Aaabbbbccccc

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Aaabbbbccccc

  1. 1. Approximation Algorithms for Problems on Networks and Streams of Data Luca Foschini - Ph.D. Defense Committee: Subhash Suri (chair), John Gilbert, Teofilo GonzalezFriday, September 7, 12
  2. 2. Why Approximation Algorithms?Friday, September 7, 12
  3. 3. Why Approximation Algorithms? Exact algorithms require many resourcesFriday, September 7, 12
  4. 4. Why Approximation Algorithms? Hardware Exact algorithms require many resources Apps DataFriday, September 7, 12
  5. 5. Why Approximation Algorithms? Hardware Exact algorithms require many resources Apps Problems solvable exactly DataFriday, September 7, 12
  6. 6. A Long History, and Work in Progress © Original ArtistFriday, September 7, 12
  7. 7. A Long History, and Work in Progress ✤ Early ‘70s - many combinatorial problems found to be NP-hard ✤ Recently - more restricting computation models proposed e.g., data stream © Original ArtistFriday, September 7, 12
  8. 8. A Long History, and Work in Progress ✤ Early ‘70s - many combinatorial problems found to be NP-hard ✤ Recently - more restricting computation models proposed e.g., data stream © Original Artist Heuristics not sufficient, provable guarantees neededFriday, September 7, 12
  9. 9. Content of the DissertationFriday, September 7, 12
  10. 10. Content of the Dissertation "Friday, September 7, 12
  11. 11. Content of the Dissertation Networks " Data StreamsFriday, September 7, 12
  12. 12. Content of the Dissertation STACS12 + Partitioning Algorithmica Networks SODA11 + Shortest Paths Algorithmica " Time Series ICDE10 Data Streams Burst Detection NSDI11Friday, September 7, 12
  13. 13. Content of the Dissertation STACS12 + ICISS08 Partitioning Algorithmica Networks ICIP11 SODA11 + Shortest Paths ALENEX10 Algorithmica " ESA11 Time Series ICDE10 Data Streams WOOT11 Burst Detection NSDI11 WAW09Friday, September 7, 12
  14. 14. Roadmap STACS12 + Partitioning Algorithmica Networks SODA11 + Shortest Paths Algorithmica " Time Series ICDE10 Data Streams Burst Detection NSDI11Friday, September 7, 12
  15. 15. k-Balanced Partitioning Problem Given: an unweighted graph G on n vertices; an integer k Find: a partition of the vertices of G into k sets Vi s.t. ✤ |Vi |  dn/ke ✤ Cut size (number of edges connecting vertices in different Vi) is minimized joint work with Andi Feldmann (ETHz) (appeared in STACS12, submitted to Algorithmica)Friday, September 7, 12
  16. 16. Motivation & Complexity ✤ Divide-and-conquer algorithms ✤ VLSI design ✤ Parallel computing ✤ NP-hard to approximate cut size within any finite value alpha [Andreev and Räcke 2006]Friday, September 7, 12
  17. 17. Related WorkFriday, September 7, 12
  18. 18. General Graphs & Trees ✤ Algorithm is !-approximation if finds a cut at most ! times optimal ✤ NP-hard to approximate cut size within any finite ! [Andreev and Räcke 2006]Friday, September 7, 12
  19. 19. General Graphs & Trees ✤ Algorithm is !-approximation if finds a cut at most ! times optimal ✤ NP-hard to approximate cut size within any finite ! [Andreev and Räcke 2006] Trees - simple instances?Friday, September 7, 12
  20. 20. General Graphs & Trees ✤ Algorithm is !-approximation if finds a cut at most ! times optimal ✤ NP-hard to approximate cut size n=31, k=8 cut size = 10 within any finite ! [Andreev and Räcke 2006] Trees - simple instances? n=31, k=9 cut size = 8Friday, September 7, 12
  21. 21. Trees Are HardFriday, September 7, 12
  22. 22. Trees Are Hard ✤ NP-hard to approx. cut size for !=nc (for any c<1) even if constant diameterFriday, September 7, 12
  23. 23. Trees Are Hard ✤ NP-hard to approx. cut size for !=nc (for any c<1) even if constant diameter ✤ APX-hard to approx. cut-size even if constant degreeFriday, September 7, 12
  24. 24. Trees Are Hard ✤ NP-hard to approx. cut size for !=nc (for any c<1) even if constant diameter ✤ APX-hard to approx. cut-size even if constant degree Most NP-hard problems become trivial on treesFriday, September 7, 12
  25. 25. Relax!Friday, September 7, 12
  26. 26. Relax! Balance constraint relaxed: |Vi |  (1 + ")dn/keFriday, September 7, 12
  27. 27. Relax! Balance constraint relaxed: |Vi |  (1 + ")dn/ke Balance relaxed Perfect balance Optimal cut size Cut size approximated !Friday, September 7, 12
  28. 28. Relax! Balance constraint relaxed: Bicriteria Approximation: cut size approximation ! measured |Vi |  (1 + ")dn/ke w.r.t perfectly balanced optimum Balance relaxed Perfect balance Optimal cut size Cut size approximated !Friday, September 7, 12
  29. 29. 0<eps<1 on general graphs ✤ eps>1 -- alpha in .... spreading metric techniques ✤ 0<eps < 1 not much improvement. 1/epsˆ2 log ^1.5 n ✤ What about trees?Friday, September 7, 12
  30. 30. Summary of PTAS for Trees ✤ Compute optimal cut size for each coarse signature using DP ✤ Pack each coarse signatures into bins of size (1 + ")dn/ke ✤ Pick solution with smallest cut size among those fitting into k bins 4 1+3d 1 log( 1 )e ✤ Total time complexity O(n (k/") " " )Friday, September 7, 12
  31. 31. Summary of PTAS for Trees ✤ Compute optimal cut size for each coarse signature using DP ✤ Pack each coarse signatures into bins of size (1 + ")dn/ke ✤ Pick solution with smallest cut size among those fitting into k bins 4 1+3d 1 log( 1 )e ✤ Total time complexity O(n (k/") " " ) Show that ! =1Friday, September 7, 12
  32. 32. Extension to General Graphs ✤ Decomposition of graph into collection of trees [Räcke, Madry], cut size worsen by at most O(log n) for at least 1 tree ✤ Apply PTAS for trees to each instance ✤ Return partition for tree with minimum cut ✤ alpha = O(log n) improvesFriday, September 7, 12
  33. 33. Tree DecompositionFriday, September 7, 12
  34. 34. Analysis of EmbeddingFriday, September 7, 12
  35. 35. Extensions & Open Problems ✤ Tree embedding techniques allow the !=1 tree PTAS to translate to a !=O(log n) approx for general weighted graphs ✤ Improves on previous best != O(log 1.5 n/"2 )Friday, September 7, 12
  36. 36. Extensions & Open Problems ✤ Tree embedding techniques allow the !=1 tree PTAS to translate to a !=O(log n) approx for general weighted graphs ✤ Improves on previous best != O(log 1.5 n/"2 )    Graphs TreesFriday, September 7, 12
  37. 37. Roadmap STACS12 + Partitioning Algorithmica Networks SODA11 + Shortest Paths Algorithmica " Time Series ICDE10 Data Streams Burst Detection NSDI11Friday, September 7, 12
  38. 38. Approximating Time Series ✤ Represent a time series with B linear segments ✤ New value arrives to the time series, need to reallocate segmentsFriday, September 7, 12
  39. 39. Approximating Time Series ✤ Represent a time series with B linear segments ✤ New value arrives to the time series, need to reallocate segmentsFriday, September 7, 12
  40. 40. Approximating Time Series ✤ Represent a time series with B linear segments ✤ New value arrives to the time series, need to reallocate segmentsFriday, September 7, 12
  41. 41. Old Algorithms, New ProofsFriday, September 7, 12
  42. 42. Old Algorithms, New Proofs ✤ We prove that a popular greedy merge scheme gives constant (bicriteria) approx. for many L_p norms. (ICDE10; joint with Gandhi, Suri)Friday, September 7, 12
  43. 43. Old Algorithms, New Proofs ✤ We prove that a popular greedy merge scheme gives constant (bicriteria) approx. for many L_p norms. (ICDE10; joint with Gandhi, Suri) ✤ Results implemented in Linux Kernel and used to detect traffic bursts in networks (NSDI11, joint with Uyeda, Suri, Varghese, Baker)Friday, September 7, 12
  44. 44. Old Algorithms, New Proofs ✤ We prove that a popular greedy merge scheme gives constant (bicriteria) approx. for many L_p norms. (ICDE10; joint with Gandhi, Suri) ✤ Results implemented in Linux Kernel and used to detect traffic bursts in networks (NSDI11, joint with Uyeda, Suri, Varghese, Baker) Next steps: Extend results in ICDE10 to other normsFriday, September 7, 12
  45. 45. Conclusion ✤ Approximation is necessary to reduce resource utilization ✤ Presented approximation algorithms for problems from different domains that we cannot afford to solve exactly ✤ Presented basic building blocks that can be used across the board to design approximation algorithmsFriday, September 7, 12

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