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The Geometry of Binary Search Trees

The Geometry of Binary Search Trees. Dion Harmon NECSI. Erik Demaine MIT. John Iacono Poly Inst. of NYU. Daniel Kane Harvard. Mihai P ă tra ş cu IBM Almaden. ASS. Point set in the plane

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The Geometry of Binary Search Trees

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  1. The Geometry ofBinary Search Trees Dion Harmon NECSI Erik Demaine MIT John Iacono Poly Inst. of NYU Daniel Kane Harvard Mihai Pătraşcu IBM Almaden

  2. ASS • Point set in the plane • ASS = any nontrivial rectangle spanned by two points contains another point(interior or on boundary)

  3. MinASS Problem • Problem: Given a point set, find the minimum ASS superset • Up to constant factors, can assume input points are in general position (no two horiz/vert aligned) original points added points

  4. MinASS in Worst Case • O(n lg n) points always suffice • Ω(n lg n) points are sometimes necessary(random; bit-reversal permutation matrix) ≤¼n ≤½n ≤¼n ≤¼n ≤½n ≤¼n ≤ n

  5. NP-completeness • Theorem: MinASS is NP-complete • OPEN:General positionMinASS

  6. Approximating MinASS • OPEN: O(1)-approximation? • Known: O(lg lg n)-approximation[and it’s not easy] original points added points

  7. GreedyASS • Imagine points arriving row by row • Add necessary points on the new rowto remain ASS • Conjecture:O(1)-approximation original points added points

  8. Binary Search Tree (BST) • Recall our good old friends: • Unit-cost operations: • Move finger up/left/right • Rotate left/right 8 4 12 2 6 10 14 1 3 5 7 9 11 13 15 4 2 2 4 5 1 1 5 3 3

  9. The Best BST • Problem: • Given (offline) access sequence S=x1, x2, …, xm • OPT(S) = minimum sequence of unit-cost opsin BST to touch x1, x2, …, xm in order • Without rotations, this problem is solved by (static) optimal BSTs of Knuth [1971] • Ultimate goal:O(1)-competitive online algorithm 8 4 12 2 6 10 14 1 3 5 7 9 11 13 15

  10. Example ofBST Execution rotate • Access sequence:1, 4, 5, 7, 6, 2, 3 4 2 6 1 3 5 7

  11. 6 Example ofBST Execution 4 7 2 5 • Access sequence:1, 4, 5, 7, 6, 2, 3 1 3

  12. 1,2,4 ASS/BST Equivalence 3 5,6 7 • In fact, any ASS point set is a BST execution! • Treap by next access time • Corollary: MinASS(S)= OPT(S) 1 2 3 4 5 6 7

  13. Attacks on OPT(S) • Splay trees [Sleator & Tarjan 1983] • Conjecture: O(1)-competitive • Many nice properties known,but no o(lg n)-competitiveness known • Tango trees [Demaine, Harmon, Iacono, Pătraşcu 2004] • O(lglg n)-competitive • GreedyASS[Lucas 1988; Munro 2000] • Proposed as offline BST algorithm(“Order by Next Request”) • No o(n)-competitiveness known

  14. Online ASS/BST Equivalence • Theorem: If ASS sequence computed online row by row (like GreedyASS), then can convert to an online BST algorithm with constant-factor slowdown • Transform to geometry and back • Main ingredient: split trees • Support any sequence of n splits in O(n) time • Splay trees take O(n (n)) time [Lucas 1988] • Splay trees take O(n *(n)) time [Pettie 20??] x ≤x >x

  15. New Dynamic Optimality Conjecture • GreedyASS is now an online algorithm! • Conjecture:GreedyASS is O(1)-competitive • Previously conjectured to be anoffline O(1)-approximation to OPT[Lucas 1988; Munro 2000]

  16. Lower Bounds • Need lower bounds on OPT(S) to compare algorithms (like GreedyASS) against • Wilber [1989] proved two lower bounds: • Wilber I used for O(lg lg n)-competitiveness of Tango trees • Wilber II used for key-independent optimality [Iacono 2002] • Conjecture: Wilber II ≥ Wilber I • Conjecture: OPT(S) = Θ(Wilber II)

  17. Independent Set Lower Bounds • Wilber I and Wilber II fall in the (new) class of independent rectangle bounds • Theorem: MinASS(S) = Ω(largest independent rectangle set) • What is the best lower bound in this class? independent dependent

  18. SignedGreedy + − Just fix empty positive-slope rectangles Just fix empty negative-slope rectangles • Max of: original points original points original points added points added points added points

  19. Lower Bounds: SignedGreedy • Theorem: SignedGreedy(S) is withina constant factor of the bestindependent rectangle bound • Corollary: • OPT(S) ≥ SignedGreedy(S) • SignedGreedy(S) = Ω(Wilber I + WilberII) • SignedGreedy (lower bound) is annoyingly similar to GreedyASS (upper bound)

  20. Open Problems • Does GreedyASS share all the nice properties of splay trees? • Recent result: [Iacono & Pătraşcu]GreedyASS satisfies access theorem (from splay trees) working-set bound entropy bound static finger bound O(lg n) amortized per operation • Anyone for dynamic finger?

  21. Open Problems • Hardness of approximability • NP-hardness of general position

  22. P.S. • ASS = Arborally Satisfied Set • [Arboral = arboreal = related to trees]

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