Lecture 7 All-Pairs Shortest Paths. All-Pairs Shortest Paths.
Dynamic Networks and Shortest Paths Takeshi Shirabe Technical University of Vienna.
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Transcript of Dynamic Networks and Shortest Paths Takeshi Shirabe Technical University of Vienna.
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Dynamic Networks and Shortest Paths
Takeshi ShirabeTechnical University of Vienna
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2Takeshi SHIRABE
Problem
/7
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
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3Takeshi SHIRABE
Problem
/7
wij = fij(t)
Time-dependent Networks
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w12 w24
w13 w34
w23
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4Takeshi SHIRABE
Problem
/7
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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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5Takeshi SHIRABE
Solution
/7
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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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6Takeshi SHIRABE
Application
/7
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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7Takeshi SHIRABE
Questions
/7
• Dynamic networks worth studying?• Any efficient solution or approximation methods?• Any applications?
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8Takeshi SHIRABE
Appendix
/7
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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9Takeshi SHIRABE
Examples
/7
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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?
1 2
3w23
Consider speed as the state…