1. Dynamic Networks and Shortest Paths Takeshi Shirabe Technical University of Vienna

2. 2 Takeshi SHIRABE Problem /7 w ij ’s are constant. Given a network, find a sequence of arcs from a source node to a sink node that has the minimum total arc weight. Shortest Path Problem 1 2 3 4 w 12 w 24 w 13 w 34 w 23

3. 3 Takeshi SHIRABE Problem /7 w ij = f ij (t) Time-dependent Networks 1 2 3 4 w 12 w 24 w 13 w 34 w 23

4. 4 Takeshi SHIRABE Problem /7 1 2 3 4 w 12 w 24 w 13 w 34 w 23 w ij = f ij (s(i),s(j)) s(j)= g ij (s(i)) Dynamic Networks where s(i) is some state of a traveler at i

5. 5 Takeshi SHIRABE Solution /7 1 2 3 4 w 12 w 24 w 13 w 34 w 23 1.Limit possible states to a finite set of values. 2 1 2 3 2 4 2 2 3 1 3 3 4 3 2 1 1 3 1 4 1  3.Draw an arc for each pair of connectable nodes and assign it a weight. 2.Duplicate each node as many as those states. S={1,2,3}

6. 6 Takeshi SHIRABE Application /7 Minimum Work Paths in Elevated Networks 1 2 3 4 w 12 w 24 w 13 w 34 w 23 s(j): level of kinetic energy at j max(s(i)-u ij -r ij, 0) w ij : amount of work required for moving from i to j max(u ij +r ij -s(i), 0) u ij : change in gravitational potential energy when moving from i to j r ij :loss of energy from friction when moving from i to j

7. 7 Takeshi SHIRABE Questions /7 Dynamic networks worth studying? Any efficient solution or approximation methods? Any applications?

8. 8 Takeshi SHIRABE Appendix /7 θ mg μmgcosθ mgcosθ yjyj yiyi x ij i j ii u ij = mg(y j -y i ) r ij = μmgcosθ(x ij /cosθ) = μmgx ij x ij : horizontal distance from i to j y i : height of i θ: incline of arc (i,j); tanθ = (y j -y i )/x ij m: mass of the traveler g: coefficient of gravitation μ: coefficient of friction

9. 9 Takeshi SHIRABE Examples /7 1 2 3 w 23 1.What if arc (2,3) is approached with excessive speed? 2.What if arc (2,3) is approached with insufficient speed? 1 2 3 w 23 Consider speed as the state…