Exploring earliest-arrival paths in large-scale time-dependent networks via combinatorial oracles Workshop on Algorithmic Aspects of Temporal Graphs II Patras, July 8 2019 Spyros Kontogiannis kontog@uoi.gr Joint work with: G. Papastavrou A. Papadopoulos A. Paraskevopoulos C. Zaroliagis D. Wagner CSE.UoI CEID.UPatras CEID.UPatras CEID.UPatras KIT
Time-Dependent Route Planning Time-Dependent Route Planning ...problem, assumptions and challenges... ...problem, assumptions and challenges... Spyros Kontogiannis (kontog@uoi.gr) Exploring earliest-arrival paths in large-scale TD networks via combinatorial oracles 2 / 37
Shortest Paths ... a fundamental problem, both in theory and in practice... o Input: ◮ Directed graph G = ( V , E ) . ◮ Arc-traversal-time values : D [ uv ] > 0. ◮ Origin-destination pair: ( o , d ) ∈ V × V . d Spyros Kontogiannis (kontog@uoi.gr) Exploring earliest-arrival paths in large-scale TD networks via combinatorial oracles 3 / 37
Shortest Paths ... a fundamental problem, both in theory and in practice... o Input: ◮ Directed graph G = ( V , E ) . ◮ Arc-traversal-time values : D [ uv ] > 0. ◮ Origin-destination pair: ( o , d ) ∈ V × V . � D [ π ] = � Output: π ∗ ∈ arg min π ∈ P o , d a ∈ π D [ a ] � d Spyros Kontogiannis (kontog@uoi.gr) Exploring earliest-arrival paths in large-scale TD networks via combinatorial oracles 3 / 37
Shortest Paths ... a fundamental problem, both in theory and in practice... o Input: ◮ Directed graph G = ( V , E ) . ◮ Arc-traversal-time values : D [ uv ] > 0. ◮ Origin-destination pair: ( o , d ) ∈ V × V . � D [ π ] = � Output: π ∗ ∈ arg min π ∈ P o , d a ∈ π D [ a ] � MOTIVATION & CHALLENGES: Routing in road networks . ◮ V = set of intersections, E = set of road segments. ◮ Non-planar , sparse ( | E | ∈ O ( | V | ) ) graphs. ◮ Very large size: | V | = millions of intersections. d Spyros Kontogiannis (kontog@uoi.gr) Exploring earliest-arrival paths in large-scale TD networks via combinatorial oracles 3 / 37
Shortest Paths ... a fundamental problem, both in theory and in practice... o Input: ◮ Directed graph G = ( V , E ) . ◮ Arc-traversal-time values : D [ uv ] > 0. ◮ Origin-destination pair: ( o , d ) ∈ V × V . � D [ π ] = � Output: π ∗ ∈ arg min π ∈ P o , d a ∈ π D [ a ] � MOTIVATION & CHALLENGES: Routing in road networks . ◮ V = set of intersections, E = set of road segments. ◮ Non-planar , sparse ( | E | ∈ O ( | V | ) ) graphs. ◮ Very large size: | V | = millions of intersections. d ...possibly the most characteristic success story of algorithm engineering... Numerous oracles and speedup techniques for static road networks. Spyros Kontogiannis (kontog@uoi.gr) Exploring earliest-arrival paths in large-scale TD networks via combinatorial oracles 3 / 37
Time-Dependent Shortest Paths ...a more challenging problem, both in theory and in practice ... o Input: ◮ Directed graph G = ( V , E ) . ◮ Arc-traversal-time functions : D [ uv ] : [ 0 , T ) �→ R > 0 . Assumption: Periodic, continuous, piecewise-linear, FIFO-compliant functions... ◮ Origin-destination-dep. time triple: ( o , d , t o ) ∈ V × V × [ 0 , T ) . d Spyros Kontogiannis (kontog@uoi.gr) Exploring earliest-arrival paths in large-scale TD networks via combinatorial oracles 4 / 37
Time-Dependent Shortest Paths ...a more challenging problem, both in theory and in practice ... o Input: ◮ Directed graph G = ( V , E ) . ◮ Arc-traversal-time functions : D [ uv ] : [ 0 , T ) �→ R > 0 . Assumption: Periodic, continuous, piecewise-linear, FIFO-compliant functions... ◮ Origin-destination-dep. time triple: ( o , d , t o ) ∈ V × V × [ 0 , T ) . Output: � D [ π ]( t o ) = � π ∗ ∈ arg min π ∈ P o , d a ∈ π D [ a ]( t o ) � d Spyros Kontogiannis (kontog@uoi.gr) Exploring earliest-arrival paths in large-scale TD networks via combinatorial oracles 4 / 37
Time-Dependent Shortest Paths ...a more challenging problem, both in theory and in practice ... o Input: ◮ Directed graph G = ( V , E ) . ◮ Arc-traversal-time functions : D [ uv ] : [ 0 , T ) �→ R > 0 . Assumption: Periodic, continuous, piecewise-linear, FIFO-compliant functions... ◮ Origin-destination-dep. time triple: ( o , d , t o ) ∈ V × V × [ 0 , T ) . Output: � D [ π ]( t o ) = � π ∗ ∈ arg min π ∈ P o , d a ∈ π D [ a ]( t o ) � MOTIVATION & CHALLENGES: Routing in road networks . d ◮ V = set of intersections, E = set of road segments. ◮ Non-planar , sparse ( | E | ∈ O ( | V | ) ) graphs. ◮ Very large size: | V | = millions of intersections. ◮ Time-Dependence : Computationally harder instances. Spyros Kontogiannis (kontog@uoi.gr) Exploring earliest-arrival paths in large-scale TD networks via combinatorial oracles 4 / 37
Time-Dependent Shortest Path: Examples Instance with ARC DELAY functions u u x+2 x+2 2x+0.1 2x+0.1 1 1 o d o d 3x 3x x+2 x+2 2x+0.1 2x+0.1 v v Q1 How would you commute as fast as possible from o to d , for a given departure time t o from o ? E.g.: Spyros Kontogiannis (kontog@uoi.gr) Exploring earliest-arrival paths in large-scale TD networks via combinatorial oracles 5 / 37
Time-Dependent Shortest Path: Examples 0.1 u u 2.1 x+2 0.1 2x+0.1 0 1 1 o d o d 0.3 3x 1.3 2 x+2 0.9 2x+0.1 v v 0.4 Q1 How would you commute as fast as possible from o to d , for a given departure time t o from o ? E.g.: t o = 0 Spyros Kontogiannis (kontog@uoi.gr) Exploring earliest-arrival paths in large-scale TD networks via combinatorial oracles 5 / 37
Time-Dependent Shortest Path: Examples 3.1 u u 5.1 x+2 2.1 2x+0.1 1 1 1 o d o d 9.3 3x 8.2 3 x+2 8.1 2x+0.1 v v 4 Q1 How would you commute as fast as possible from o to d , for a given departure time t o from o ? E.g.: t o = 1 Spyros Kontogiannis (kontog@uoi.gr) Exploring earliest-arrival paths in large-scale TD networks via combinatorial oracles 5 / 37
Time-Dependent Shortest Path: Examples Instance with ARC DELAY functions u u x+2 x+2 2x+0.1 2x+0.1 1 1 o d o d 3x 3x x+2 x+2 2x+0.1 2x+0.1 v v Q1 How would you commute as fast as possible from o to d , for a given departure time t o from o ? E.g.: Q2 What if you are not sure about the departure time? Spyros Kontogiannis (kontog@uoi.gr) Exploring earliest-arrival paths in large-scale TD networks via combinatorial oracles 5 / 37
Time-Dependent Shortest Path: Examples Instance with ARC- DELAY ARRIVAL functions u u 2x+2 x+2 3x+0.1 2x+0.1 x+1 1 o d o d 4x 3x 2x+2 x+2 3x+0.1 2x+0.1 v v Q1 How would you commute as fast as possible from o to d , for a given departure time t o from o ? E.g.: Q2 What if you are not sure about the departure time? Spyros Kontogiannis (kontog@uoi.gr) Exploring earliest-arrival paths in large-scale TD networks via combinatorial oracles 5 / 37
Time-Dependent Shortest Path: Examples Instance with ARC- DELAY ARRIVAL functions Arr[ovud](t o ) = Arr[ud](Arr[vu](Arr[ov](t o ))) = 4t o +8 Arr[oud](t o ) = Arr[ud](Arr[ou](t o )) = 6t o + 2.2 u u 2x+2 2x+2 3x+0.1 3x+0.1 x+1 x+1 o d o d 4x 4x 2x+2 2x+2 3x+0.1 3x+0.1 v v Arr[ovd](t o ) = Arr[vd](Arr[ov](t o )) = 6t o + 6.1 Arr[ouvd](t o ) = Arr[vd](Arr[uv](Arr[ou](t o ))) = 36t o +1.3 Q1 How would you commute as fast as possible from o to d , for a given departure time t o from o ? E.g.: Q2 What if you are not sure about the departure time? Spyros Kontogiannis (kontog@uoi.gr) Exploring earliest-arrival paths in large-scale TD networks via combinatorial oracles 5 / 37
Time-Dependent Shortest Path: Examples Instance with ARC- DELAY ARRIVAL functions Arr[ovud](t o ) = Arr[ud](Arr[vu](Arr[ov](t o ))) = 4t o +8 Arr[oud](t o ) = Arr[ud](Arr[ou](t o )) = 6t o + 2.2 u u 2x+2 2x+2 3x+0.1 3x+0.1 x+1 x+1 o d o d 4x 4x 2x+2 2x+2 3x+0.1 3x+0.1 v v Arr[ovd](t o ) = Arr[vd](Arr[ov](t o )) = 6t o + 6.1 Arr[ouvd](t o ) = Arr[vd](Arr[uv](Arr[ou](t o ))) = 36t o +1.3 Q1 How would you commute as fast as possible from o to d , for a given departure time t o from o ? E.g.: Q2 What if you are not sure about the departure time? t o ∈ [ 0 , 0 . 03 ) orange path , earliest-arrival (path) function = t o ∈ [ 0 . 03 , 2 . 9 ) A yellow path , t o ∈ [ 2 . 9 , + ∞ ) purple path , Spyros Kontogiannis (kontog@uoi.gr) Exploring earliest-arrival paths in large-scale TD networks via combinatorial oracles 5 / 37
Time-Dependent Shortest Path: Definitions INPUT: Directed graph G = ( V , A ) , n = | V | . v D[uv](t u ) = Arr[uv](t u ) u Arc travel-time / arrival-time functions: = t u + D[uv](t u ) D [ uv ]( t u ) Arr [ uv ]( t u ) = t u + D [ uv ]( t u ) Spyros Kontogiannis (kontog@uoi.gr) Exploring earliest-arrival paths in large-scale TD networks via combinatorial oracles 6 / 37
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