de dictionariis dynamicis pauco spatio utentibus

play

De Dictionariis Dynamicis Pauco Spatio Utentibus ( lat. On Dynamic - PowerPoint PPT Presentation

Jun 21, 2023 •147 likes •495 views

De Dictionariis Dynamicis Pauco Spatio Utentibus ( lat. On Dynamic Dictionaries Using Little Space) Erik D. Demaine 1 Friedhelm Meyer auf der Heide 2 Rasmus Pagh 3 cu 1 Mihai P atras 1 MIT 2 University of Paderborn 3 IT University of

De Dictionariis Dynamicis Pauco Spatio Utentibus ( lat. On Dynamic Dictionaries Using Little Space) Erik D. Demaine 1 Friedhelm Meyer auf der Heide 2 Rasmus Pagh 3 ¸cu 1 Mihai Pˇ atras 1 MIT 2 University of Paderborn 3 IT University of Copenhagen LATIN, March 2006 Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Dictionaries Maintain S ⊂ [ u ] , | S | ≤ n under: INSERT ( x ) : S ← S ∪ { x } DELETE ( x ) : S ← S \ { x } MEMBER ( x ) : is x ∈ S ? OPUS CLASSICUM : [FKS’82] query: O ( 1 ) worst-case − → update: O ( 1 ) expected w.h.p. [DMadH’90] O ( n lg u space: O ( n lg u ) bits − → n ) bits [RR’03] We achieve both, through new techniques. Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Dictionaries Maintain S ⊂ [ u ] , | S | ≤ n under: INSERT ( x ) : S ← S ∪ { x } DELETE ( x ) : S ← S \ { x } MEMBER ( x ) : is x ∈ S ? OPUS CLASSICUM : [FKS’82] query: O ( 1 ) worst-case − → update: O ( 1 ) expected w.h.p. [DMadH’90] O ( n lg u space: O ( n lg u ) bits − → n ) bits [RR’03] We achieve both, through new techniques. Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Dictionaries Maintain S ⊂ [ u ] , | S | ≤ n under: INSERT ( x ) : S ← S ∪ { x } DELETE ( x ) : S ← S \ { x } MEMBER ( x ) : is x ∈ S ? OPUS CLASSICUM : [FKS’82] query: O ( 1 ) worst-case − → update: O ( 1 ) expected w.h.p. [DMadH’90] O ( n lg u space: O ( n lg u ) bits − → n ) bits [RR’03] We achieve both, through new techniques. Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Dictionaries Maintain S ⊂ [ u ] , | S | ≤ n under: INSERT ( x ) : S ← S ∪ { x } DELETE ( x ) : S ← S \ { x } MEMBER ( x ) : is x ∈ S ? OPUS CLASSICUM : [FKS’82] query: O ( 1 ) worst-case − → update: O ( 1 ) expected w.h.p. [DMadH’90] O ( n lg u space: O ( n lg u ) bits − → n ) bits [RR’03] We achieve both, through new techniques. Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Dictionaries Maintain S ⊂ [ u ] , | S | ≤ n under: INSERT ( x ) : S ← S ∪ { x } DELETE ( x ) : S ← S \ { x } MEMBER ( x ) : is x ∈ S ? OPUS CLASSICUM : [FKS’82] query: O ( 1 ) worst-case − → update: O ( 1 ) expected w.h.p. [DMadH’90] O ( n lg u space: O ( n lg u ) bits − → n ) bits [RR’03] We achieve both, through new techniques. Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Dictionaries Maintain S ⊂ [ u ] , | S | ≤ n under: INSERT ( x ) : S ← S ∪ { x } DELETE ( x ) : S ← S \ { x } MEMBER ( x ) : is x ∈ S ? OPUS CLASSICUM : [FKS’82] query: O ( 1 ) worst-case − → update: O ( 1 ) expected w.h.p. [DMadH’90] O ( n lg u space: O ( n lg u ) bits − → n ) bits [RR’03] We achieve both, through new techniques. Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Basic Idea How to get w.h.p. from expected bounds? independence Idea: √ n independent tables permute universe randomly: π 1 distribute elements by hi ( π ( x )) 2 store lo ( π ( x )) in hash table 3 Permutation needed! To be information-efficient, need to store just lo ( π ( x )) . Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Basic Idea How to get w.h.p. from expected bounds? independence Idea: √ n independent tables permute universe randomly: π 1 distribute elements by hi ( π ( x )) 2 store lo ( π ( x )) in hash table 3 Permutation needed! To be information-efficient, need to store just lo ( π ( x )) . Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Basic Idea How to get w.h.p. from expected bounds? independence Idea: √ n independent tables x ∈ [ u ] permute universe randomly: π 1 π distribute elements by hi ( π ( x )) 2 store lo ( π ( x )) in hash table 3 hi lo · · · · · · · · · Permutation needed! To be information-efficient, need to store just lo ( π ( x )) . Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Basic Idea How to get w.h.p. from expected bounds? independence Idea: √ n independent tables x ∈ [ u ] permute universe randomly: π 1 π distribute elements by hi ( π ( x )) 2 store lo ( π ( x )) in hash table 3 hi lo · · · · · · · · · Permutation needed! To be information-efficient, need to store just lo ( π ( x )) . Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Basic Idea How to get w.h.p. from expected bounds? independence Idea: √ n independent tables x ∈ [ u ] permute universe randomly: π 1 π distribute elements by hi ( π ( x )) 2 store lo ( π ( x )) in hash table 3 hi lo · · · · · · · · · Permutation needed! To be information-efficient, need to store just lo ( π ( x )) . Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Basic Idea How to get w.h.p. from expected bounds? independence Idea: √ n independent tables x ∈ [ u ] permute universe randomly: π 1 π distribute elements by hi ( π ( x )) 2 store lo ( π ( x )) in hash table 3 hi lo · · · · · · · · · Permutation needed! To be information-efficient, need to store just lo ( π ( x )) . Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Basic Idea How to get w.h.p. from expected bounds? independence Idea: √ n independent tables x ∈ [ u ] permute universe randomly: π 1 π distribute elements by hi ( π ( x )) 2 store lo ( π ( x )) in hash table 3 hi lo · · · · · · · · · Permutation needed! To be information-efficient, need to store just lo ( π ( x )) . Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Highly Independent Permutations? We need k -independent permutations. Wait in line. . . Idea: construct permutations with good k th moment bounds. Tools: Siegel’s highly independent hash functions 1 many tricks for reusing randomness 2 Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Highly Independent Permutations? We need k -independent permutations. Wait in line. . . Idea: construct permutations with good k th moment bounds. Tools: Siegel’s highly independent hash functions 1 many tricks for reusing randomness 2 Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Highly Independent Permutations? We need k -independent permutations. Wait in line. . . Idea: construct permutations with good k th moment bounds. Tools: Siegel’s highly independent hash functions 1 many tricks for reusing randomness 2 Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Highly Independent Permutations? We need k -independent permutations. Wait in line. . . Idea: construct permutations with good k th moment bounds. Tools: Siegel’s highly independent hash functions 1 many tricks for reusing randomness 2 Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
The Dictionary Structure break universe into n / lg n segments 1 minidictionary in each segment with capacity 2 lg n 2 fall back to high performance dictionary: 3 Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
The Dictionary Structure break universe into n / lg n segments 1 minidictionary in each segment with capacity 2 lg n 2 fall back to high performance dictionary: 3 Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
The Dictionary Structure break universe into n / lg n segments 1 minidictionary in each segment with capacity 2 lg n 2 fall back to high performance dictionary: 3 Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
The Dictionary Structure break universe into n / lg n segments 1 minidictionary in each segment with capacity 2 lg n 2 fall back to high performance dictionary: 3 elements overflowing capacity of minidictionary elements for which the minidictionary failed Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
The Dictionary Structure break universe into n / lg n segments 1 minidictionary in each segment with capacity 2 lg n 2 fall back to high performance dictionary: 3 exp. O ( 1 ) bad elements from each segment ⇒ O ( n / lg n ) total w.h.p. Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Life is tough Minidictionary stores O ( lg n ) elements in arbitrary order ⇒ Ω( lg lg n ) bits per element! For small universes, need to do something crazy. . . � O ( lg lg n ) levels of filters all packed in a word! on disjoint segments of the universe Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis
Dictionary? What dictionary? What is the query? membership: is x ∈ S ? � u = Ω( n lg u � requires ≥ lg n ) bits n retrieval: return data ( x ) if x ∈ S very useful without membership (trust me) 1 membership lower bound does not apply! 2 optimal space Θ( n lg lg u n ) [sublinear!] perf hashing: return unique, immutable ID for any x ∈ S roughly same as retrieval Demaine, Meyer auf der Heide, Pagh, Pˇ atras ¸cu Dictionariis Dynamicis

Download Presentation

Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

More recommend