# Inferring Spatial Distance Rankings with Partial Knowledge on Routing Networks

## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Our Contribution

- A weighted directed graph $(V,A,c)$ with vertices V, arcs $A\subset V\times V$, and a cost function $c:A\to \mathbb{R}$, and
- A set of target vertices $\mathcal{T}\subset V$,

**Input:**Query vertex $s\in V$;**Output:**List L of all target vertices sorted with respect to the shortest distances from s, i.e.,- $\left|L\right|=\left|\mathcal{T}\right|$,
- $L\left[i\right]\in \mathcal{T}$, and
- $\rho (s,L[i\left]\right)<\rho (s,L[i+1\left]\right)$ for all $i\in [1..|\mathcal{T}|-1]$,

- where $\rho $ is the quasimetric induced by the cost function c.

#### 2.2. Structure of the Paper

## 3. Properties of Quasimetric Spaces

**Definition**

**1**

**Lemma**

**1.**

**Proof.**

**Remark**

**1.**

**Lemma**

**2.**

**Proof.**

**Remark**

**2.**

**Lemma**

**3.**

**Proof.**

**Corollary**

**1.**

**Proposition**

**1.**

**Proof.**

**Example**

**1.**

**Lemma**

**4**

**Proof.**

**Example**

**2.**

## 4. Routing Networks

**Definition**

**2**

**Definition**

**3**

**Lemma**

**5.**

**Proof.**

- From $l\ge 0$, we get the non-negativity of $\rho $.
- As $\ell \left(P\right)>0$ for all walks from u to $v\in V,u\ne v$, we yield by definition $\rho \left(u,v\right)>0$ for all $u,v\in V,u\ne v$. As $c(v,v)=0$ for each $v\in V$, we can conclude that $\ell \left((v,v)\right)=0$ for each $v\in V$. Thus, we yield the positive definiteness of $\rho $.
- If we define the concatenation of walks by$$({v}_{1},\dots ,{v}_{n})\circ ({w}_{1},\dots ,{w}_{n}):=\left\{\begin{array}{cc}({v}_{1},\dots ,{v}_{n},{w}_{2},\dots ,{w}_{n})\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}{v}_{n}={w}_{1},\hfill \\ \phantom{\rule{4.pt}{0ex}}\mathrm{undefined}\phantom{\rule{4.pt}{0ex}}\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

**Definition**

**4**

**Remark**

**3**

**Definition**

**5**

**Problem**

**1**

**Remark**

**4.**

**Solution**

**1.**

Algorithm 1 An implementation on quasimetric networks |

Algorithm 2 Complete the ranking that is represented by both graphs |

**Example**

**3.**

**Remark**

**5**

**Remark**

**6**

**Remark**

**7**

#### Possible Extensions

**Remark**

**8**

**Remark**

**9**(Validity regions)

**.**

**Lemma**

**6**

**Proof.**

**Remark**

**10.**

**Remark**

**11.**

**Problem**

**2**

## 5. Time-Dependent Routing Networks

**Definition**

**6**

**Lemma**

**7.**

**Proof.**

**Example**

**4.**

**Definition**

**7**

**Lemma**

**8.**

**Proof.**

**Definition**

**8**

**Example**

**5.**

- If we can pre-compute and store values of routing distances, we can create the quasimetric network $(V,A,{c}_{min})$ with a new function ${c}_{min}:={inf}_{t\in I}{c}^{t}$. Then, the induced quasimetric ${\rho}_{min}$ is a lower bound of ${\rho}^{t}$ for all $t\in I$.
- If ${c}^{t}(u,v)$ measures the time it takes for an object o to get from $u\in V$ to $w\in V$ when starting at time t, then there exists some mapping ${d}_{2}:V\times V\to {\mathbb{R}}^{n}$ with $n\in \mathbb{N}$ so that ${d}_{2}(u,w)$ measures the distance from u to w. If we know the maximum speed ${v}_{max}(u,w)$ of o while moving from u to w, we can take ${d}^{t}(u,w):=\frac{{d}_{2}(u,w)}{{v}_{max}(u,w)}\le {c}^{t}(u,w)$ as a time-independent lower bounding metric.

**Definition**

**9**

- $t\mapsto {c}_{a}\left(t\right)\in \mathcal{C}$ for every $a\in A$ where $\mathcal{C}\subset \left\{f:I\to I\right\}$ is a class of functions for which $\mathcal{C}$ is closed under composition and the evaluation of a function of $\mathcal{C}$ is reasonably fast.
- It is possible to divide I into $n\in \mathbb{N}$ disjoint intervals$$[{t}_{1},{t}_{2})\cup [{t}_{2},{t}_{3})\cup \dots \cup [{t}_{n-1},{t}_{n})=I.$$
- For all $u,v\in V$ exist some functions ${f}_{1}^{(u,v)},\dots ,$${f}_{n}^{(u,v)}\in \mathcal{C}$ such that for every $k\in \left\{1,\dots ,n-1\right\}$, we have ${\rho}^{t}\left(u,v\right)={f}_{k}^{(u,v)}\left(t\right)$ for all $t\in [{t}_{k},{t}_{k+1})$.

**Remark**

**12.**

- The values of c are discrete, i.e., $t\mapsto {c}_{a}\left(t\right)$ is a step function for all $a\in A$. Then, $t\mapsto {\rho}^{t}\left(u,v\right)$ is a step function for all $u,v\in V$ (as V is finite). We can conclude that we can find a partition $[{t}_{1},{t}_{2})\cup [{t}_{2},{t}_{3})\cup \dots \cup [{t}_{n-1},{t}_{n})=I$ such that for every $k\in \left\{1,\dots ,n-1\right\}$, we have ${\rho}^{u}={\rho}^{v}$ for all $u,v\in [{t}_{k},{t}_{k+1})$. Hence, we can define ${f}_{k}^{(u,v)}\left(t\right):={\rho}^{t}\left(u,v\right)$ as constant functions for all $u,v\in V,t\in [{t}_{k},{t}_{k+1})$.
- The class of polynomial splines is closed under composition and can be factorized after composition for optimized evaluations.

**Problem**

**3**

**Solution**

**2.**

**Remark**

**13.**

## 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

**Remark**

**14**

#### 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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**Figure 1.**Geometrical illustration of the inequality of Lemma 1 for simplicity on the standard Euclidean space.

**Figure 2.**Unsuitable condition for the $\delta $-criterion. The success of the $\delta $-criterion basically depends on the distance between query vertex s and pivot p. Here, although u is closer to s than v, the $\delta $-criteria is in this situation indecisive.

**Figure 3.**Statistical evaluation of the distances of the target vertices on the San Francisco dataset. (

**a**) Correlation between $\delta $ and the minimum distance between all targets. (

**b**) Anti-correlation between $\left|\mathcal{T}\right|$ and the minimum distance between all targets, caused by the uniform distribution.

**Figure 4.**Flowchart of our asynchronous algorithm determining the ranks of the target vertices. The flowchart shows asynchronous (dashed arrows) and synchronous (solid arrows) execution of tasks (rounded rectangles). Tasks of different threads are grouped by color: The main thread is blue, both shortest-path-finding threads are colored in green and the callback thread is yellow.

**Figure 5.**Synthetically generated graphs considered in Section 6.4.1 and Section 6.4.2. Dashed lines visualize simple paths. On the left, the query vertex s is in the center, and $\mathcal{T}=\{{u}_{1},\dots ,{u}_{15}\}$. On the right, we assign each vertex an index by enumeration.

**Figure 6.**Evaluation of a star-shaped graph with the query vertex s in the center. Each axis of length ℓ of the star is assigned exactly one target vertex ${u}_{i}$ whose distance to s is $\ell \xb7{i}^{2}/{\left|\mathcal{T}\right|}^{2}$. Since the arcs all have unit costs, the length between two nodes $(u,v)$ is the number of nodes of the shortest path from u to v. Naive and Imp. are measuring Dijkstra’s algorithm and Dijkstra’s algorithm combined with our solution, respectively.

**Figure 7.**Evaluation of a star-shaped graph similar to Figure 6. Here, the distance between s and a target vertex ${u}_{i}$ is ${i}^{2}+\left|\mathcal{T}\right|/2$. We evaluated Dijkstra’s algorithm (Naive), our implementation using Dijkstra’s algorithm without caching (No Cache), and when caching a query vertex whose distance to s is $1/4$, $1/2$, and $2/3$ of the distance from s to s’s closest target. We call these instances Cached 1/4, Cached 1/2, and Cached 2/3, respectively, in the legend.

**Figure 8.**Evaluation of a line graph with two target vertices u and v having the indices 0 and 250, respectively. We promote vertices with an index from 10 to 225 to the query vertex s; the x-axis reflects this index (i.e., the start position is the index of the start vertex). Naive measures the time for Dijkstra’s algorithm to find the shortest paths to all target vertices. The curves No Cache and With Cache measure our solution when running the computation without cache and running the computation with the node in the middle of the line graph cached, respectively.

**Figure 9.**Evaluation of a line graph with $\left|\mathcal{T}\right|=13$ equidistantly distributed target vertices; the vertex with index 0 is a target vertex. As in Figure 8, we successively promote vertices with an index from 0 to 400 to a query vertex (this index is reflected by the x-axis, i.e., the start position), and run a query. We do not promote any target vertex to a start vertex, except for index 0. The curves Dij. and A* measure the time for Dijkstra’s algorithm and the A* algorithm for finding the shortest paths to all target vertices. The curves imp. Dij. and imp. A* measure our implementation using Dijkstra’s algorithm and the A* algorithm, respectively.

**Figure 10.**Visualization of the Munich dataset via OpenStreetMap, covering the indexed area [48.14, 48.145] north latitude and [11.54, 11.543] east longitude from www.openstreetmap.org (accessed on 19 December 2021) with a scale factor of 16. The map is oriented north-up. © OpenStreetMap contributors.

**Figure 11.**Random queries on the Munich dataset. The start vertices are enumerated by start positions (the x-axis in the plot), which are ranked ascendingly with respect to the time consumed by our algorithm in its first round. The curves Dijkstra measure the time for Dijkstra’s algorithm to find the shortest paths to all target vertices. The other curves with prefixes imp. Dij. denote our solution using Dijkstra’s algorithm. The implementation with suffix NoCache does not cache past queries, while Round 1 caches query vertices, and Round 2 makes use of the cache of Round 1.

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**MDPI and ACS Style**

Köppl, D.
Inferring Spatial Distance Rankings with Partial Knowledge on Routing Networks. *Information* **2022**, *13*, 168.
https://doi.org/10.3390/info13040168

**AMA Style**

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 Style**

Köppl, Dominik.
2022. "Inferring Spatial Distance Rankings with Partial Knowledge on Routing Networks" *Information* 13, no. 4: 168.
https://doi.org/10.3390/info13040168