Temporal paths discovery with multiple constraints in attributed dynamic graphs

Abstract

Temporal path discovery in dynamic graphs is significant in many applications, such as the trip planning in transportation networks, and the disease progression tracking in gene networks. Attributed Dynamic Graph (ADG) contains multiple value-changed attributes on edges, such as the speed of a bus between two stations, and the price of the bus ticket in different time periods. The traditional methods for temporal path discovery in ADGs assume users only consider a single constraint on the attributes in their models, such as finding the fast route to reach the destination under the constraint of a given budget, which makes the path discovery problem simple though it still suffers expensive time cost. However, such an assumption is too strict in real applications, where users can specify multiple constraints, such as finding the fast route under the constraints of a given budget and the total number of stopovers. In such a situation, the temporal path discovery becomes more complicated as it subsumes the classical NP-Complete Multi-Constraint Path (MCP) problem, and thus the traditional methods cannot work for finding a new type of Temporal Path with Multiple Constraints (TPMC). In this paper, we propose a set of new Two-Pass approximation algorithms to bi-directionally search ADGs to find TPMC results. To the best of our knowledge, our Two-Pass algorithms are the first algorithms to support the discovery of the temporal paths with multiple constraints in ADGs. The experimental results on 12 real-world dynamic graphs demonstrate that our algorithms outperform the state-of-the-art methods in both efficiency and effectiveness.

Appendix

The proof of Theorem 1

Assume that the delivered temporal path $p^{M}_{v_{s}, v_{d}}(t)$ is not feasible, and $p^{M^{\prime }}_{v_{s}, v_{d}}(t)$ is a feasible temporal path. Then it must be true that $\delta (p^{M}_{v_{s}, v_{d}}(t)) \leq \delta (p^{M^{\prime }}_{v_{s}, v_{t}}(t)$. In addition, we have $\delta (p^{M^{\prime }}_{v_{s}, v_{d}}(t)) \leq K$ due to ${\Pi }^{j}(p^{M^{\prime }}_{v_{s}, v_{d}} (t)) \leq \lambda _{j}$ for 1 ≤ j ≤ K. Thus, we have $\delta (p^{M}_{v_{s}, v_{d}}(t)) \leq K$. Assume ${\Pi }^{j}(p^{M}_{v_{s}, v_{d}}(t)) > \lambda _{j}$ for all j^′s, then $\delta (p^{M}_{v_{s}, v_{d}}(t)) > K$, which contradicts $\delta (p^{M}_{v_{s}, v_{d}}(t)) \leq K$, so there is at least one j making ${\Pi }^{k}(p^{M}_{v_{s}, v_{d}}(t)) \leq \lambda _{j}$, then attribute (a) in Theorem 1 is proven. Assume that for at least one λ_j, we have ${\Pi }^{j}(p^{M}_{v_{s}, v_{d}}(t)) > \sqrt [\theta ]{K}\lambda _{j}$, so that $(\frac {{\Pi }^{j}(p^{M}_{v_{s}, v_{d}}(t))}{\lambda _{j}})^{\theta } > K$, which leads to $\delta (p^{M}_{v_{s}, v_{d}}(t)) > K$. But this contradicts $\delta (p^{M}_{v_{s}, v_{d}}(t)) \leq K$. Then, attribute (b) in Theorem 1 is proven. □

The proof of Corollary 1

Follows immediately from Theorem 1, ${\Pi }^{j}(p^{M}_{v_{s}, v_{d}}(t)) \leq \sqrt [\theta +\varepsilon ]{K}\lambda _{j} < \sqrt [\theta ]{K}\lambda _{j}$, for any ε > 0, 1 ≤ j ≤ K. □

The proof of Theorem 2

Assume $p^{Back}_{v_{s}, v_{d}}(t^{\prime }_{s})$ consists of n + 2 vertices v_s, v₁,...,v_n, v_d. In the Forward Pass procedure, TPA-EA searches the decedents of v_s and chooses v₁ from these vertices when a combined temporal path from v_s to v_d via v₁ is feasible and the current path from v_s to v₁ has the earliest arrival time. This step is repeated at all the vertices between v₁ and v_n until a temporal path $p^{E}_{v_{s}, v_{d}}(t_{s})$ is identified. If at each search step, only one vertex of {v₁,...,v_n} can lead to a feasible combined temporal path, then $p^{E}_{v_{s}, v_{d}}(t_{s})$ is the only feasible temporal path between v_s and v_d in the ADG, i.e., $p^{E}_{v_{s}, v_{d}}(t_{s}) = p^{Back}_{v_{s}, v_{d}}(t^{\prime }_{s})$. Thus, $end(p^{E}_{v_{s}, v_{d}}(t_{s})) = end(p^{Back}_{v_{s}, v_{d}}(t^{\prime }_{s}))$. Otherwise, if $p^{E}_{v_{s}, v_{d}}(t_{s}) \neq p^{Back}_{v_{s}, v_{d}}(t^{\prime }_{s})$, this can lead to $end(p^{E}_{v_{s}, v_{d}}(t_{s})) < end(p^{Back}_{v_{s}, v_{d}}(t^{\prime }_{s}))$ by minimizing the arrival time in all candidate vertices which have feasible combined temporal paths by using the Dijkstra-based algorithm. Therefore, Theorem 2 is proven. □

The proof of Theorem 3

Based on Theorem 2, for each distinct departure time of v_s within [t_α, t_β], TPA-F can deliver an EA-TPMC. Then at each departure time point t, $end(p^{F}_{v_{s}, v_{d}}(t)) - start(p^{F}_{v_{s}, v_{d}}(t)) \leq end(p^{Back}_{v_{s},v_{d}}(t_{d})) - start(p^{Back}_{v_{s},v_{d}}(t_{d}))$. TPA-F selects the fastest path from all EA-TPMC results with the departure time point t_s, and then $end(p^{F}_{v_{s}, v_{d}}(t_{s})) - start(p^{F}_{v_{s},v_{d}} (t_{s})) \leq end(p^{F}_{v_{s}, v_{d}}(t)) - start(p^{Back}_{v_{s},v_{d}}(t_{d}))$. Thus, $end(p^{F}_{v_{s}, v_{d}}(t_{s})) - start(p^{F}_{v_{s}, v_{d}} (t_{s})) \leq end(p^{Back}_{v_{s},v_{d}}(t_{d})) - start(p^{Back}_{v_{s},v_{d}}(t_{d}))$. Therefore, Theorem 3 is proven. □

The proof of Lemma 1

Suppose that within $[t_{v_{i}}, t_{\beta }]$, there exists another path $p^{Back}_{v_{i}, v_{d}}(t^{\prime }_{i})$ from v_i to v_d, where $\delta (p^{Back}_{v_{i}, v_{d}}(t^{\prime }_{i}))<\delta (p^{Back}_{v_{i}, v_{d}}(t_{v_{i}}))$. Since $t^{\prime }_{i} \geq t_{v_{i}}$, we can have $p^{Back}_{v_{k}, v_{d}}(t^{\prime }_{k})$ by replacing $p^{Back}_{v_{i}, v_{d}}(t_{v_{i}})$ with $p^{Back}_{v_{i}, v_{d}} (t^{\prime }_{i})$, where $\delta (p^{Back}_{v_{k}, v_{d}}(t^{\prime }_{i}))<\delta (p^{Back}_{v_{k}, v_{d}}(t_{v_{k}}))$. But this contradicts the fact that $p^{Back}_{v_{i}, v_{d}}(t_{v_{i}})$ is a path with the minimal objective function value. Therefore Lemma 1 is proven. □

The proof of Lemma 2

Let $p^{Back}_{v_{i},v_{d}}(t^{\prime }_{i})$ be a suffix-subpath of $p^{Back}_{v_{s},v_{d}}(t^{\prime }_{s})$. Since $t_{i} \geq t^{\prime }_{i}$, we can use $p^{Back}_{v_{i},v_{d}}(t_{i})$ to replace $p^{Back}_{v_{i},v_{d}}(t^{\prime }_{i})$ to form a new temporal path as $p^{Back}_{v_{s},v_{d}}(t_{s})$. As $\delta (p^{Back}_{v_{i},v_{d}}(t_{i})) \leqslant \delta (p^{Back}_{v_{i},v_{d}}(t^{\prime }_{i}))$, $\delta (p_{v_{s},v_{d}}(t_{s})) \leqslant \delta (p_{v_{s},v_{d}}(t^{\prime }_{s}))$, and thus $p^{Back}_{v_{i},v_{d}}(t^{\prime }_{i})$ can be pruned without affecting δ_min and $\delta ^{L}_{min}$. Therefore Lemma 2 is proven. □

The proof of Lemma 3

Suppose $p^{E}_{v_{s}, v_{i}}(t_{s})$ is not the EA-TPMC from v_s to v_i. Then we can have another path $p^{E}_{v_{s}, v_{i}}(t^{\prime }_{s})$ that is the EA-TPMC. As $end(p^{E}_{v_{s}, v_{i}}(t^{\prime }_{s})) < end(p^{E}_{v_{s}, v_{i}} (t_{s}))$, we can have $p^{E}_{v_{s}, v_{d}}(t^{\prime }_{s})$ by combining $p^{E}_{v_{s}, v_{i}}(t^{\prime }_{s})$ and the path from v_i to v_d in $p^{E}_{v_{s}, v_{d}}(t_{s})$. Thus, $end(p^{E}_{v_{s}, v_{d}}(t^{\prime }_{s})) < end(p^{E}_{v_{s}, v_{d}}(t_{s}))$, which contradicts that $p^{E}_{v_{s}, v_{d}}(t_{s})$ is the EA-TPMC. Then Lemma 3 is proven. □