Defense

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  • Defense

    1. 1. Approximation Algorithms for Problemson Networks and Streams of DataLuca Foschini - Ph.D. DefenseCommittee: Subhash Suri (chair), John Gilbert, Teofilo Gonzalez
    2. 2. Why Approximation Algorithms?
    3. 3. Why Approximation Algorithms?Exact algorithms require many resources
    4. 4. Why Approximation Algorithms? HardwareExact algorithms require many resources Apps Data
    5. 5. Why Approximation Algorithms? HardwareExact algorithms require many resources Apps Problems solvable exactly Data
    6. 6. A Long History,and Work in Progress © Original Artist
    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 Artist
    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 needed
    9. 9. Content of the Dissertation
    10. 10. Content of the Dissertation"
    11. 11. Content of the Dissertation Networks" Data Streams
    12. 12. Content of the Dissertation STACS12 + Partitioning Algorithmica Networks SODA11 + Shortest Paths Algorithmica" Time Series ICDE10 Data Streams Burst Detection NSDI11
    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 WAW09
    14. 14. Roadmap STACS12 + Partitioning Algorithmica Networks SODA11 + Shortest Paths Algorithmica " Time Series ICDE10 Data Streams Burst Detection NSDI11
    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)
    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]
    17. 17. Related Work
    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]
    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?
    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 = 8
    21. 21. Trees Are Hard
    22. 22. Trees Are Hard✤ NP-hard to approx. cut size for !=nc (for any c<1) even if constant diameter
    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 degree
    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 trees
    25. 25. Relax!
    26. 26. Relax! Balance constraint relaxed: |Vi |  (1 + ")dn/ke
    27. 27. Relax! Balance constraint relaxed: |Vi |  (1 + ")dn/ke Balance relaxedPerfect balanceOptimal cut size Cut size approximated !
    28. 28. Relax! Balance constraint relaxed: Bicriteria Approximation: cut size approximation ! measured |Vi |  (1 + ")dn/ke w.r.t perfectly balanced optimum Balance relaxedPerfect balanceOptimal cut size Cut size approximated !
    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?
    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/") " " )
    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 ! =1
    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) improves
    33. 33. Tree Decomposition
    34. 34. Analysis of Embedding
    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 )
    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 Trees
    37. 37. Roadmap STACS12 + Partitioning Algorithmica Networks SODA11 + Shortest Paths Algorithmica " Time Series ICDE10 Data Streams Burst Detection NSDI11
    38. 38. Approximating Time Series✤ Represent a time series with B linear segments✤ New value arrives to the time series, need to reallocate segments
    39. 39. Approximating Time Series✤ Represent a time series with B linear segments✤ New value arrives to the time series, need to reallocate segments
    40. 40. Approximating Time Series✤ Represent a time series with B linear segments✤ New value arrives to the time series, need to reallocate segments
    41. 41. Old Algorithms, New Proofs
    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)
    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)
    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 norms
    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 algorithms

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