the online labeling problem
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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The Online Labeling Problem
Jan Bulánek(Institute of Math, Prague)
Martin Babka (Charles University)Vladimír Čunát (Charles University)
Michal Koucký (Institute of Math, Prague)Michael Saks (Rutgers University)
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Sorted Arrays Basis of many algorithms Easy to work with
Dynamization?Online Labeling
Storing elements in the array
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3 1915117 12
1 -5 32 7… 14
Stream of n elements
Array of size Θ(n)
Gaps in the arrayMuze pohnout co chce
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Online labelingInput:
A stream of n numbers 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)
Naïve solution O(n) per insertion
ApplicationsMany applications, e.g.:
[Bender, Demaine, Farach-Colton ’00] Cache-oblivous B-trees
[Emek, Korman ’11] Distributed Controllers Lower bounds
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Linear array algorithm [Itai, Konheim, Rodeh ’81] O(log2 n) per insertion, amortized[Itai, Katriel ’07] Simpler algorithm
Basic ideas Small gaps Spread items evenly Density threshold function
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Algorithm for linear arrays – cont.How to find segment to rearrange
Too denseGood densityRearrange items evenly
Array size (m) Amortized insertion costm=n O(log3 n) [Z 93]
m=Θ(n) O(log2 n) [IKR 81][W92, BCD+02]*m=n1+o(1) O( ) [IKR 81]m=n1+ℇ O(log n)m=nΩ(log n) O( ) [BKS 12]
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Upper bounds
TIGHT!
!
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Lower Bounds[Zhang ’93] m=O(n) Ω(log2 n) per insertion, amortized Only smooth strategies
[Dietz, Seiferas, Zhang ’94] m=n1+Θ(1) Ω(log n) per insertion, amortized Proof contains a gap
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Lower Bounds – cont.[B., Koucký, Saks STOC’12] All strategies Uses some ideas from [Zhang 93]
m=n Ω(log3 n)m=Θ(n) Ω(log2 n)
Lower Bounds – proof techniqueAdversary Generates input stream Reacts on the state of the array Inserts to dense areas
Only deterministic case
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Lower Bounds – cont.[Babka, B., Čunát, Koucký, Saks ESA’12] All strategies Fills the gap in [DSZ ’04] and extends their result Tight bounds for the bucketing game
m=n1+Θ(1) Ω()m=n1+Ω (1) Ω( )
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Lower Bounds – cont.[Babka, B., Čunát, Koucký, Saks 12, manuscript] All strategies Extends results of [BKS 12]
m=n1+o(1) Ω( )
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Lower Bounds – Sumary
Array size (m) Insertion cost
m=n+a(n) Ω(log2 n )m=cn Ω()m=n∙f(n)f(n)∊o(n) Ω( )m=ne(n)e(n)∊Ω(1) Ω( )
Trivial for r<mLimited universe
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m …
1 2 3 4 … r-1 rU
Maybe easier for r small
Limited universe – cont.
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3 4 …
1 2 3 4 … r-1 rU
Limited universe – cont.
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Open problems Randomized algorithms? Limited universe m log n
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The End!