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Simultaneous Nearest Neighbor Search Piotr Indyk Robert Kleinberg MIT Cornell Sepideh Mahabadi Yang Yuan MIT Cornell Nearest Neighbor Dataset of points in a metric space (, ) 6/17/2016 2 Nearest Neighbor

  1. Simultaneous Nearest Neighbor Search Piotr Indyk Robert Kleinberg MIT Cornell Sepideh Mahabadi Yang Yuan MIT Cornell

  2. Nearest Neighbor β€’ Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, 𝑒 π‘Œ ) 6/17/2016 2

  3. Nearest Neighbor β€’ Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, 𝑒 π‘Œ ) β€’ A query point comes online π‘Ÿ π‘Ÿ 6/17/2016 3

  4. Nearest Neighbor β€’ Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, 𝑒 π‘Œ ) β€’ A query point comes online π‘Ÿ π‘Ÿ β€’ Goal: π‘ž βˆ— β€’ Find the nearest data-set point π‘ž βˆ— 6/17/2016 4

  5. Nearest Neighbor β€’ Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, 𝑒 π‘Œ ) β€’ A query point comes online π‘Ÿ π‘Ÿ β€’ Goal: π‘ž βˆ— β€’ Find the nearest data-set point π‘ž βˆ— β€’ Do it in sub-linear time 6/17/2016 5

  6. Approximate Nearest Neighbor β€’ Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, 𝑒 π‘Œ ) β€’ A query point comes online π‘Ÿ π‘ž π‘Ÿ β€’ Goal: π‘ž βˆ— β€’ Find the nearest data-set point π‘ž βˆ— β€’ Do it in sub-linear time β€’ Approximate Nearest Neighbor 6/17/2016 6

  7. What if We have multiple queries We need the results of the queries to be related. 6/17/2016 7

  8. What if We have multiple queries We need the results of the queries to be related. Example: β€’ Noisy image β€’ For each pixel find the true color β€’ Neighboring pixels have similar color 6/17/2016 8

  9. Simultaneous NN Problem 6/17/2016 Sepideh Mahabadi 9

  10. The SNN problem ( Felzenszwalb’15) β€’ Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, 𝑒 π‘Œ ) 6/17/2016 10

  11. The SNN problem ( Felzenszwalb’15) β€’ Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, 𝑒 π‘Œ ) β€’ Query comes online and contains π‘Ÿ 2 β€’ 𝑙 query points 𝑅 = (π‘Ÿ 1 , … , π‘Ÿ 𝑙 ) π‘Ÿ 3 π‘Ÿ 1 6/17/2016 11

  12. The SNN problem ( Felzenszwalb’15) β€’ Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, 𝑒 π‘Œ ) β€’ Query comes online and contains π‘Ÿ 2 β€’ 𝑙 query points 𝑅 = (π‘Ÿ 1 , … , π‘Ÿ 𝑙 ) π‘Ÿ 3 π‘Ÿ 1 β€’ A compatibility graph 𝐻 = 𝑅, 𝐹 𝐻 6/17/2016 12

  13. The SNN problem ( Felzenszwalb’15) β€’ Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, 𝑒 π‘Œ ) β€’ Query comes online and contains π‘Ÿ 2 β€’ 𝑙 query points 𝑅 = (π‘Ÿ 1 , … , π‘Ÿ 𝑙 ) π‘Ÿ 3 π‘Ÿ 1 β€’ A compatibility graph 𝐻 = 𝑅, 𝐹 𝐻 6/17/2016 13

  14. The SNN problem (Felzenszwalb’15) β€’ Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, 𝑒 π‘Œ ) β€’ Query comes online and contains π‘Ÿ 2 β€’ 𝑙 query points 𝑅 = (π‘Ÿ 1 , … , π‘Ÿ 𝑙 ) π‘Ÿ 3 π‘Ÿ 1 β€’ A compatibility graph 𝐻 = 𝑅, 𝐹 𝐻 6/17/2016 14

  15. The SNN problem ( Felzenszwalb’15) β€’ Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, 𝑒 π‘Œ ) π‘ž 3 β€’ Query comes online and contains π‘ž 2 π‘Ÿ 2 β€’ 𝑙 query points 𝑅 = (π‘Ÿ 1 , … , π‘Ÿ 𝑙 ) π‘Ÿ 3 π‘ž 1 π‘Ÿ 1 β€’ A compatibility graph 𝐻 = 𝑅, 𝐹 𝐻 β€’ Goal is to report (π‘ž 1 , … , π‘ž 𝑙 ) , π‘ž 𝑗 ∈ 𝑄 , that minimizes 𝒍 𝒋=𝟐 𝒆 𝒀 (𝒓 𝒋 , 𝒒 𝒋 ) + 𝒓 𝒋 ,𝒓 π’Œ βˆˆπ‘­ 𝑯 𝒆 𝒀 (𝒒 𝒋 , 𝒒 π’Œ ) 6/17/2016 15

  16. The Generalized SNN β€’ Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, 𝑒 π‘Œ ) π‘ž 3 β€’ Query comes online and contains π‘ž 2 π‘Ÿ 2 β€’ 𝑙 query points 𝑅 = (π‘Ÿ 1 , … , π‘Ÿ 𝑙 ) π‘Ÿ 3 π‘ž 1 π‘Ÿ 1 β€’ A compatibility graph 𝐻 = 𝑅, 𝐹 𝐻 β€’ Goal is to report (π‘ž 1 , … , π‘ž 𝑙 ) , π‘ž 𝑗 ∈ 𝑄 , that minimizes 𝒍 𝒋=𝟐 𝝀 𝒋 𝑒 𝒁 (𝒓 𝒋 , 𝒒 𝒋 ) + 𝒓 𝒋 ,𝒓 π’Œ βˆˆπ‘­ 𝑯 𝝁 𝒋,π’Œ 𝒆 𝒀 (𝒒 𝒋 , 𝒒 π’Œ ) 6/17/2016 16

  17. Independent NN Algorithm 6/17/2016 Sepideh Mahabadi 17

  18. Independent NN Algorithm INN Algorithm β€’ For each query point π‘Ÿ 𝑗 ∈ 𝑅 π‘Ÿ 2 π‘Ÿ 3 π‘Ÿ 1 6/17/2016 18

  19. Independent NN Algorithm INN Algorithm β€’ For each query point π‘Ÿ 𝑗 ∈ 𝑅 β€’ Independently find a (approximate) nearest neighbor 𝒒 𝒋 (Searching step) π‘ž 2 π‘Ÿ 2 π‘Ÿ 3 π‘ž 1 π‘Ÿ 1 π‘ž 3 6/17/2016 19

  20. Independent NN Algorithm INN Algorithm β€’ For each query point π‘Ÿ 𝑗 ∈ 𝑅 β€’ Independently find a (approximate) nearest neighbor 𝒒 𝒋 (Searching step) β€’ Replace the label set 𝑄 with the reduced set 𝑸 = { 𝒒 𝟐 , … , 𝒒 𝒍 } (Pruning step) π‘ž 2 π‘Ÿ 2 π‘Ÿ 3 π‘ž 1 π‘Ÿ 1 π‘ž 3 6/17/2016 20

  21. Independent NN Algorithm INN Algorithm β€’ For each query point π‘Ÿ 𝑗 ∈ 𝑅 β€’ Independently find a (approximate) nearest neighbor 𝒒 𝒋 (Searching step) β€’ Replace the label set 𝑄 with the reduced set 𝑸 = { 𝒒 𝟐 , … , 𝒒 𝒍 } (Pruning step) β€’ Solve the problem for 𝑄 π‘ž 2 π‘Ÿ 2 π‘Ÿ 3 π‘ž 1 π‘Ÿ 1 π‘ž 3 6/17/2016 21

  22. Independent NN Algorithm INN Algorithm β€’ For each query point π‘Ÿ 𝑗 ∈ 𝑅 β€’ Independently find a (approximate) nearest neighbor 𝒒 𝒋 (Searching step) β€’ Replace the label set 𝑄 with the reduced set 𝑸 = { 𝒒 𝟐 , … , 𝒒 𝒍 } (Pruning step) β€’ Solve the problem for 𝑄 π‘ž 2 π‘Ÿ 2 οƒ˜ Reduces the size of labels from π‘œ down to 𝑙 π‘Ÿ 3 π‘ž 1 π‘Ÿ 1 π‘ž 3 6/17/2016 22

  23. Independent NN Algorithm INN Algorithm β€’ For each query point π‘Ÿ 𝑗 ∈ 𝑅 β€’ Independently find a (approximate) nearest neighbor 𝒒 𝒋 (Searching step) β€’ Replace the label set 𝑄 with the reduced set 𝑸 = { 𝒒 𝟐 , … , 𝒒 𝒍 } (Pruning step) β€’ Solve the problem for 𝑄 π‘ž 2 π‘Ÿ 2 οƒ˜ Reduces the size of labels from π‘œ down to 𝑙 π‘Ÿ 3 π‘ž 1 π‘Ÿ 1 π‘ž 3 οƒ˜ The optimal value increases by a factor 𝜷 οƒ˜ pruning gap 6/17/2016 23

  24. Independent NN Algorithm INN Algorithm β€’ For each query point π‘Ÿ 𝑗 ∈ 𝑅 β€’ Independently find a (approximate) nearest neighbor 𝒒 𝒋 (Searching step) β€’ Replace the label set 𝑄 with the reduced set 𝑸 = { 𝒒 𝟐 , … , 𝒒 𝒍 } (Pruning step) β€’ Solve the problem for 𝑄 π‘ž 2 π‘Ÿ 2 οƒ˜ Reduces the size of labels from π‘œ down to 𝑙 π‘Ÿ 3 π‘ž 1 π‘Ÿ 1 π‘ž 3 οƒ˜ The optimal value increases by a factor 𝜷 οƒ˜ pruning gap οƒ˜ Any metric labeling 𝛾 -approximate algorithm can be used on the reduced set , giving us (𝛽 β‹… 𝛾) -approximate algorithm. 6/17/2016 24

  25. Results 6/17/2016 Sepideh Mahabadi 25

  26. Results β€’ Prove bounds for the pruning gap 6/17/2016 26

  27. Results β€’ Prove bounds for the pruning gap 𝐦𝐩𝐑 𝒍 β€’ 𝜷 = 𝑷 𝐦𝐩𝐑 𝐦𝐩𝐑 π₯ 6/17/2016 27

  28. Results β€’ Prove bounds for the pruning gap 𝐦𝐩𝐑 𝒍 β€’ 𝜷 = 𝑷 𝐦𝐩𝐑 𝐦𝐩𝐑 π₯ β€’ 𝜷 = 𝛁 𝐦𝐩𝐑 𝒍 6/17/2016 28

  29. Results β€’ Prove bounds for the pruning gap 𝐦𝐩𝐑 𝒍 β€’ 𝜷 = 𝑷 𝐦𝐩𝐑 𝐦𝐩𝐑 π₯ β€’ 𝜷 = 𝛁 𝐦𝐩𝐑 𝒍 β€’ For 𝑠 -sparse graph: 𝜷 = 𝑷 𝒔 6/17/2016 29

  30. Results β€’ Prove bounds for the pruning gap 𝐦𝐩𝐑 𝒍 β€’ 𝜷 = 𝑷 𝐦𝐩𝐑 𝐦𝐩𝐑 π₯ β€’ 𝜷 = 𝛁 𝐦𝐩𝐑 𝒍 β€’ For 𝑠 -sparse graph: 𝜷 = 𝑷 𝒔 β€’ Graphs with pseudo-arboricity 𝑠 : each edge can be mapped to a vertex such that at most 𝑠 edges are mapped to any vertex 6/17/2016 30

  31. Results β€’ Prove bounds for the pruning gap 𝐦𝐩𝐑 𝒍 β€’ 𝜷 = 𝑷 𝐦𝐩𝐑 𝐦𝐩𝐑 π₯ β€’ 𝜷 = 𝛁 𝐦𝐩𝐑 𝒍 β€’ For 𝑠 -sparse graph: 𝜷 = 𝑷 𝒔 β€’ Graphs with pseudo-arboricity 𝑠 : each edge can be mapped to a vertex such that at most 𝑠 edges are mapped to any vertex β€’ Would mean constant approximation factor for trees, grids, planar graphs , …, and in particular 𝑃(𝑠) -approximation for 𝑠 - degree graphs 6/17/2016 31

  32. Results β€’ Prove bounds for the pruning gap 𝐦𝐩𝐑 𝒍 β€’ 𝜷 = 𝑷 𝐦𝐩𝐑 𝐦𝐩𝐑 π₯ β€’ 𝜷 = 𝛁 𝐦𝐩𝐑 𝒍 β€’ For 𝑠 -sparse graph: 𝜷 = 𝑷 𝒔 β€’ Graphs with pseudo-arboricity 𝑠 : each edge can be mapped to a vertex such that at most 𝑠 edges are mapped to any vertex β€’ Would mean constant approximation factor for trees, grids, planar graphs , …, and in particular 𝑃(𝑠) -approximation for 𝑠 - degree graphs β€’ 𝜷 is very close to one in experiments 6/17/2016 32

  33. Results β€’ Prove bounds for the pruning gap 𝐦𝐩𝐑 𝒍 β€’ 𝜷 = 𝑷 𝐦𝐩𝐑 𝐦𝐩𝐑 π₯ β€’ 𝜷 = 𝛁 𝐦𝐩𝐑 𝒍 β€’ For 𝑠 -sparse graph: 𝜷 = 𝑷 𝒔 β€’ Graphs with pseudo-arboricity 𝑠 : each edge can be mapped to a vertex such that at most 𝑠 edges are mapped to any vertex β€’ Would mean constant approximation factor for trees, grids, planar graphs , …, and in particular 𝑃(𝑠) -approximation for 𝑠 - degree graphs β€’ 𝜷 is very close to one in experiments 6/17/2016 33

  34. Overview of the proof for 𝐦𝐩𝐑 𝒍 𝜷 = 𝑷 𝐦𝐩𝐑 𝐦𝐩𝐑 𝒍 6/17/2016 Sepideh Mahabadi 34

  35. 0-Extension Problem [Kar98] 6/17/2016 35

  36. 0-Extension Problem [Kar98] β€’ The input: β€’ a graph 𝐼 π‘Š, 𝐹 6/17/2016 36

  37. 0-Extension Problem [Kar98] β€’ The input: β€’ a graph 𝐼 π‘Š, 𝐹 β€’ a weight function π‘₯ 𝑓 1 2 2 1 2 6/17/2016 37

  38. 0-Extension Problem [Kar98] β€’ The input: β€’ a graph 𝐼 π‘Š, 𝐹 β€’ a weight function π‘₯ 𝑓 β€’ a set of terminals π‘ˆ βŠ‚ π‘Š 1 2 2 1 2 6/17/2016 38

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