Facility Location in Dynamic Geometric Data Streams

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Facility Location in Dynamic Geometric Data
Streams
Christiane Lammersen Christian Sohler
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Dynamic Geometric Data Streams

• Streams of geometric data arise in
• Mobile networks
• Sensor networks
• Continuously changing data
• Mobile networks position of nodes
• Sensor networks measured data
• Communication in form of update operations
• Update consists of ID of node, old value, new
  value

IITK Workshop on Algorithms for
Christiane Lammersen Processing Massive Data Sets
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Hierarchical Communication Systems

• upper layer offers lower layer a certain service
• each node can be a server
• cost for server ? access time

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Hierarchical Communication Systems

• upper layer offers lower layer a certain service
• each node can be a server
• cost for server ? access time

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Dynamic Geometric Data Streams

• m insert and delete operations
• points in low-dimensional, discrete space 1,
  ..., Dd
• polylog(D, m) memory space, one pass

Indyk 04
D
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Dynamic Uniform FLP

• point set P
• facilities have uniform opening cost f
• clients have uniform demand b
• goal maintaining F ? P, so as to minimize

• FLP related to k-Median but
• F can be Q(P)
• problem in streaming
• approximation of the cost

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Related Work

• P. Indyk Algorithms for Dynamic Geometric
  Problems over Data Streams, STOC 04
• O(log2D)-approximation for cost of FLP
• Idea nested squared grids, open facility in all
  heavy
• cells
• G. Frahling and C. Sohler Coresets in Dynamic
  Geometric Data Streams, STOC 05
• space partition based on heavy cells

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Construction of Our Streaming Method
deterministic method Edet(P) Q(OPT(P))
randomized method Erand(P) Q(Edet(P))
streaming method Estream(P) Q(Erand(P))
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Deterministic Method

• Impose log(D)1 nested squared grids
• In each grid, identify the heavy cells
• Partition the input space based on the heavy
  cells
• For each cell size, count the number of points
  within cells of that size
• gt estimator for cost
• Indyk 04, Frahling and Sohler 05

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Deterministic Method

• Impose log(D)1 nested squared grids
• In each grid, identify the heavy cells
• Partition the input space based on the heavy
  cells
• For each cell size, count the number of points
  within cells of that size
• gt estimator for cost

Idea Open one facility in each heavy cell in the
space partition.
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Deterministic Method

• Impose log(D)1 nested squared grids
• In each grid, identify the heavy cells
• Partition the input space based on the heavy
  cells
• For each cell size, count the number of points
  within cells of that size
• gt estimator for cost

Idea Open one facility in each heavy cell in the
space partition.
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Nested Grids

• Impose log(D)1
• nested squared grids

D 16 Level 4
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Nested Grids

• Impose log(D)1
• nested squared grids

D 16 Level 3
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Nested Grids

• Impose log(D)1
• nested squared grids

D 16 Level 2
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Nested Grids

• Impose log(D)1
• nested squared grids

D 16 Level 1
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Nested Grids

• Impose log(D)1
• nested squared grids

D 16 Level 0
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Deterministic Method

• Impose log(D)1 nested squared grids
• In each grid, identify the heavy cells
• Partition the input space based on the heavy
  cells
• For each cell size, count the number of points
  within cells of that size
• gt estimator for cost

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Space Partition

• In each grid, identify the
• heavy cells
• Partition the input space
• based on the heavy cells

Cell in level i is heavy if it contains f / 2i
points.
f 8 D 16 Level 4
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Space Partition

• In each grid, identify the
• heavy cells
• Partition the input space
• based on the heavy cells

Cell in level i is heavy if it contains f / 2i
points.
f 8 D 16 Level 3
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Space Partition

• In each grid, identify the
• heavy cells
• Partition the input space
• based on the heavy cells

Cell in level i is heavy if it contains f / 2i
points.
f 8 D 16 Level 3
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Space Partition

• In each grid, identify the
• heavy cells
• Partition the input space
• based on the heavy cells

Cell in level i is heavy if it contains f / 2i
points.
f 8 D 16 Level 3
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Space Partition

• In each grid, identify the
• heavy cells
• Partition the input space
• based on the heavy cells

Cell in level i is heavy if it contains f / 2i
points.
f 8 D 16 Level 3
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Space Partition

• In each grid, identify the
• heavy cells
• Partition the input space
• based on the heavy cells

Cell in level i is heavy if it contains f / 2i
points.
f 8 D 16 Level 2
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Space Partition

• In each grid, identify the
• heavy cells
• Partition the input space
• based on the heavy cells

Cell in level i is heavy if it contains f / 2i
points.
f 8 D 16 Level 2
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Space Partition

• In each grid, identify the
• heavy cells
• Partition the input space
• based on the heavy cells

Cell in level i is heavy if it contains f / 2i
points.
f 8 D 16 Level 2
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Space Partition

• In each grid, identify the
• heavy cells
• Partition the input space
• based on the heavy cells

Cell in level i is heavy if it contains f / 2i
points.
f 8 D 16 Level 2
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Space Partition

• In each grid, identify the
• heavy cells
• Partition the input space
• based on the heavy cells

Cell in level i is heavy if it contains f / 2i
points.
f 8 D 16 Level 1
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Space Partition

• In each grid, identify the
• heavy cells
• Partition the input space
• based on the heavy cells

Cell in level i is heavy if it contains f / 2i
points.
f 8 D 16 Level 1
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Space Partition

• In each grid, identify the
• heavy cells
• Partition the input space
• based on the heavy cells

Cell in level i is heavy if it contains f / 2i
points.
f 8 D 16 Level 1
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Space Partition

• In each grid, identify the
• heavy cells
• Partition the input space
• based on the heavy cells

Cell in level i is heavy if it contains f / 2i
points.
f 8 D 16 Level 0
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Space Partition

• In each grid, identify the
• heavy cells
• Partition the input space
• based on the heavy cells

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Deterministic Method

• Impose log(D)1 nested squared grids
• In each grid, identify the heavy cells
• Partition the input space based on the heavy
  cells
• For each cell size, count the number of points
  within cells of that size
• gt estimator for cost

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Cost Estimator

• For each cell size, count
• the number of points
• within cells of that size
• gt estimator for cost

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Cost Estimator

• For each cell size, count
• the number of points
• within cells of that size
• gt estimator for cost

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Cost Estimator

• For each cell size, count
• the number of points
• within cells of that size
• gt estimator for cost

9 points
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Cost Estimator

• For each cell size, count
• the number of points
• within cells of that size
• gt estimator for cost

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Cost Estimator

• For each cell size, count
• the number of points
• within cells of that size
• gt estimator for cost

7 points
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Value of Cost Estimator is W(OPT(P))

• Contribution of heavy
• cell C in level i is at most
• Contribution of light cell
• C in level i is at most

• A heavy cell in level i
• contains Q( f / 2i) points.
• The space partition is balanced.
• The distance of a cell in
• level i to heavy cell is O(2i).

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Value of Cost Estimator is O(OPT(P))

• Contribution of distant
• cell C in level i is at
• least n(C) .2i-1
• OPT(P) ? f . FOPT
• Estimated cost for near
• cell C in level i is
• n(C) .2i O( f )
• There is a constant
• number of near cells.
• Estimated cost for near
• cells is O( f . FOPT)

level i
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Deterministic Method

• Impose log(D)1 nested squared grids
• In each grid, identify the heavy cells
• Partition the input space based on the heavy
  cells
• For each cell size, count the number of points
  within cells of that size
• gt estimator for cost

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Randomized Method

• Idea
• Heavy cell in level i contains at least f /2i
  points
• Sample a point in level i with probability 2i/f
• Problem coin flips delete operations
• Solution
• Hash function hi 1,, Dd ? 1,, ? f / 2i ?
  
• Sample set Si p? P hi( p) 1

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Randomized Method

• for each level i do
• F(i) ? set of all marked cells C in level i such
  that
• no subcell of C is marked
• no smaller cell within a distance of less than
  2i-1 is marked
• return

Erand(P) Q(Edet(P))
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Streaming Method

• Idea Reduction to counting distinct elements
• Implementation
• For each level i count distinct elements in
• DE1(i) CC is in level i and marked?CC is
  in level i and a) or b) fails
• and DE2(i) CC is in level i and a) or b)
  fails
• Output difference as cost for level i

DE1(i)
DE2(i)
DE1(i1)
DE2(i1)
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Conclusion Future Work

• Streaming Algorithm for Dynamic FLP
• constant factor approximation of cost
• update-time O(log(1/d) . polylog(D))
• space O(log(1/d) . polylog(D))
• failure probability d
• Future Work
• approximation factor not exponential in d
• (1e)-approximation algorithm

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Thank you for your attention!
Department of Computer Science Technische
Universität Dortmund Otto-Hahn-Str. 14 44221
Dortmund, Germany Phone [phone]
Fax. [phone] Email
christiane.lammersen_at_tu-dortmund.de http//ls2-ww
w.cs.uni-dortmund.de/lammersen/