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The Online Labeling Problem

The Online Labeling Problem. Jan Bul ánek ( Institute of Math , Prague) M artin Babka (Charles University) Vladimír Čunát (Charles University ) Michal Kouck ý ( Institute of Math , Prague ) Michael Saks (Rutgers University). Sorted Arrays. Basis of many algorithms

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The Online Labeling Problem

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  1. The Online LabelingProblem Jan Bulánek (Institute of Math, Prague) Martin Babka (Charles University) Vladimír Čunát(Charles University) Michal Koucký (Institute of Math, Prague) Michael Saks(Rutgers University)

  2. Sorted Arrays • Basis of many algorithms • Easy to work with • Dynamization? Online Labeling

  3. Storing elements in the array 12 Gaps in the array Muzepohnout co chce 1 -5 32 7 14 … Stream of nelements 12 7 11 15 Array of size Θ(n)

  4. Online labeling Input: • A streamofnnumbers • An array of size m • For the size Θ(n) File maintenance problem Want: • maintain a sorted array of all already seen items • minimize the total number of item moves (cost) Rictzediry mi sami o sobenestaci Naïve solution O(n)per insertion

  5. Applications Many applications, e.g.: [Bender, Demaine, Farach-Colton ’00] • Cache-oblivous B-trees [Emek, Korman’11] • Distributed Controllers • Lower bounds

  6. Linear array algorithm [Itai, Konheim, Rodeh ’81] • O(log2n)per insertion, amortized [Itai, Katriel ’07] • Simpler algorithm Basic ideas • Small gaps • Spread items evenly • Density threshold function

  7. Algorithm for linear arrays – cont. How to find segment to rearrange Too dense Rearrange items evenly Good density

  8. Upper bounds TIGHT!! Anderssonlai

  9. Lower Bounds [Zhang ’93] • m=O(n) • Ω(log2n) per insertion, amortized • Only smooth strategies [Dietz, Seiferas, Zhang ’94] • m=n1+Θ(1) • Ω(logn) per insertion, amortized • Proof contains a gap

  10. Lower Bounds – cont. [B., Koucký, SaksSTOC’12] • Allstrategies • Uses some ideas from [Zhang 93]

  11. Lower Bounds – proof technique Adversary • Generates input stream • Reacts on the state of the array • Inserts to dense areas Only deterministic case

  12. Lower Bounds – cont. [Babka, B., Čunát, Koucký, SaksESA’12] • Allstrategies • Fillsthe gap in [DSZ ’04] and extends their result • Tight bounds for the bucketing game

  13. Lower Bounds – cont. [Babka, B., Čunát, Koucký, Saks 12, manuscript] • Allstrategies • Extends results of [BKS 12]

  14. Lower Bounds –Sumary

  15. Limited universe • Trivial for r<m U m

  16. Limited universe – cont. • Maybe easier for r small U

  17. Limited universe – cont.

  18. Open problems • Randomized algorithms? • Limited universe m log n The End!

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