x1x1 x2x2 top-k y 3-sided x1x1 x2x2 External Memory Three-Sided Range Reporting and Top-k Queries with Sublogarithmic Updates Gerth Stølting Brodal Aarhus University arxiv.org/abs/ k = 3 “the result is obtained by combining already existing techniques (and no new techniques are introduced)” - anonymous reviewer

Updates O(log n) 3-sided O(log n + k) Updates O(log n) 3-sided & top-k O(log n + k) A ++ ++ ++ ++ ++ ++ y x1x1 x2x2 Internal Memory – Priority Search Trees Properties  leaves x-sorted  point p stored on leaf p-to-root path  y-values satisfy heap-order McCreight 1985 Frederickson 1993 BCDEFGHIJKLMNOP B EHIK O P DGJN CM L F 3-sided M D IJKL IK J N M L EFGH EH G F s = O(log n) heap-ordered top-k select k+s largest top-9 -- -- -- -- -- -- -- -- QA Q A B C D E F G H I J K L M N O P A Q in O(k+s) time

External Memory Model CPU Internal External M IO of B consecutive cells Aggarwal & Vitter 1988 B-tree Bayer & McCreight 1972 height O(log Δ (n/B)) degree Δ ≤ B Δ-1 search keys leaves Θ(B) elements Insert, Delete, Search O(log Δ (n/B)) IOs Buffered B-tree Arge 1995 Search O(log Δ (n/B)) IOs Updates amortized O(Δ/B·log Δ (n/B)) IOs buffer of ≤ B delayed updates

External Memory Results UpdatesQuery Ramaswamy, Subramanian 1995 O A (log n·log B)O(log B n + k/B) Subramanian, Ramaswamy 1995 O A (log B n + (log B n) 2 /B)O(log B n + k/B + log ** B) Arge et al. 1999O(log B n)O(log B n + k/B) NEWO A (1/(εB 1-ε ) · log B n)O A (1/ε · log B n + k/B) Afshani et al. 2011(static)O(log B n + k/B) Sheng, Tao 2012O A ((log B n) 2 )O(log B n + k/B) Tao 2014O A (log B n)O(log B n + k/B) NEWO A (1/(εB 1-ε ) · log B n)O A (1/ε · log B n + k/B) O A = amortized 3-sided top-k k = 3 NEW result : Combination of Arge 1995, Arge et al. 1999, Frederickson 1993, Blum et al. 1973

External Memory 3-sided Data Structure x-sorted degree Δ = B ε 3-sided external memory priority search tree root stores topmost Θ(B) points PvPv buffers of O(B) delayed insertions and deletions below v IvIv DvDv v CvCv efficient structure for accessing children’s P sets y-sorted  Insertions / deletions : Update root P v or add to delayed update buffer I v / D v  Update buffer overflow : Flush recursively to child with most updates (≥ B 1-ε )  Leaf overflow : split leaf, and recursively split ancestors of degree Δ+1  Underflowing point buffer P v : pull elements recursively from children using C v  3-sided query : i) Identify nodes to visit using C v structures. ii) flush updates down from ancestors of visited nodes. iii) report from nodes using P v, C v and update buffers x1x1 x2x2

Child Structure C v degree Δ = B ε 3-sided PvPv IvIv DvDv v CvCv  Capacity : B 1+ε  Insetion /deletion buffer O(B) points  O(B ε ) blocks  Catalog block  y-samples block (new) Arge et al y b1b1 b8b8 b7b7 b6b6 b5b5 b4b4 b3b3 b2b2 b 6,7 b 1,2 b 6,8 b 3,4 b 1,8 b 1,5 b 3,5 Insert / delete s points : O(1 + s/B 1-ε ) IOs 3-sided query : O(1 + k/B) IOs y-samples for range [x 1,x 2 ] : O(1) IOs (new)

External Memory Top-k – Overall Approach 3-sided top-k k = 3 x1x1 x2x2 x1x1 x2x2 y find magic y 3-sided query select k-largest Superset O(B·log Δ (n/B) + k) Answer Blum et al Construct (on demand) a binary heap over the samples of every Θ(B)’th element in the C v structures – and select the Θ(log Δ (n/B) + k/B)’th element using Frederickson 1993 All steps require O(log Δ (n/B) + k/B) IOs The 3-sided query is amortized

Summary O A = amortized 3-sided top-k k = 3 Open problem : Remove amortization ? – The End UpdatesQuery Ramaswamy, Subramanian 1995 O A (log n·log B)O(log B n + k/B) Subramanian, Ramaswamy 1995 O A (log B n + (log B n) 2 /B)O(log B n + k/B + log ** B) Arge et al. 1999O(log B n)O(log B n + k/B) NEWO A (1/(εB 1-ε ) · log B n)O A (1/ε · log B n + k/B) Afshani et al. 2011(static)O(log B n + k/B) Sheng, Tao 2012O A ((log B n) 2 )O(log B n + k/B) Tao 2014O A (log B n)O(log B n + k/B) NEWO A (1/(εB 1-ε ) · log B n)O A (1/ε · log B n + k/B)