10}^{-4}$ | 29.4 | 6.1 | 82.1 | 12.2 | ||

${10}^{-5}$ | 71.9 | 16.4 | 211.6 | 30.1 | ||

G4 | 160,092 | ${10}^{-2}$ | 9.2 | 0.8 | 10.2 | 1.3 |

${10}^{-3}$ | 11.4 | 2.2 | 27.9 | 4.1 | ||

${10}^{-4}$ | 24.8 | 6.3 | 84.8 | 12.4 | ||

${10}^{-5}$ | 58.1 | 16.6 | 215.0 | 30.3 |

Imai and Iri | Douglas Peucker | |||
---|---|---|---|---|

Time [s] | # bps | Time [s] | # bps | |

G1 | 0.9 | 561,853 | 1.8 | 1,122,458 |

G2 | 3.0 | 1,429,080 | 5.6 | 2,779,762 |

G3 | 9.3 | 3 855,536 | 18.8 | 7,352,745 |

G4 | 14.1 | 5,298,280 | 28.8 | 10,057,055 |

1 Interval, 1 Thread | 4 Intervals, 1 Thread | 4 Intervals, 4 Threads |
---|---|---|

8.6 s | 7.6 s | 2.6 s |

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**MDPI and ACS Style**

Kolovský, F.; Ježek, J.; Kolingerová, I.
The *ε*-Approximation of the Time-Dependent Shortest Path Problem Solution for All Departure Times. *ISPRS Int. J. Geo-Inf.* **2019**, *8*, 538.
https://doi.org/10.3390/ijgi8120538

**AMA Style**

Kolovský F, Ježek J, Kolingerová I.
The *ε*-Approximation of the Time-Dependent Shortest Path Problem Solution for All Departure Times. *ISPRS International Journal of Geo-Information*. 2019; 8(12):538.
https://doi.org/10.3390/ijgi8120538

**Chicago/Turabian Style**

Kolovský, František, Jan Ježek, and Ivana Kolingerová.
2019. "The *ε*-Approximation of the Time-Dependent Shortest Path Problem Solution for All Departure Times" *ISPRS International Journal of Geo-Information* 8, no. 12: 538.
https://doi.org/10.3390/ijgi8120538