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Facility Location in Dynamic Geometric Data Streams

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

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Facility Location in Dynamic Geometric Data Streams

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

  2. 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 2

  3. Hierarchical Communication Systems upper layer offers lower layer a certain service each node can be a server cost for server ↔ access time 3 3 3

  4. Hierarchical Communication Systems upper layer offers lower layer a certain service each node can be a server cost for server ↔ access time

  5. Dynamic Geometric Data Streams m insert and delete operations points in low-dimensional, discrete space {1, ..., D}d polylog(D, m) memory space, one pass [Indyk ‘04] D 5

  6. Dynamic Uniform FLP • FLP related to k-Median but • |F| can be Q(|P|) • problem in streaming • approximation of the cost point set P facilities have uniform opening cost f clients have uniform demand b goal: maintaining FP, so as to minimize 6 6 6

  7. 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 7 7 7

  8. 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))

  9. 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 => estimator for cost: [Indyk ’04, Frahling and Sohler ‘05] Deterministic Method

  10. 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 => estimator for cost: Deterministic Method Idea: Open one facility in each heavy cell in the space partition. 10

  11. 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 => estimator for cost: Deterministic Method Idea: Open one facility in each heavy cell in the space partition. 11

  12. Nested Grids • Impose log(D)+1 • nested squared grids D = 16 Level: 4

  13. Nested Grids • Impose log(D)+1 • nested squared grids D = 16 Level: 3

  14. Nested Grids • Impose log(D)+1 • nested squared grids D = 16 Level: 2

  15. Nested Grids • Impose log(D)+1 • nested squared grids D = 16 Level: 1

  16. Nested Grids • Impose log(D)+1 • nested squared grids D = 16 Level: 0

  17. 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 => estimator for cost:

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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

  29. 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

  30. 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

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

  32. 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 => estimator for cost:

  33. Cost Estimator • For each cell size, count • the number of points • within cells of that size • => estimator for cost:

  34. Cost Estimator • For each cell size, count • the number of points • within cells of that size • => estimator for cost:

  35. Cost Estimator • For each cell size, count • the number of points • within cells of that size • => estimator for cost: 9 points

  36. Cost Estimator • For each cell size, count • the number of points • within cells of that size • => estimator for cost:

  37. Cost Estimator • For each cell size, count • the number of points • within cells of that size • => estimator for cost: 7 points

  38. 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 ito heavy cell is O(2i).

  39. Value of Cost Estimator is O(OPT(P)) radius 2i-1 • 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

  40. 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 => estimator for cost:

  41. 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,…, D}d → { 1,…,  f / 2i  } Sample set Si = { pP | hi(p) = 1} hi … 1 2 3 4 41

  42. Randomized Method for each level ido 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))

  43. Streaming Method • Idea: Reduction to counting distinct elements • Implementation: • For each level i count distinct elements in • DE1(i) = {C|C is in level i and marked}{C|C is in level i and a) or b) fails} • and DE2(i) = {C|C is in level i and a) or b) fails} • Output difference as cost for level i DE1(i) DE2(i) DE1(i+1) DE2(i+1)

  44. 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 • (1+e)-approximation algorithm 44 44 44

  45. Thank you for your attention! Department of Computer Science Technische Universität Dortmund Otto-Hahn-Str. 14 44221 Dortmund, Germany Phone: +49 231 755-4762 Fax.: +49 231 755-2047 Email: christiane.lammersen@tu-dortmund.de http://ls2-www.cs.uni-dortmund.de/~lammersen/

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