(Data) STRUCTURES Mihai P ă trașcu. LSD dyn. marked ancestor reachability oracles in the butterfly...
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Transcript of (Data) STRUCTURES Mihai P ă trașcu. LSD dyn. marked ancestor reachability oracles in the butterfly...
![Page 1: (Data) STRUCTURES Mihai P ă trașcu. LSD dyn. marked ancestor reachability oracles in the butterfly partial match(1+ε)-ANN ℓ 1, ℓ 2 NN in ℓ 1, ℓ 2 3-ANN.](https://reader038.fdocument.org/reader038/viewer/2022102818/56649ced5503460f949b9bfb/html5/thumbnails/1.jpg)
(Data) STRUCTURES
Mihai Pătrașcu
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LSD
dyn. marked ancestor
reachability oracles in the butterfly
partial match (1+ε)-ANN ℓ1, ℓ2
NN in ℓ1, ℓ23-ANN in ℓ∞
worst-caseunion-find dyn. trees, graphs
dyn. 1D stabbing
partial sums
2D stabbing
4D reporting 2D counting
dyn. NN in 2D
dyn. 2D reporting
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LSD
dyn. marked ancestor
reachability oracles in the butterfly
partial match (1+ε)-ANN ℓ1, ℓ2
NN in ℓ1, ℓ23-ANN in ℓ∞
worst-caseunion-find dyn. trees, graphs
dyn. 1D stabbing
partial sums
2D stabbing
4D reporting 2D counting
dyn. NN in 2D
dyn. 2D reportingbetter bound
simpler, same bound
simpler, loses lglg n
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How Hard is Range Reporting?SELECT * FROM employeesWHERE salary <= 70000 AND startdate <= 1998
70000
69000
68000
71000
Static, 2D: O(lglg n)[Alstrup, Brodal, Rauhe
FOCS’00]
Dynamic, 2D: Ω(lg n / lglg n)[Alstrup,Husfeldt, Rauhe
FOCS’98]
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How Hard is Range Reporting?Static, 2D: O(lglg n)
[Alstrup, Brodal, Rauhe FOCS’00]
Dynamic, 2D: Ω(lg n / lglg n)[Alstrup,Husfeldt, Rauhe
FOCS’98]Marked Ancestor: * mark(v) / unmark(v)
* query(v): ? ∃marked ancestor
Dyn. 1D Stabbing: * insert/delete segment * query(x): does
x stab a segment?
Dyn. 2D Range Reporting
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How Hard is Range Reporting?Static, 2D: O(lglg n)
[Alstrup, Brodal, Rauhe FOCS’00]
Dynamic, 2D: Ω(lg n / lglg n)[Alstrup,Husfeldt, Rauhe
FOCS’98]
Static 3D ≈ Dynamic 2D ?
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Persistence
Persistent : { static data struct. } ⟼ { dynamic data struct. }
Given problem with ℙ updateℙ(x), queryℙ(y)
Persistent( ) = the problem ℙ“preprocess (x1, x2, …, xN) to answer:
query(y, t): answer queryℙ(y) after updateℙ(x1), …, updateℙ(xt)”
Persistent(incremental 2D range reporting)= static 3D range reporting
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How Hard is Range Reporting?Static, 2D: O(lglg n)
[Alstrup, Brodal, Rauhe FOCS’00]
Dynamic, 2D: Ω(lg n / lglg n)[Alstrup,Husfeldt, Rauhe FOCS’98]
Static 3D = Persistent(Dynamic 2D)
But: Persistent(Marked Ancestor) has O(1) solution…
[Nekrich SoCG’07] static 3D in O(lg2lg n)…
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How Could 4D be Hard?
Fully-persistent data structures:
FullyPersistent(Marked Ancestor)≤ Static 2D Stabbing≤ Static 4D Reporting
v0
v1 v2
v3
v6
v4v5
v7
query(y)?
update(y)
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FullyPersistent (Marked Ancestor)
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FullyPersistent (Marked Ancestor)
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FullyPersistent (Marked Ancestor)
“reachabilityoracles”
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Lopsided Set Disjointness
Review: Communication Complexity
set Sset T
“is S ∩ T = ?”∅
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LSD Lower Bounds
May assume universe is of size 2T… split in blocks of size B=2T/S
Say Alice sends εS lg B∙ bits ⇒ in average block, Alice’s value is uncertain in a set of B1-ε’
⇒ Bob must identify his elements in uncertainty sets,sending Ω(S B∙ 1-ε’) bits
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LSD Reachability Oracles➙
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LSD Reachability Oracles➙
T = { } = deleted edges
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LSD Reachability Oracles➙
T = { } = deleted edgesS = { } = one out-edge / node
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LSD
dyn. marked ancestor
reachability oracles in the butterfly
partial match (1+ε)-ANN ℓ1, ℓ2
NN in ℓ1, ℓ23-ANN in ℓ∞
worst-caseunion-find dyn. trees, graphs
dyn. 1D stabbing
partial sums
2D stabbing
4D reporting 2D counting
dyn. NN in 2D
dyn. 2D reporting
The End