International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
Competitive Maintenance of
Minimum Spanning Trees in
Dynamic Graphs
Miroslaw Dynia
Miroslaw Korzeniowski
Jaroslaw Kutylowski
Jaroslaw Kutylowski
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International Graduate
School of Dynamic
Intelligent Systems
The Problem
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
Graph contains edges of same weight
many different Minimum Spanning Trees possible
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Jaroslaw Kutylowski
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The Question
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
Are we able to choose the Minimum Spanning Tree
so that we are not forced to change it very often?
(assuming that the underlying graph changes)
Jaroslaw Kutylowski
3
Model
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
Modelled as an online problem
• Adversary – Dynamics in Graph
– Changes weight of edges in rounds
– In one round changes one edge by +1/-1
• Algorithm
– Choose Minimum Spanning Tree after each round
• Cost Measure
– Number of changed edges between MSTs in
consecutive rounds
Jaroslaw Kutylowski
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International Graduate
School of Dynamic
Intelligent Systems
The Problem
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
• Algorithm uses red MST – cost 1
• Algorithm uses green MST – cost 0
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Jaroslaw Kutylowski
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Motivation
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
• General idea: how MSTs behave on dynamic graphs
– Data structures known
– No competitive analysis
– Important when computation is cheap, changes are
costly
• Concrete scenario
– Robots on large area
– Robots in teams
– Teams need communication with each other
– Cost of changing communication paths extremely large
Jaroslaw Kutylowski
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International Graduate
School of Dynamic
Intelligent Systems
Motivation
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
Communication
paths with relay
stations
Network
component
Jaroslaw Kutylowski
7
International Graduate
School of Dynamic
Intelligent Systems
Motivation
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
• Adversary changes weights of edges
– Network components move
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Jaroslaw Kutylowski
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International Graduate
School of Dynamic
Intelligent Systems
Motivation
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
• Algorithm has to maintain a MST
– Number of relay stations used is proportional to
length of edge
– Use the least possible number of relay stations
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Jaroslaw Kutylowski
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International Graduate
School of Dynamic
Intelligent Systems
Motivation
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
• Costs for changed edges in MST
– Relay stations have to change their location – very
costly!
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Jaroslaw Kutylowski
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Results
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
Deterministic algorithms
– Upper bound O(n2)
– Matching lower bound Ω(n2)
Randomized algorithms (against oblivious adversaries)
– Lower bound Ω(log n)
– Upper bound O(n log n)
– On planar graphs upper bound O(log n)
Jaroslaw Kutylowski
11
Agenda
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
1. Problem definition
2. Motivation
3. Lower bound for deterministic algorithms
4. Sketch of deterministic algorithm MSTMark
5. Idea of lower bound for randomized algorithms
6. Sketch of randomized algorithm RandMST
Jaroslaw Kutylowski
12
Deterministic
lower bound
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
Deterministic lower bound
No deterministic algorithm can have competitive
ratio better than Ω(n2)
Jaroslaw Kutylowski
13
Deterministic
lower bound
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
n2/4 edges
Weight 1
n/2 nodes
n/2 nodes
Weight n
Jaroslaw Kutylowski
14
Deterministic
upper bound
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
Deterministic upper bound
Algorithm MSTMark has O(n2) competitive ratio
Jaroslaw Kutylowski
15
Deterministic
upper bound
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
MSTMark algorithm – edge weight increase
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e
2
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2
alternatives
for e
Jaroslaw Kutylowski
16
Deterministic
upper bound
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
MSTMark algorithm – edge weight decrease
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2
1
e
3
2
Cycle of e in M
Jaroslaw Kutylowski
17
Deterministic
upper bound
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
We mark edges with PRESENCE …
– we know that at moment of marking OPT uses that edge
… or ABSENCE
– we know that at moment of marking OPT does not use
that edge
We can show
• Phase changes after we have proven that OPT has
changed at least one edge in its MST
• ALG pays at most O(n2) in one phase
Jaroslaw Kutylowski
18
Deterministic
upper bound
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
A
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A
Jaroslaw Kutylowski
19
Randomized
lower bound
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
Randomized lower bound
No randomized algorithm can have competitive
ratio better than Ω(log n) against
oblivious adversary
Jaroslaw Kutylowski
20
Randomized
lower bound
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
Same graph as in deterministic lower bound
n2/4 edges
Weight 1
n/2 nodes
n/2 nodes
Weight n
Jaroslaw Kutylowski
21
Randomized
lower bound
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
• adversary does not know which edge ALG uses
• adversary chooses edges to increase at random
• in expectation adversary must increase half of edges
to hit ALG
n2/4 edges
Jaroslaw Kutylowski
22
Randomized
upper bound
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
Randomized upper bound
Randomized algorithm RandMST
has competitive ratio
• O(n log n) for general graphs
• O(log n) for planar graphs
(only increasing edge weights)
Jaroslaw Kutylowski
23
Randomized
upper bound
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
Divide graph in infinite number of levels
• Level corresponds to edge weight w
What we can do
• Assign ALG‘s and OPT‘s cost to levels
• Separate the levels from each other
• Approximate ALG‘s and OPT‘s cost on each level
Jaroslaw Kutylowski
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Randomized
upper bound
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
Level w
<w
edge
set
w
>w
fixed component
realm
realm
Jaroslaw Kutylowski
25
Randomized
upper bound
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
Fixed components splits
• Edge e with weight w increases inside of fixed component
• Alternatives of e form a new edge set
• Edge sets get splitted
Realm splits
• Weight of all edges between two components >w
• Realms do not increase number of fixed components and
edge sets
Jaroslaw Kutylowski
26
Randomized
upper bound
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
It holds
• OPT‘s cost lower bounded by number of
fixed component splits
• ALG‘s cost upper bounded by number of edge sets
– adversary can have some information which edge sets ALG uses
– choice of a specific edge inside of an edge set is uniform
• General graphs: O(n log n) cost per fixed component split
• Planar graphs: O(log n) cost per fixed component split
Jaroslaw Kutylowski
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Conclusions
& Future work
International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
We know a little more about MSTs in dynamic graphs
• Deterministic algorithms are hopeless
• Randomized algorithms are fine on restricted graphs
Future work
• Improve randomized upper bound for increase/decrease
• Finer notions of adversary
• Possibility for adversary to change many edges per round
Jaroslaw Kutylowski
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International Graduate
School of Dynamic
Intelligent Systems
HEINZ NIXDORF INSTITUT
University of Paderborn
Algorithms and Complexity
Thank you for your
attention!
Jaroslaw Kutylowski
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