Finding Temporal Paths under Waiting Time Constraints Philipp - PowerPoint PPT Presentation

Finding Temporal Paths under Waiting Time Constraints Philipp Zschoche TU Berlin July 7 2020, Algorithmic Aspects of Temporal Graphs III Based on joint work with Arnaud Casteigts, Anne-Sophie Himmel, and Hendrik Molter.

Motivational Example: Disease Control Source: The University of Hong Kong Philipp Zschoche (TU Berlin) τ := lifetime 2 / 12

Motivational Example: Disease Control Source: The University of Hong Kong Infectious diseases are often transmitted via physical contact. Philipp Zschoche (TU Berlin) τ := lifetime 2 / 12

Motivational Example: Disease Control Source: The University of Hong Kong Infectious diseases are often transmitted via physical contact. Contact Tracing : (1) Identify and isolate infected persons (2) Isolate all potentially infected persons (by known cases). Philipp Zschoche (TU Berlin) τ := lifetime 2 / 12

Motivational Example: Disease Control Source: The University of Hong Kong Infectious diseases are often transmitted via physical contact. Contact Tracing : (1) Identify and isolate infected persons (2) Isolate all potentially infected persons (by known cases). At time of identification: a person may started long infection chains . Philipp Zschoche (TU Berlin) τ := lifetime 2 / 12

Motivational Example: Disease Control Can s have (indirectly) infected z ? Static graph: z s Philipp Zschoche (TU Berlin) τ := lifetime 3 / 12

Motivational Example: Disease Control Can s have (indirectly) infected z ? Static graph: z s Day 1: z s Philipp Zschoche (TU Berlin) τ := lifetime 3 / 12

Motivational Example: Disease Control Can s have (indirectly) infected z ? Static graph: z s Day 1: z s Philipp Zschoche (TU Berlin) τ := lifetime 3 / 12

Motivational Example: Disease Control Can s have (indirectly) infected z ? Static graph: z s Day 1: Day 2: z z s s Philipp Zschoche (TU Berlin) τ := lifetime 3 / 12

Motivational Example: Disease Control Can s have (indirectly) infected z ? Static graph: z s Day 1: Day 2: Day 3: z z z s s s Philipp Zschoche (TU Berlin) τ := lifetime 3 / 12

Motivational Example: Disease Control Can s have (indirectly) infected z ? Static graph: z s Day 1: Day 2: Day 3: z z z s s s ⇒ Time information is crucial for infection transmission routes. Philipp Zschoche (TU Berlin) τ := lifetime 3 / 12

Temporal Graphs – Formal Definition A temporal graph G = ( V , ( E i ) i ∈ [ τ ] ) is defined as vertex set V with a list of edge sets E 1 ,..., E τ over V , where τ is the lifetime of G . Philipp Zschoche (TU Berlin) τ := lifetime 4 / 12

Temporal Graphs – Formal Definition A temporal graph G = ( V , ( E i ) i ∈ [ τ ] ) is defined as vertex set V with a list of edge sets E 1 ,..., E τ over V , where τ is the lifetime of G . 2 3 G : 1 1 2 3 1 , 2 Philipp Zschoche (TU Berlin) τ := lifetime 4 / 12

Temporal Graphs – Formal Definition A temporal graph G = ( V , ( E i ) i ∈ [ τ ] ) is defined as vertex set V with a list of edge sets E 1 ,..., E τ over V , where τ is the lifetime of G . 2 3 G : 1 1 2 3 1 , 2 G 1 : layers Philipp Zschoche (TU Berlin) τ := lifetime 4 / 12

Temporal Graphs – Formal Definition A temporal graph G = ( V , ( E i ) i ∈ [ τ ] ) is defined as vertex set V with a list of edge sets E 1 ,..., E τ over V , where τ is the lifetime of G . 2 3 G : 1 1 2 3 1 , 2 G 1 : G 2 : layers Philipp Zschoche (TU Berlin) τ := lifetime 4 / 12

Temporal Graphs – Formal Definition A temporal graph G = ( V , ( E i ) i ∈ [ τ ] ) is defined as vertex set V with a list of edge sets E 1 ,..., E τ over V , where τ is the lifetime of G . 2 3 G : 1 1 2 3 1 , 2 G 1 : G 2 : G 3 : layers Philipp Zschoche (TU Berlin) τ := lifetime 4 / 12

Temporal Graphs – Formal Definition A temporal graph G = ( V , ( E i ) i ∈ [ τ ] ) is defined as vertex set V with a list of edge sets E 1 ,..., E τ over V , where τ is the lifetime of G . 2 3 G : 1 1 2 3 1 , 2 G 1 : G 2 : G 3 : G ↓ : layers underlying graph Philipp Zschoche (TU Berlin) τ := lifetime 4 / 12

Motivation: Disease Spreading Source: Robert Koch-Institut Philipp Zschoche (TU Berlin) τ := lifetime 5 / 12

Motivation: Disease Spreading S usceptible Source: Robert Koch-Institut Philipp Zschoche (TU Berlin) τ := lifetime 5 / 12

Motivation: Disease Spreading S usceptible I nfectious Source: Robert Koch-Institut Philipp Zschoche (TU Berlin) τ := lifetime 5 / 12

Motivation: Disease Spreading S usceptible I nfectious R ecovered / R esistent Source: Robert Koch-Institut Philipp Zschoche (TU Berlin) τ := lifetime 5 / 12

Motivation: Disease Spreading S usceptible • Infectious period: 5 days. I nfectious R ecovered / R esistent Source: Robert Koch-Institut Philipp Zschoche (TU Berlin) τ := lifetime 5 / 12

Motivation: Disease Spreading S usceptible • Infectious period: 5 days. I nfectious R ecovered / R esistent Source: Robert Koch-Institut Day 1: z s Philipp Zschoche (TU Berlin) τ := lifetime 5 / 12

Motivation: Disease Spreading S usceptible • Infectious period: 5 days. I nfectious R ecovered / R esistent Source: Robert Koch-Institut Day 1: Day 4: z z s s Philipp Zschoche (TU Berlin) τ := lifetime 5 / 12

Motivation: Disease Spreading S usceptible • Infectious period: 5 days. I nfectious R ecovered / R esistent Source: Robert Koch-Institut Day 1: Day 4: Day 7: z z z s s s Philipp Zschoche (TU Berlin) τ := lifetime 5 / 12

Motivation: Disease Spreading S usceptible • Infectious period: 5 days. I nfectious • Disease spreads along paths with bounded waiting times . R ecovered / R esistent Source: Robert Koch-Institut Day 1: Day 4: Day 7: z z z s s s Philipp Zschoche (TU Berlin) τ := lifetime 5 / 12

Restless Paths – Formal Definition A temporal ( s , z ) -path is a list of edges labeled with non-decreasing time steps that Philipp Zschoche (TU Berlin) τ := lifetime 6 / 12

Restless Paths – Formal Definition A temporal ( s , z ) -path is a list of edges labeled with non-decreasing time steps that • the edges form an ( s , z ) -path in the underlying graph Philipp Zschoche (TU Berlin) τ := lifetime 6 / 12

Restless Paths – Formal Definition A ∆ -restless temporal ( s , z ) -path is a list of edges labeled with non-decreasing time steps that • the edges form an ( s , z ) -path in the underlying graph and • consecutive time steps differ by at most ∆ . Philipp Zschoche (TU Berlin) τ := lifetime 6 / 12

Restless Paths – Formal Definition A ∆ -restless temporal ( s , z ) -path is a list of edges labeled with non-decreasing time steps that • the edges form an ( s , z ) -path in the underlying graph and • consecutive time steps differ by at most ∆ . temporal ( s , z ) -paths: 1-restless temporal ( s , z ) -path: 2 3 2 3 s z s z 1 1 2 3 1 1 2 3 1 1 Philipp Zschoche (TU Berlin) τ := lifetime 6 / 12

Restless Paths – Formal Definition A ∆ -restless temporal ( s , z ) -path is a list of edges labeled with non-decreasing time steps that • the edges form an ( s , z ) -path in the underlying graph and • consecutive time steps differ by at most ∆ . temporal ( s , z ) -paths: 1-restless temporal ( s , z ) -path: 2 3 2 3 s z s z 1 1 2 3 1 1 2 3 1 1 Philipp Zschoche (TU Berlin) τ := lifetime 6 / 12

Restless Paths – Formal Definition A ∆ -restless temporal ( s , z ) -path is a list of edges labeled with non-decreasing time steps that • the edges form an ( s , z ) -path in the underlying graph and • consecutive time steps differ by at most ∆ . temporal ( s , z ) -paths: 1-restless temporal ( s , z ) -path: 2 3 2 3 s z s z 1 1 2 3 1 1 2 3 1 1 Philipp Zschoche (TU Berlin) τ := lifetime 6 / 12

Restless Paths – Formal Definition A ∆ -restless temporal ( s , z ) -path is a list of edges labeled with non-decreasing time steps that • the edges form an ( s , z ) -path in the underlying graph and • consecutive time steps differ by at most ∆ . temporal ( s , z ) -paths: 1-restless temporal ( s , z ) -path: 2 3 2 3 s z s z 1 1 2 3 1 1 2 3 1 1 Philipp Zschoche (TU Berlin) τ := lifetime 6 / 12

Restless Paths – Formal Definition A ∆ -restless temporal ( s , z ) -path is a list of edges labeled with non-decreasing time steps that • the edges form an ( s , z ) -path in the underlying graph and • consecutive time steps differ by at most ∆ . temporal ( s , z ) -paths: 1-restless temporal ( s , z ) -path: 2 3 2 3 s z s z 1 1 2 3 1 1 2 3 1 1 • Temporal Paths: Xuan et al. [IJFCS ’03], Wu et al. [IEEE TKDE ’16] Philipp Zschoche (TU Berlin) τ := lifetime 6 / 12

Restless Paths – Formal Definition A ∆ -restless temporal ( s , z ) -path is a list of edges labeled with non-decreasing time steps that • the edges form an ( s , z ) -path in the underlying graph and • consecutive time steps differ by at most ∆ . temporal ( s , z ) -paths: 1-restless temporal ( s , z ) -path: 2 3 2 3 s z s z 1 1 2 3 1 1 2 3 1 1 • Temporal Paths: Xuan et al. [IJFCS ’03], Wu et al. [IEEE TKDE ’16] • Restless Temporal Walks: Himmel et al. [Complex Networks ’19] Philipp Zschoche (TU Berlin) τ := lifetime 6 / 12