Dynamic Networks and Shortest Paths

Takeshi ShirabeTechnical University of Vienna

2Takeshi SHIRABE

Problem

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wij’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

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w12 w24

w13 w34

w23

3Takeshi SHIRABE

Problem

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wij = fij(t)

Time-dependent Networks

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w12 w24

w13 w34

w23

4Takeshi SHIRABE

Problem

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1

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w12 w24

w13 w34

w23wij = fij(s(i),s(j))

s(j) = gij(s(i))

Dynamic Networks

where s(i) is some state of a traveler at i

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Solution

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w12 w24

w13 w34

w23

1. Limit possible states to a finite set of values.

2 21 2 3 2 4 2

2 31 3 3 3 4 3

2 11 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}

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Application

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Minimum Work Paths in Elevated Networks

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w12 w24

w13 w34

w23

s(j): level of kinetic energy at j max(s(i)-uij-rij, 0)

wij: amount of work required for moving from i to j

max(uij+rij-s(i), 0)

uij: change in gravitational potential energy when moving from i to j

rij: loss of energy from friction when moving from i to j

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Questions

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• Dynamic networks worth studying?• Any efficient solution or approximation methods?• Any applications?

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Appendix

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θmg

μmgcosθ mgcosθ

yj

yi

xij

i

j

i i

uij = mg(yj-yi)

rij = μmgcosθ(xij/cosθ) = μmgxij

xij: horizontal distance from i to jyi: height of iθ: incline of arc (i,j); tanθ = (yj-yi)/xijm: mass of the travelerg: coefficient of gravitationμ: coefficient of friction

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Examples

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w23

1. What if arc (2,3) is approached with excessive speed?

2. What if arc (2,3) is approached with insufficient speed?

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3w23

Consider speed as the state…