Inferring Spatial Distance Rankings with Partial Knowledge on Routing Networks
Abstract
:1. Introduction
2. Related Work
2.1. Our Contribution
- A weighted directed graph with vertices V, arcs , and a cost function , and
- A set of target vertices ,
- Input: Query vertex ;
- Output: List L of all target vertices sorted with respect to the shortest distances from s, i.e.,
- ,
- , and
- for all ,
- where is the quasimetric induced by the cost function c.
2.2. Structure of the Paper
3. Properties of Quasimetric Spaces
4. Routing Networks
- From , we get the non-negativity of .
- As for all walks from u to , we yield by definition for all . As for each , we can conclude that for each . Thus, we yield the positive definiteness of .
- If we define the concatenation of walks by
Algorithm 1 An implementation on quasimetric networks |
Algorithm 2 Complete the ranking that is represented by both graphs |
Possible Extensions
5. Time-Dependent Routing Networks
- If we can pre-compute and store values of routing distances, we can create the quasimetric network with a new function . Then, the induced quasimetric is a lower bound of for all .
- If measures the time it takes for an object o to get from to when starting at time t, then there exists some mapping with so that measures the distance from u to w. If we know the maximum speed of o while moving from u to w, we can take as a time-independent lower bounding metric.
- for every where is a class of functions for which is closed under composition and the evaluation of a function of is reasonably fast.
- It is possible to divide I into disjoint intervals
- For all exist some functions such that for every , we have for all .
- The values of c are discrete, i.e., is a step function for all . Then, is a step function for all (as V is finite). We can conclude that we can find a partition such that for every , we have for all . Hence, we can define as constant functions for all .
- The class of polynomial splines is closed under composition and can be factorized after composition for optimized evaluations.
6. Evaluation
6.1. Statistics
6.2. Design and Overview
- 1.
- Choose a fixed function at the beginning of A*; or
- 2.
- Update the heuristic whenever another specific target vertex is found. Updating means reevaluating the heuristic estimate for each vertex that is currently in the open front. This takes time and shows that Dijkstra is faster in situations where the target vertices are arbitrarily distributed on the network.
6.3. Implementation
6.4. Experimental Results
6.4.1. Star Graph
6.4.2. Line Graph
6.4.3. Show Case: City of Munich
7. Discussion
7.1. Star Graph
7.2. Line Graph
7.3. Show Case
8. Conclusions and Future Work
8.1. Theoretical Conclusions
8.2. Practical Conclusions
8.3. Future Work
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Köppl, D. Inferring Spatial Distance Rankings with Partial Knowledge on Routing Networks. Information 2022, 13, 168. https://doi.org/10.3390/info13040168
Köppl D. Inferring Spatial Distance Rankings with Partial Knowledge on Routing Networks. Information. 2022; 13(4):168. https://doi.org/10.3390/info13040168
Chicago/Turabian StyleKöppl, Dominik. 2022. "Inferring Spatial Distance Rankings with Partial Knowledge on Routing Networks" Information 13, no. 4: 168. https://doi.org/10.3390/info13040168