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Competitive Maintenance of Minimum Spanning Trees in Dynamic Graphs

Competitive Maintenance of Minimum Spanning Trees in Dynamic Graphs. Miroslaw Dynia Miroslaw Korzeniowski Jaroslaw Kutylowski. The Problem. Graph contains edges of same weight many different Minimum Spanning Trees possible. 5. 5. 5. 5. 5. 5. 6. 5. The Question.

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Competitive Maintenance of Minimum Spanning Trees in Dynamic Graphs

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  1. Competitive Maintenance of Minimum Spanning Trees in Dynamic Graphs Miroslaw Dynia Miroslaw Korzeniowski Jaroslaw Kutylowski

  2. The Problem Graph contains edges of same weight many different Minimum Spanning Trees possible 5 5 5 5 5 5 6 5

  3. The Question • 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)

  4. Model • 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

  5. The Problem • Algorithm uses red MST – cost 1 • Algorithm uses green MST – cost 0 5 5 5 5 5 5 6 5

  6. Motivation • 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

  7. Motivation Communication paths with relay stations Networkcomponent

  8. Motivation • Adversary changes weights of edges • Network components move 5 5 8 5

  9. Motivation • Algorithm has to maintain a MST • Number of relay stations used is proportional tolength of edge • Use the least possible number of relay stations 5 5 7 5

  10. Motivation • Costs for changed edges in MST • Relay stations have to change their location – very costly! 5 5 5 5

  11. Results • Deterministic algorithms • Upper bound O(n2) • Matching lower boundΩ(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)

  12. Agenda • Problem definition • Motivation • Lower bound for deterministic algorithms • Sketch of deterministic algorithm MSTMark • Idea of lower bound for randomized algorithms • Sketch of randomized algorithm RandMST

  13. Deterministic lower bound • Deterministic lower bound • No deterministic algorithm can have competitive ratio better than Ω(n2)

  14. Deterministiclower bound n2/4 edges Weight 1 n/2 nodes n/2 nodes Weight n

  15. Deterministicupper bound • Deterministic upper bound • Algorithm MSTMark has O(n2) competitive ratio

  16. Deterministic upper bound MSTMark algorithm – edge weight increase 2 3 e 2 2 2 2 alternatives for e

  17. Deterministic upper bound MSTMark algorithm– edge weight decrease 3 2 1 e 3 2 Cycle of e in M

  18. Deterministic upper bound • We mark edges withPRESENCE … • we know that at moment of marking OPT uses that edge • … orABSENCE • 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

  19. Deterministic upper bound A A A A A A A A A A A A A

  20. Randomized lower bound • Randomized lower bound • No randomized algorithm can have competitive ratio better than Ω(log n) against oblivious adversary

  21. Randomized lower bound • Same graph as in deterministic lower bound n2/4 edges Weight 1 n/2 nodes n/2 nodes Weight n

  22. Randomizedlower bound • 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

  23. Randomized upper bound • 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)

  24. Randomizedupper bound • 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

  25. Randomized upper bound • Level w <w edgeset w >w fixed component realm realm

  26. Randomized upper bound • 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

  27. Randomizedupper bound • 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

  28. Conclusions & Future work • 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

  29. Thank you for your attention!

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