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On Cartesian Trees, Lowest Common Ancestors, and Range Minimum Queries 0 2 0 4 3 5 1 7 6 1 2 min? 3 6 4 5 7 LCA? Parallel Computing Day Ben-Gurion University Tuesday, October 20, 2009 RMQ 2 0 4 3 5 1 7 6 Tuesday,

  1. Parallel RMQ [Berkman and Vishkin 1993] • Min of n elements in O(1) time using n 2 processors [Valiant 1975] 1 0 1 2 3 2 1 0 1 2 1 2 3 4 5 4 3 2 3 4 Tuesday, October 20, 2009

  2. Parallel RMQ [Berkman and Vishkin 1993] • Min of n elements in O(1) time using n 2 processors [Valiant 1975] • O(1) RMQ using n 4 processors 1 0 1 2 3 2 1 0 1 2 1 2 3 4 5 4 3 2 3 4 Tuesday, October 20, 2009

  3. Parallel RMQ [Berkman and Vishkin 1993] • Min of n elements in O(1) time using n 2 processors [Valiant 1975] • O(1) RMQ using n 4 processors • O(1) RMQ using n 2.5 processors √ n √ n √ n √ n 0 0 1 2 1 0 1 2 3 2 1 0 1 2 1 2 3 4 5 4 3 2 3 4 Tuesday, October 20, 2009

  4. Parallel RMQ [Berkman and Vishkin 1993] • Min of n elements in O(1) time using n 2 processors [Valiant 1975] • O(1) RMQ using n 4 processors • O(1) RMQ using n 2.5 processors • O(1/ ε ) RMQ using n 1+ ε processors √ n √ n √ n √ n 0 0 1 2 1 0 1 2 3 2 1 0 1 2 1 2 3 4 5 4 3 2 3 4 Tuesday, October 20, 2009

  5. Parallel RMQ 1 [Berkman and Vishkin 1993] 1 0 1 2 3 2 1 0 1 2 1 2 3 4 5 4 3 2 3 4 Tuesday, October 20, 2009

  6. Parallel RMQ 1 [Berkman and Vishkin 1993] • Min of n integers each between 1 and n in O(1) time using n processors [Fich, Ragde and Wigderson 1984] 1 0 1 2 3 2 1 0 1 2 1 2 3 4 5 4 3 2 3 4 Tuesday, October 20, 2009

  7. Parallel RMQ 1 [Berkman and Vishkin 1993] • Min of n integers each between 1 and n in O(1) time using n processors [Fich, Ragde and Wigderson 1984] 1 0 1 2 3 2 1 0 1 2 1 2 3 4 5 4 3 2 3 4 Tuesday, October 20, 2009

  8. Parallel RMQ 1 [Berkman and Vishkin 1993] • Min of n integers each between 1 and n in O(1) time using n processors [Fich, Ragde and Wigderson 1984] 1 0 1 2 3 2 1 0 1 2 1 2 3 4 5 4 3 2 3 4 Tuesday, October 20, 2009

  9. Parallel RMQ 1 [Berkman and Vishkin 1993] • Min of n integers each between 1 and n in O(1) time using n processors [Fich, Ragde and Wigderson 1984] 1 0 1 2 3 2 1 0 1 2 1 2 3 4 5 4 3 2 3 4 Tuesday, October 20, 2009

  10. Parallel RMQ 1 [Berkman and Vishkin 1993] • Min of n integers each between 1 and n in O(1) time using n processors [Fich, Ragde and Wigderson 1984] 1 0 1 2 3 2 1 0 1 2 1 2 3 4 5 4 3 2 3 4 Tuesday, October 20, 2009

  11. Parallel RMQ 1 [Berkman and Vishkin 1993] • Min of n integers each between 1 and n in O(1) time using n processors [Fich, Ragde and Wigderson 1984] 1 0 1 2 3 2 1 0 1 2 1 2 3 4 5 4 3 2 3 4 Tuesday, October 20, 2009

  12. Parallel RMQ 1 [Berkman and Vishkin 1993] • Min of n integers each between 1 and n in O(1) time using n processors [Fich, Ragde and Wigderson 1984] 1 0 1 2 3 2 1 0 1 2 1 2 3 4 5 4 3 2 3 4 • n log 3 n processors, O(1) time, O(1) query Tuesday, October 20, 2009

  13. Parallel RMQ 1 [Berkman and Vishkin 1993] • Min of n integers each between 1 and n in O(1) time using n processors [Fich, Ragde and Wigderson 1984] 1 0 1 2 3 2 1 0 1 2 1 2 3 4 5 4 3 2 3 4 • n log 3 n processors, O(1) time, O(1) query • n α k ( n ) processors, O(k) time, O(k) query Tuesday, October 20, 2009

  14. Problems with RMQ 1 • RMQ LCA RMQ 1 2 0 4 3 5 4 7 0 5 6 1 4 8 6 7 3 4 2 5 4 Tuesday, October 20, 2009

  15. Problems with RMQ 1 • RMQ LCA RMQ 1 • inefficient in parallel • inefficient in terms of cache-misses (can’t be done via scans) 2 0 4 3 5 4 7 0 5 6 1 4 8 6 7 3 4 2 5 4 Tuesday, October 20, 2009

  16. Problems with RMQ 1 • RMQ LCA RMQ 1 • inefficient in parallel • inefficient in terms of cache-misses (can’t be done via scans) ¼ log n ¼ log n ¼ log n ¼ log n 0 0 1 2 2 0 4 3 5 4 7 0 5 6 1 4 8 6 7 3 4 2 5 4 • # different Blocks = # different Cartesian trees = 4 ¼ log n = √ n [Fischer, Heun 2006] Tuesday, October 20, 2009

  17. Problems with RMQ 1 • RMQ LCA RMQ 1 • inefficient in parallel • inefficient in terms of cache-misses (can’t be done via scans) ¼ log n ¼ log n ¼ log n ¼ log n 0 0 1 2 2 0 4 3 5 4 7 0 5 6 1 4 8 6 7 3 4 2 5 4 • # different Blocks = # different Cartesian trees = 4 ¼ log n = √ n [Fischer, Heun 2006] • Lookup table: index, construct Tuesday, October 20, 2009

  18. Cache-Oblivious RMQ [Demaine, Landau and W. 2009] • An optimal RMQ solution that only makes sequential scans O( n ) prep. O(1) query (serial algorithm) Tuesday, October 20, 2009

  19. A Cache-Oblivious Cartesian Tree ¼ log n 2 0 4 3 5 1 7 6 Tuesday, October 20, 2009

  20. A Cache-Oblivious Cartesian Tree 0 1 2 ¼ log n 3 6 2 0 4 3 5 1 7 6 4 5 7 Tuesday, October 20, 2009

  21. A Cache-Oblivious Cartesian Tree 0 1 2 ¼ log n 3 6 2 0 4 3 5 1 7 6 4 5 7 • cache-oblivious stack holds rightmost path Tuesday, October 20, 2009

  22. A Cache-Oblivious Cartesian Tree 0 1 2 ¼ log n 3 6 2 0 4 3 5 1 7 6 4 5 7 • cache-oblivious stack holds rightmost path ... • when we climb (pop) i vertices, concatenate 0111 11 { i Tuesday, October 20, 2009

  23. A Cache-Oblivious Cartesian Tree 0 1 2 ¼ log n 3 6 2 0 4 3 5 1 7 6 4 5 7 ... ... ... ... 011 1011 1011 1011 1 { { { { i 4 i 1 i 3 i 2 • cache-oblivious stack holds rightmost path ... • when we climb (pop) i vertices, concatenate 0111 11 { i Tuesday, October 20, 2009

  24. A Cache-Oblivious Cartesian Tree 0 1 2 ¼ log n 3 6 2 0 4 3 5 1 7 6 4 5 7 [ ] i 4 ∈ ... ... ... ... 011 1011 1011 1011 1 √ n { { { { i 1 i 3 i 2 • cache-oblivious stack holds rightmost path ... • when we climb (pop) i vertices, concatenate 0111 11 { i Tuesday, October 20, 2009

  25. A Cache-Oblivious Cartesian Tree 0 1 2 ¼ log n 3 6 2 0 4 3 5 1 7 6 4 5 7 min? • ∀ binary string and ∀ query: [ ] i 4 ∈ ... ... ... ... 011 1011 1011 1011 1 √ n { { { { i 1 i 3 i 2 • cache-oblivious stack holds rightmost path ... • when we climb (pop) i vertices, concatenate 0111 11 { i Tuesday, October 20, 2009

  26. A Cache-Oblivious Cartesian Tree 0 1 2 ¼ log n 3 6 2 0 4 3 5 1 7 6 4 5 7 min? • ∀ binary string and ∀ query: [ ] i 4 ∈ 011 ⋄⋄⋄ 1011 ⋄⋄⋄ 1011 ⋄⋄⋄ 1011 ⋄⋄⋄ 1 √ n { { { { i 1 i 3 i 2 • cache-oblivious stack holds rightmost path ... • when we climb (pop) i vertices, concatenate 0111 11 { i Tuesday, October 20, 2009

  27. RMQ Generalization I : A Cartesian Tree of a Tree [Demaine, Landau and W. 2009] Tuesday, October 20, 2009

  28. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph Tuesday, October 20, 2009

  29. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph • query asks for max flow that can be routed between u,v along any simple path Tuesday, October 20, 2009

  30. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph • query asks for max flow that can be routed between u,v along any simple path • Undirected graphs: solve problem on the Maximum Spanning Tree [Hu 1961] Tuesday, October 20, 2009

  31. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph • query asks for max flow that can be routed between u,v along any simple path • Undirected graphs: solve problem on the Maximum Spanning Tree [Hu 1961] 5 1 2 6 3 6 3 Tuesday, October 20, 2009

  32. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph • query asks for max flow that can be routed between u,v along any simple path • Undirected graphs: solve problem on the Maximum Spanning Tree [Hu 1961] 5 1 2 6 v 3 6 3 u Tuesday, October 20, 2009

  33. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph • query asks for max flow that can be routed between u,v along any simple path • Undirected graphs: solve problem on the Maximum Spanning Tree [Hu 1961] 5 1 2 6 v 3 6 3 u Tuesday, October 20, 2009

  34. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph • query asks for max flow that can be routed between u,v along any simple path • Undirected graphs: solve problem on the Maximum Spanning Tree [Hu 1961] • The Minimum Spanning Tree Verification problem 5 1 2 6 v 3 6 3 u Tuesday, October 20, 2009

  35. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph • query asks for max flow that can be routed between u,v along any simple path • Undirected graphs: solve problem on the Maximum Spanning Tree [Hu 1961] • The Minimum Spanning Tree Verification problem 5 1 2 6 v 3 6 3 u 5 Tuesday, October 20, 2009

  36. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph • query asks for max flow that can be routed between u,v along any simple path • Undirected graphs: solve problem on the Maximum Spanning Tree [Hu 1961] • The Minimum Spanning Tree Verification problem 5 1 2 6 6 v 3 6 3 u 5 Tuesday, October 20, 2009

  37. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph • query asks for max flow that can be routed between u,v along any simple path • Undirected graphs: solve problem on the Maximum Spanning Tree [Hu 1961] • The Minimum Spanning Tree Verification problem O( n α k ( n )) prep. O( k ) query [Alon and Schieber 1987] 5 1 Ω ( n α k ( n )) prep. for O( k ) query [Pettie 2002] 2 6 3 6 3 Tuesday, October 20, 2009

  38. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph • query asks for max flow that can be routed between u,v along any simple path • Undirected graphs: solve problem on the Maximum Spanning Tree [Hu 1961] • The Minimum Spanning Tree Verification problem 5 1 2 6 3 6 3 Tuesday, October 20, 2009

  39. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph • query asks for max flow that can be routed between u,v along any simple path • Undirected graphs: solve problem on the Maximum Spanning Tree [Hu 1961] • The Minimum Spanning Tree Verification problem 1 5 1 2 6 3 6 3 Tuesday, October 20, 2009

  40. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph • query asks for max flow that can be routed between u,v along any simple path • Undirected graphs: solve problem on the Maximum Spanning Tree [Hu 1961] • The Minimum Spanning Tree Verification problem 1 1 5 1 2 6 3 6 3 Tuesday, October 20, 2009

  41. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph • query asks for max flow that can be routed between u,v along any simple path • Undirected graphs: solve problem on the Maximum Spanning Tree [Hu 1961] • The Minimum Spanning Tree Verification problem 1 1 5 1 2 6 3 6 3 Tuesday, October 20, 2009

  42. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph • query asks for max flow that can be routed between u,v along any simple path • Undirected graphs: solve problem on the Maximum Spanning Tree [Hu 1961] • The Minimum Spanning Tree Verification problem 1 1 5 1 2 6 3 6 3 Tuesday, October 20, 2009

  43. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph • query asks for max flow that can be routed between u,v along any simple path • Undirected graphs: solve problem on the Maximum Spanning Tree [Hu 1961] • The Minimum Spanning Tree Verification problem 1 1 5 1 2 6 3 6 3 Tuesday, October 20, 2009

  44. RMQ Generalization I : A Cartesian Tree of a Tree • The Bottleneck Edge Query problem (RMQ on graphs\trees): • preprocess an edge-weighted graph • query asks for max flow that can be routed between u,v along any simple path • Undirected graphs: solve problem on the Maximum Spanning Tree [Hu 1961] • The Minimum Spanning Tree Verification problem 1 1 5 1 2 6 3 6 3 Tuesday, October 20, 2009

  45. RMQ Generalization I : A Cartesian Tree of a Tree • Construct Cartesian tree of an input tree in O( n + sort(edges)) 1 1 5 1 2 6 3 6 3 Tuesday, October 20, 2009

  46. RMQ Generalization I : A Cartesian Tree of a Tree • Construct Cartesian tree of an input tree in O( n + sort(edges)) • Tight lower bound 1 1 5 1 2 6 3 6 3 Tuesday, October 20, 2009

  47. RMQ Generalization I : Compared to AlonSchieber • Construct Cartesian tree of an input tree in O( n + sort(edges)) • Tight lower bound 1 1 5 1 2 6 3 6 3 Tuesday, October 20, 2009

  48. RMQ Generalization I : Compared to AlonSchieber • Construct Cartesian tree of an input tree in O( n + sort(edges)) • Tight lower bound • Linear-time if edge-weights are sorted or integers 1 1 5 1 2 6 3 6 3 Tuesday, October 20, 2009

  49. RMQ Generalization I : Compared to AlonSchieber • Construct Cartesian tree of an input tree in O( n + sort(edges)) • Tight lower bound • Linear-time if edge-weights are sorted or integers • Otherwise we get linear-space, O( n log [ k ] n ) prep. O( k ) query 1 1 5 1 2 6 3 6 3 Tuesday, October 20, 2009

  50. RMQ Generalization I : Compared to AlonSchieber • Construct Cartesian tree of an input tree in O( n + sort(edges)) • Tight lower bound • Linear-time if edge-weights are sorted or integers • Otherwise we get linear-space, O( n log [ k ] n ) prep. O( k ) query • Maintain dynamic Cartesian tree and LCA info 1 1 5 1 2 6 3 6 3 Tuesday, October 20, 2009

  51. RMQ Generalization I : Compared to AlonSchieber • Construct Cartesian tree of an input tree in O( n + sort(edges)) • Tight lower bound • Linear-time if edge-weights are sorted or integers • Otherwise we get linear-space, O( n log [ k ] n ) prep. O( k ) query • Maintain dynamic Cartesian tree and LCA info • Distributed: LCA labeling of the Cartesian tree 1 1 5 1 2 6 3 6 3 Tuesday, October 20, 2009

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