Streaming Algorithms for Geometric Problems

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Streaming Algorithms for Geometric Problems

• Piotr Indyk
• MIT

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Streaming The Model

• Single pass over the data e1, e2, ,en
• Bounded storage
• Fast processing time per element

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Recap Norm estimation

• Norm estimation
• Stream elements (i,b) , i1m
• Interpretation xixib
• Want to (1?)-approximate xp
• Note the frequency moment Fp is a special case
• Algorithms with (log n1/?)O(1) space
• p2 Alon-Matias-Szegedy96
• p?(0,2 Indyk00
• Idea maintain Ax

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Recap Other algorithms

• Compute the number of distinct elements F0
• Exactly ?(n) bits of space
• (1?) -approximation O(1/?2 log n) bits
  Flajolet-Martin, JCSS85 ,
• Compute the median
• Exactly ?(n)
• (50 ? ?) -approximation O(1/? polylog n)
  Paterson-Munro, TCS80 ,

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Geometric Data Stream Algorithms as Data
Structures

• Data structures that support
• Insert(p) to P
• Possibly Delete(p) from P
• Compute-Some-Property(P)
• Use space that is sub-linear in P

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Dichotomy

• Insertions Deletions
• Randomized linear
• mappings FM85,
• AMS96
• Randomized

• Insertions
• Merge and Reduce
• MP80,
• core-sets
• Deterministic

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Insertions and Deletions
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Minimum Weight Bi-chromatic Matching

• Estimate the cost of MWBM

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Minimum Weight Matching

• Estimate the cost of MWM

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Minimum Spanning Tree

• Estimate the cost of MST

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Facility Location

• Goal choose a set F of facilities to minimize
  the
• sum of the distances to nearest facility plus
• the number of facilities times f
• Again, report the cost

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Approach

• Assume P?1?2
• Reduce to vector problems
• Impose square grids G0Gk, with side lengths
  20,21, , 2k , shifted at random.
• For each square cell c in Gi, let nP(c) be the
  number of points from P in c.
• The algorithms will maintain certain statistics
  over nP(.), which will allow it to approximately
  solve the problems

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Estimators

• MST ?i 2i ?c ?Gi nP(c)gt0
• MWM ?i 2i ?c ?Gi nP(c) is odd
• MWBM ?i 2i ?c ?Gi nG(c)-nB(c)
• Fac. Loc. ?i 2i ?c ?Gi minnP(c), Ti
• K-median ?i 2i ?c ?Gi - B(Q, 2i) nP(c)
• (const. factor)
• Maintain non-zero entries in nP FM85
• Maintain L1 difference I00

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Results Indyk04
? 1? Frahling-Indyk-Sohler05
Space (log ? log n)O(1)
follows from Charikar02,Indyk-Thaper03
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Probabilistic embeddings into HSTs
T
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• Known Bartal96, Charikar-Chekuri-Goel-Guha-Plotk
  in98
• p-q Dtree (p,q)
• E Dtree(p,q) p-q O(log ?)

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MST
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• ECost(MST in T) O(log ?) Cost(MST)
• Cost(MST in T) ? Cost(T)
• How to compute Cost(T) ?
• Sum over all levels i, of the nodes at i, times
  2i
• Node c exists iff ni(c)gt0

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Matching

• Algorithm
• Match what you can at the current level
• Odd leftovers wait for the next level
• Repeat
• Optimal on the HST
• Cost?i 2i ?c ?Gi nP(c) is odd

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Insertions-only
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k-median/k-center

• k is given
• Goal choose k medians/centers to minimize
• k-median the sum of the distances
• k-center the max distance

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Metric clustering problems

• k-center Charikar-Chekuri-Feder-Motwani97
• k-median Guha-Mishra-Motwani-OCallaghan00,
  Meyerson01, Charikar-OCallaghan-Panigrahy03
  
• Bounds
• Poly(K,log n) space
• O(1)-approximation

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Geometric Problems

• Diameter, Minimum Enclosing Ball
  Agarwal-Har-Peled01, Feigenbaum-Kannan-Zhang02
  (Algorithmica), Hershberger-Suri04
• K-center AHP01
• K-median Har-Peled-Mazumdar04
• Range searching via ?-approximations
• Suri-Toth-Zhou04
• Bagchi-Chaudhary-Eppstein-Goodrich04

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Dominant Approach Merge and Reduce

• Main ideas
• Design an (off-line) algorithm that computes a
  sketch of the input
• Small size
• Sufficient to solve the problem
• A sketch of sketches is a sketch

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Tree Computation
p1
p2
p3
p4
p5
p7
p6
p8
p9
p10
p11
p12
p13
p15
p14
p16
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Algorithm

• Space (sketch size)log n
• Time sketch computation time
• Question Where do sketches come from ?

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Idea I solutionsketch

• Consider k-median
• Guha-Mishra-Motwani-OCallaghan00
  approximate k-median of approximate weighted
  k-medians is an approximate k-median
• Result
• Constant depth tree
• Space kn? , ?gt0
• O(1) -approximation
• Works for any metric space

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Use the solution, ctd.

• ?-Approximations find a subset S?P , such that
  for any rectangle/halfspace/etc R,
• R?S/S R?P/P ??
• Matousek approximation of a union of
  approximations is an approximation
• BCEG04 convert it into streaming algorithm,
  applications
• ?1/?2 space
• STZ04 better/optimal bounds for rectangles
  and halfspaces

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Idea 2 Core-Sets AHP01

• Assume we want to minimize CP(o)
• S?P is an ?-core-set for P, if for any o, and a
  set T
• CP?T (o) lt (1?) CS?T (o)
• Note this must hold for all o, not just the
  optimal one
• The latter is called a weak coreset
• Badoiu-HarPeled-Indyk02

o
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Example Core-set for MEB

• Compute extremal points
• Choose densely spaced direction v1 vk
• I.e., for any u there is vi such that uvi
  u2 / (1?)
• For each direction maintain extremal point
• kO(1/?)(d-1)/2 suffice

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Stream Algorithms via Core-sets

• Diameter/MEB/width O(1/?)(d-1)/2 log n space
  AHPV010204
• k-center O(k/?d) log n HP01
• k-median O(k/?d) log n HPM04
• (kdlog n1/?)O(1) Chen06
• Faster algorithms and other results Chan04,
  Suri-Hershberger03

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Core-sets Randomized Maps

• (1?)-approximate k-median under insertions and
  deletions
• Frahling-Sohler05

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Insertions and Deletions, ctd
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Histograms

• View x as a function x1n ? 1M
• Approximate it using piecewise constant function
  h, with B pieces (buckets)
• Problem can be formulated in 2D as well (buckets
  become rectangular tiles)

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Results 1D

• Gilbert-Guha-Indyk-Kotidis-Muthukrishnan-Strauss,
  STOC02
• Maintains h with B pieces such that
• x-h2 (1?)x-hOPT2
• Under increments/decrements of x
• Space poly(B,1/?,log n)
• Time poly(B,1/?,log n)

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Results 2D

• Thaper-Guha-Indyk-Koudas, SIGMOD02
• Maintains h with B log (nM) tiles such that
• x-h2 (1?)x-hOPT2
• Under increments/decrements of x
• Space/Update time poly(B,1/?,log n)
• Histogram reconstruction time poly(B,1/?, n)
• Muthukrishnan-Strauss, FSTTCS03
• Maintains h with 4B tiles
• Time poly(B,1/?, log(nM))

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General Approach

• Maintain sketches Ax of x
• This allows us to estimate the error of any given
  h, via x-h ? Ax-Ah
• Construct h
• Enumeration
• Greedy
• Dynamic Programming

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Conclusions

• Algorithms for geometric data streams
• Insertions-only merge and reduce
• Insertions and deletions randomized linear
  embeddings

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Open Problems

• High dimensions
• Diameter
• 21/2-approx, O(d2 n1/2 ) space, follows from
  Goel-Indyk-Varadarajan, SODA01
• c-approx, O( dn1/(c2 - 1) ) Indyk, SODA03
• Conjecture ?21/2-approx, O(d polylog n) space
• Min-width cylinder 18-approx, O(d) space
  Chan04
• Other problems ?

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Open Problems

• Range queries
• General lower bounds ? (Not just for ?-
  approximations)
• ?(1/?2) -bit bound for general queries follows
  from LB for dot product Indyk-Woodruff, FOCS03
  , and is tight (for randomized algorithms)
• What about e.g., half-space queries ? O(1/?4/3)
  is known STZ04
• Other problems STZ04

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Open Problems

• Matchings, Facility Location, etc
• Replace log ? by O(1) or even 1?
• Possible for MST Frahling-Indyk-Sohler??
• Related to computing bi-chromatic matching
  Agarwal-Varadarajan04
• Min-sum clustering ?