dynamic flows dynamic transshipment & evolving graphs 2/28/2012 tcs group seminar 1
TRANSCRIPT
TCS Group Seminar
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Dynamic FlowsDynamic Transshipment&Evolving Graphs
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Seminar outline
Earliest Arrival Flows• reminder & example• evacuation problems
Dynamic Transshipment & Evolving Graphs
Lexicographically Maximal Flows
Push-Relabel framework
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Earliest Arrival Flows
•Example
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Earliest Arrival Flows
•Time-expanded Graph▫(Ford-Fulkerson ’58)
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t=1t=2t=3t=4t=5
t=7
t=0
t=6
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Earliest Arrival Flows
1. Compute distance labels in residual graph
1. it defines a cut
2. no augmenting path can arrive before
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t=1t=2t=3t=4t=5
t=7
t=0
t=6
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0 1 2 3
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Earliest Arrival Flows
1. Compute distance labels in residual graph
2. Add shortest path
3. Repeat
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t=1t=2t=3t=4t=5
t=7
t=0
t=6
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0 3 4 70 1 2 3
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Earliest Arrival Flows
1. Compute distance labels in residual graph
2. Add shortest path
3. Repeat
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t=1t=2t=3t=4t=5
t=7
t=0
t=6
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0 3 4 7
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Earliest Arrival Flows
•Several sources: evacuation problem
•See: works from Skutella, Minieka, and students.
•Maybe interesting for extracting maximum information from a short-lived WSN▫battlezone▫vulcano, nuclear reactor...
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Seminar outline
Earliest Arrival Flows
Dynamic Transshipment & Evolving Graphs• definitions• equivalence• submodularity
Lexicographically Maximal Flows
Push-Relabel framework
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Dynamic transshipment
•Several sources with a fixed supply•Several sinks with a fixed demand
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S2
A B S-S1
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Evolving graph
•Edges have a schedule [t1;t2], [t3,t4],...
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Dynamic transshipment= Flow in evolving graph
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S2
A B S-S1
[-2;0]
[-3;0]
S
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Flow in evolving graph= dynamic transshipment
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[t1;t2]
•similar to capacitated max flow = uncapacitated transshipment in static graphs.
demand: t2-t1delay: T-t2
supply: t2-t1delay: t1
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Submodularity
•The dual of the dynamic transshipment problem is to find a subset of sources/sinks and a minimum cut in the time-expanded graph between those subsets.▫(solve a min-cost flow for each
sources/sinks subset)•The min-cut function is submodular on
sources/sinks subsets.
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Submodularity
•Minimizing a submodular function can be done with a variant of the Ellipsoid method▫convex function on convex sets▫P, but not practical
•Test feasibility of dynamic transshipment with a submodular oracle (Hoppe&Tardos ’95)
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Seminar outline
Earliest Arrival Flows
Dynamic Transshipment & Evolving Graphs
Lexicographically Maximal Flows• definition• algorithm• building a solution
Push-Relabel framework
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Lexicographically Maximal Flows•Given a sequence of sources/sinks
▫(a,b,c,d,e...)•A lexicographically maximal flow
maximizes the amount of flow▫from a to (b,c,d,e...)▫from (a,b) to (c,d,e...)▫from (a,b,c) to (d,e...)▫etc.
•It exists and is easily computable (Megiddo ’74)
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Lexicographically maximaldynamic flows1. Put distance labels at 0 for sources and
at T for sinks2. Compute min-cost flow (= max dynamic
flow) from {a,b,...,x,y} to {z}3. Compute min-cost augmenting flow from
{a,b,...,x} to {y,z}......................................................................
..........27.Compute min-cost augmenting flow
from {a} to {b,...,x,y,z}
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Lexicographically maximaldynamic flows•At each step, the subset of sources
decreases▫distance labels can only increase▫augmenting flows yield a valid dynamic
solution
•The labels at a given step indicate a minimum cut for the current subset of sources▫the final solution saturates that cut▫the actual proof is rather technical (see
Hoppe&Tardos ’00)
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Solution with submodular oracle• Do a complex dichotomic search with
the help of the oracle in order to1. restrict the capacities of edges that exit the
sources/enter the sinks2. order the sources and sinks
• The obtained lexicographically maximal dynamic flow answers the dynamic transshipment problem
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Seminar outline
Earliest Arrival Flows
Dynamic Transshipment & Evolving Graphs
Lexicographically Maximal Flows
Push-relabel framework• similarities & problems• fractional solution• integral solution
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Push-relabel similarities & problems •A lexicographically maximal flow is
actually a giant saturating push with labels▫a:26, b:25, c:24, d:23, ...., z:1
•Idea: dynamic push-relabel algorithm•Difficulties:
▫non-saturating pushes▫several vertices at the same level
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Push-relabel similarities & problems •Non-saturating push problem
▫Having a minimum cost flow (= maximum dynamic flow) is vital for coherent distance labels and coherent solution
•Same level problem:▫pushing from a vertex may send flow to other vertices at same level a,b,{c,d,e},f
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Push-relabel framework
•All sources & sinks start at potential 0.•The algorithm maintains a
lexicographically maximal dynamic flow from potential 26 down to potential 0
•When a node has excess flow, increase its potential by 1 and▫recompute the lex-max dynamic flow▫(1 min-cost flow computation)▫= saturating push
•What if it’s too much ???
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Fractional push-relabel•Fractional push
▫a node is at potential P in 0.72 of the solution and at potential P+1 in 0.28 of the solution
•When a node has excess flow, try to increase its potential to a full number. ▫if it still has excess flow, fine.▫if it has a deficit, make a linear combination of
(full push/no action) to have zero excess•Nodes on a same level:
▫find a linear combination for all nodes (doable)
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Fractional push-relabel• Min-cost flow corresponding to potential P/P+1
is not affected by fractions of other potentials▫ (a,{a,b(0.27)},{a,b,c(0.3),d(0.5)},{a,b,c,d,e})
• At a given level, try to push all potentials to full number:▫ (a,{a,b(0.27)},{a,b,c,d},{a,b,c,d,e})▫effect is c:+4 unit, d -1 unit, e-3 units▫select a fraction so that c and d are non negative,
and c or d is at zero▫push the other node alone.
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Staggered push-relabel
•Natural approach for integral solutions:▫dichotomic search on source/sink
capacities (i.e. size of the hose)•A node has full capacity at potential P, and
partial capacity at potential P+1:▫(a, {a,b},{a,b,c(partial)},{a,b,c,d})
•Problem with multiple nodes at same level:▫multiple dichotomic search is actually
exponential.
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Staggered push-relabel
•Assign unique potentials to each node:▫a: 0, 52,
104,▫b: 0,1, 51, 53, 103,▫c: 0, 2, 50, 54, 102,
•Maximum number of pushes unchanged▫(still 26 per node)
•Saturating push: increase node level.•Non-saturating push: increase node
capacity on top level
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Conclusion
•It is possible to augment/modify a dynamic flow under the condition of strictly increasing distance labels.
•A lex-max dynamic flow is actually a configuration in a push/relabel scheme.
•Non-saturating push can be done while maintaining feasibility by:1. using fractional solutions2. using unique potentials and restricted
capacities
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