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

*Reviewer 1:*Anonymous

*Reviewer 2:*Anonymous

*Reviewer 3:*Anonymous

**Round 1**

*Reviewer 1 Report*

The paper proposed a new variant of LCA for time-dependent shortest path problem based on profile search computation approximation. Overall the paper is well-written, the problem is well-laid out and methodology is clearly presented. Some suggestions:

I personally consider Figure 1 the quintessence of the proposed work. Some labels or legends may help the potential readers to better digest figure 1 right off the bat, thus better digest the paper. Line 100, as the authors state that road network is far from the time complexity boundary, some further insights should be provided. Line 126, since DP and II will be used as shorthand for these two methods in this paper, it is a practice to formally define them here. Line 251, the disadvantage of the proposed algorithms should be lifted from conclusion and be discussed more in details in chapter three. More literature surveys should be carried out to highlight more modern variants of LCA. Comparing the proposed algorithms to the vanilla version of LCA doesn’t really hammers out the significance of this work.

*Author Response*

Review 1

I personally consider Figure 1 the quintessence of the proposed work. Some labels or legends may help the potential readers to better digest figure 1 right off the bat, thus better digest the paper. – Labels and legend were added.

Line 100, as the authors state that road network is far from the time complexity boundary, some further insights should be provided. – The following sentence was added:* However, a real road network is far from the worst case, because it is close to planar graph and the cost of edges is the travel time, i.e. is bounded by physics rules, and thus the shortest path is near to the direct line.*

Line 126, since DP and II will be used as shorthand for these two methods in this paper, it is a practice to formally define them here. - The shorthands were added.

Line 251, the disadvantage of the proposed algorithms should be lifted from conclusion and be discussed more in details in chapter three. - The text was moved to chapter 3 (now it is the last paragraph in section 3.2) and it was extended. Now the paragraph is: *The ε-LCA-BS has one main disadvantage. If there are a lot of edges with the arrival function where the maximum slope is too big, the algorithm will perform a lot of backSearch procedure (Algorithm 3). The backSearch procedure is time-consuming so the whole algorithm will be slow in such a case. But, in a real road network the maximum slopes of AFs are not too steep [10,14]. So the main disadvantage is not a too big problem.*

More literature surveys should be carried out to highlight more modern variants of LCA. - The following paragraph was added to the section 2.4: *There are a lot of other variants of LCA. Those variants differ in the type of data structure - a queue (FIFO, First-In-First-Out) [8], a priority queue (as in our case) [5,7,10], a stack (LIFO, Last-In-First-Out) [9]. The algorithms also differ in the insertion strategy into the data structure. Some innovative insertion strategies are presented in [11] and [12]. The choice of LCA variant depends on the type of the target graph.*

*Reviewer 2 Report*

Very interesting classic approach to the search for the optimal path in road transport. The approach is based on the analysis of traditional but still valid solutions such as the Dijkstra method, Douglas-Peucker algorithm and the less known Imai and Iri algorithm. The results of an interesting experiment conducted in Paris were presented. In the next papers it is worth to analyze and compare the classical solutions of artificial intelligence method, including ant colony algorithm.

*Author Response*

The review is very positive and there is no suggestion to change anything in the paper.

*Reviewer 3 Report*

**- The article's main concept(s)**

This paper two algorithms to solve the shortest path problem (profile search). It deals with time-dependent travel time analysis. Computes the arrival function from a source node to all the other nodes. It handles the *time-depended shortest path problem *(TDSP).

Dijkstra’s algorithm deals with the search for the shortest path between a graph node. In this case, the authors discuss the ɛ-approximation profile search computation based on a label correcting modification of Dijkstra’s algorithm.

Their idea performs a simplification of the arrival functions during the computation with a suitable maximum absolute error so that the relative error ɛ is maintained. By splitting (Dijkstra’s LCA) the original departure time interval into subintervals will diminish the number of edges (nodes) relaxation.

If the starting node plus the distance/time value travelled is less, then the final node value – then the relaxation computes the final node new value with the first node plus the distance/time/cost value. The authors performed some simplifications using Douglas-Peucker or Imai&Iri algorithm. Based on Dijktra’s algorithm and some simplifications, they developed two algorithms. ɛ-LCA algorithm and the ɛ-LCA-BS algorithm (using backsearch), the first one is a diversion of the second one.

The authors performed some experiments with real road networks, with real speeds profiles computed from GPS data from city of Paris.

**- Overall Comment**

In overall,

This document shows a reasonable understanding of the time-dependent shortest path problem with the ɛ-approximation topics and its key factors. The work has a good theoretical base, using useful references and information of general knowledge.

The introducing of ɛ-approximation gives innovation in TDSP problem. Uses of profile search and back search in a diversion of a well-known algorithm.

The work uses a deep reinforcement learning, specifically a Deep Q-Network (DQN) to tackle the multi-criteria/decision in autonomous driving.

It presents a new and very simple algorithm (2) where the authors use common knowledge to tests the performance of shortest path algorithms for all departure times.

The manuscript has a simple structure but some minor English errors/typos. For a journal paper, it would be great if it expands the mathematical modelling, probably expanding the results partly to some other tests/experiments/simulations and creating a better bridge between theoretic and simulations.

The conclusions resume the paper contribution and explain common sense knowledge. But it seems very short – the work could have better conclusions.

The appearance of some small empty boxes on page 3 – on Theorem 1

Great news: Results proven that the subdivision can even save total computation time.

The authors used a Java based programing language: Scala.

Figure 4 is confusing – maybe using minus data would improve what the authors would like to show.

**- Weak and Strong points **

Strengths

Very good resume/comparison of the TDSP and profile search algorithms; Experimental results and tests with a real road network (Paris); Epsilon approximation – to a profile search in time-dependent shortest path problem The maximum relative error has a good comparison between accuracy and performance; Creation of two algorithms for epsilon-approximation of TDSP that reduce computational time/memory. .Weakness

No major novelty – just a diversion of Dijktra’s algorithm; Creation of two algorithms that are interconnected; The results/experiments sections – need some revision; The conclusions – are too small! If the arrival function has high slopes, then it became too slow to compute – too many calls to backsearch; …*Author Response*

Review 3

For a journal paper, it would be great if it expands the mathematical modelling – We are a little bit confused about this note as it does not correspond to another note from the same review: “The work has a good theoretical base, using useful references and information of general knowledge”. Could you provide more specific comment? We think that the problem is thoroughly mathematically described in the paper.

probably expanding the results partly to some other tests/experiments/simulations and creating a better bridge between theoretic and simulations. - Unfortunately, we perform the tests on real data in Paris (as the you noted in the strengths). We have not any other dataset for testing. The access to this kind of real data is very difficult, because the companies that own the datasets sell it for a lot of money or they have the data only for their own use. Therefore, we are not able to enlarge our experiments to other data.

The conclusions resume the paper contribution and explain common sense knowledge. But it seems very short – the work could have better conclusions. - We wrote the maximum knowledge that we were able to derive from the results. What is more, the previous reviewer asked to move some part of conclusion into Section 3, which is the opposite opinion than presented in the current review. Therefore, we can not process this opponent’s note.

The appearance of some small empty boxes on page 3 – on Theorem 1 - The small empty boxes on page 3 are rendered by the “proof” environment in Latex. The box marks the end of the proof. The Latex style used in the paper is the one provided by the journal.

Figure 4 is confusing – maybe using minus data would improve what the authors would like to show. - Less number of speed profiles was rendered to the chart for better understanding.