Solving Evacuation Problems in Polynomial Space Miriam Schlter & - PowerPoint PPT Presentation

Solving Evacuation Problems in Polynomial Space Miriam Schlöter & Martin Skutella √ Solving Evacuation Problems in Polynomial Space

Flows over Time Definition Solving Evacuation Problems in Polynomial Space

Flows over Time Definition Many real life problems crucially depend on time : - logistic - public transport - evacuation problems Solving Evacuation Problems in Polynomial Space

Flows over Time Definition Many real life problems crucially depend on time : - logistic - public transport - evacuation problems Flows over time are like… …like classical (static) network flows + time component : • flow needs time to travel through an arc a : arc a has a transit time 𝜐 a (length) • a bounded amount of flow can enter an arc a per time unit: arc a has a capacity u a (width) Solving Evacuation Problems in Polynomial Space

Flows over Time Definition Solving Evacuation Problems in Polynomial Space

Flows over Time Definition dynamic network 𝒪 = ( G =( V,A ) ,u, 𝜐 ,S + ,S - ): digraph G = ( V,A ) with capacities u, transit times 𝜐 , sources S + ⊂ V and sinks S - ⊂ V Solving Evacuation Problems in Polynomial Space

Flows over Time Definition dynamic network 𝒪 = ( G =( V,A ) ,u, 𝜐 ,S + ,S - ): digraph G = ( V,A ) with capacities u, transit times 𝜐 , sources S + ⊂ V and sinks S - ⊂ V t = 0 𝒪 2 2 1 1 2 2 Solving Evacuation Problems in Polynomial Space

Flows over Time Definition dynamic network 𝒪 = ( G =( V,A ) ,u, 𝜐 ,S + ,S - ): digraph G = ( V,A ) with capacities u, transit times 𝜐 , sources S + ⊂ V and sinks S - ⊂ V t = 1 𝒪 2 2 1 1 2 2 Solving Evacuation Problems in Polynomial Space

Flows over Time Definition dynamic network 𝒪 = ( G =( V,A ) ,u, 𝜐 ,S + ,S - ): digraph G = ( V,A ) with capacities u, transit times 𝜐 , sources S + ⊂ V and sinks S - ⊂ V t = 2 𝒪 2 2 1 1 2 2 Solving Evacuation Problems in Polynomial Space

Flows over Time Definition dynamic network 𝒪 = ( G =( V,A ) ,u, 𝜐 ,S + ,S - ): digraph G = ( V,A ) with capacities u, transit times 𝜐 , sources S + ⊂ V and sinks S - ⊂ V t = 3 𝒪 2 2 1 1 2 2 Solving Evacuation Problems in Polynomial Space

Flows over Time Definition dynamic network 𝒪 = ( G =( V,A ) ,u, 𝜐 ,S + ,S - ): digraph G = ( V,A ) with capacities u, transit times 𝜐 , sources S + ⊂ V and sinks S - ⊂ V t = 4 𝒪 2 2 1 1 2 2 Solving Evacuation Problems in Polynomial Space

Flows over Time Definition dynamic network 𝒪 = ( G =( V,A ) ,u, 𝜐 ,S + ,S - ): digraph G = ( V,A ) with capacities u, transit times 𝜐 , sources S + ⊂ V and sinks S - ⊂ V t = 5 𝒪 2 2 1 1 2 2 Solving Evacuation Problems in Polynomial Space

Flows over Time Definition dynamic network 𝒪 = ( G =( V,A ) ,u, 𝜐 ,S + ,S - ): digraph G = ( V,A ) with capacities u, transit times 𝜐 , sources S + ⊂ V and sinks S - ⊂ V t = 6 𝒪 2 2 1 1 2 2 Solving Evacuation Problems in Polynomial Space

Flows over Time Definition dynamic network 𝒪 = ( G =( V,A ) ,u, 𝜐 ,S + ,S - ): digraph G = ( V,A ) with capacities u, transit times 𝜐 , sources S + ⊂ V and sinks S - ⊂ V t = 7 𝒪 2 2 1 1 2 2 Solving Evacuation Problems in Polynomial Space

Flows over Time Definition dynamic network 𝒪 = ( G =( V,A ) ,u, 𝜐 ,S + ,S - ): digraph G = ( V,A ) with capacities u, transit times 𝜐 , sources S + ⊂ V and sinks S - ⊂ V t = 8 𝒪 2 2 1 1 2 2 Solving Evacuation Problems in Polynomial Space

Earliest Arrival Flows single source / single sink Given: 𝒪 = ( G = ( V , A ), u , 𝜐 , s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink Solving Evacuation Problems in Polynomial Space

Earliest Arrival Flows single source / single sink Given: 𝒪 = ( G = ( V , A ), u , 𝜐 , s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐 ( P ) ≤ T occurring in successive shortest path algorithm from s to t T = 8 1 s t 2 Solving Evacuation Problems in Polynomial Space

Earliest Arrival Flows single source / single sink Given: 𝒪 = ( G = ( V , A ), u , 𝜐 , s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐 ( P ) ≤ T occurring in successive shortest path algorithm from s to t T = 8 1 s t 2 Solving Evacuation Problems in Polynomial Space

Earliest Arrival Flows single source / single sink Given: 𝒪 = ( G = ( V , A ), u , 𝜐 , s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐 ( P ) ≤ T occurring in successive shortest path algorithm from s to t T = 8 1 s t 2 Solving Evacuation Problems in Polynomial Space

Earliest Arrival Flows single source / single sink Given: 𝒪 = ( G = ( V , A ), u , 𝜐 , s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐 ( P ) ≤ T occurring in successive shortest path algorithm from s to t T = 8 1 s t 2 Solving Evacuation Problems in Polynomial Space

Earliest Arrival Flows single source / single sink Given: 𝒪 = ( G = ( V , A ), u , 𝜐 , s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐 ( P ) ≤ T occurring in successive shortest path algorithm from s to t T = 8 1 s t 2 Solving Evacuation Problems in Polynomial Space

Earliest Arrival Flows single source / single sink Given: 𝒪 = ( G = ( V , A ), u , 𝜐 , s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐 ( P ) ≤ T occurring in successive shortest path algorithm from s to t T = 8 1 s t 2 Solving Evacuation Problems in Polynomial Space

Earliest Arrival Flows single source / single sink Given: 𝒪 = ( G = ( V , A ), u , 𝜐 , s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐 ( P ) ≤ T occurring in successive shortest path algorithm from s to t T = 8 1 s t 2 Solving Evacuation Problems in Polynomial Space

Earliest Arrival Flows single source / single sink Given: 𝒪 = ( G = ( V , A ), u , 𝜐 , s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐 ( P ) ≤ T occurring in successive shortest path algorithm from s to t T = 8 1 s t 2 Solving Evacuation Problems in Polynomial Space

Earliest Arrival Flows single source / single sink Given: 𝒪 = ( G = ( V , A ), u , 𝜐 , s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐 ( P ) ≤ T occurring in successive shortest path algorithm from s to t T = 8 1 s t 2 Solving Evacuation Problems in Polynomial Space

Earliest Arrival Flows single source / single sink Given: 𝒪 = ( G = ( V , A ), u , 𝜐 , s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐 ( P ) ≤ T occurring in successive shortest path algorithm from s to t T = 8 1 s t 2 Solving Evacuation Problems in Polynomial Space

Earliest Arrival Flows single source / single sink Given: 𝒪 = ( G = ( V , A ), u , 𝜐 , s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐 ( P ) ≤ T occurring in successive shortest path algorithm from s to t T = 8 1 s t 2 Solving Evacuation Problems in Polynomial Space

Earliest Arrival Flows single source / single sink Given: 𝒪 = ( G = ( V , A ), u , 𝜐 , s, t ) and time horizon T Aim: Flow over time f in 𝒪 s.t. at each point in time θ ≤ T as much flow as possible has reached the sink Solution [Minieka ’71, Wilkinson ’71]: Send flow along paths P with 𝜐 ( P ) ≤ T occurring in successive shortest path algorithm from s to t T = 8 1 s t 2 Solving Evacuation Problems in Polynomial Space

Earliest Arrival Flows multiple sources / single sink Given: 𝒪 = ( G = ( V , A ), u , 𝜐 , S + , t ) with supplies v on the sources Aim: Flow over time f in 𝒪 that respects the supplies such that at each point in time as much flow as possible has reached the sink Solving Evacuation Problems in Polynomial Space