1. Flight Itinerary Problem ICS 311 Fall 2006 Matt Freeburg

2. Problem Formulation: Graph  Airports as vertices  Flights as directed edges  Flight time as edge weight ORD LAX HNL 8 hrs. 54 min.

3. Advantages  Easy to understand  Existing mathematics  Existing algorithms  Existing implementations

4. Special Considerations  Edge weight is more than flight time  Later flights cannot connect to earlier flights  Two flights without enough layover time cannot connect either

5. Input  A graph G = (V,E)  A set of vertices V, one for each airport  A set of edges E, one for each flight  A starting vertex v s, an ending vertex v f, and a starting time t s

6. Input (cont.)  A function A(e) that returns the airline a i for edge e  A function T d (e) that returns the departure time for edge e  A function T a (e) that returns the arrival time for edge e  A function C(v, a i, a j ) that returns the minimum layover time, C ≥ 0

7. Input (cont.)  The weight of an edge f is defined as the sum of the flight time and the layover time, given by  w(f) = ((T a (f) – T d (f)) + (T d (f) – T a (e))  where e is the previous edge in a path  w(f) reduces to  w(f) = T a (f) – T a (e)

8. Input Constraint  Two edges e and f are considered adjacent if and only if  e = (x, y) and f = (y, z)  T d (f) – T a (e) ≥ C(y, A(e), A(f))

9. Output  A path P = {e 1, e 2, e 3, …,e m } consisting of m edges, beginning at v s and ending at v f, where the total weight of the path is minimized among all such possible paths

10. Output (cont.)  The total weight of P is given by m  W(P) = ∑ ( T a (e i ) – T a (e i-1 ) ) i = 1 i = 1  where T a (e 0 ) = t s