Algorithms and Data Structures Group. Gerth Stølting Brodal

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1 Algorithms and Data Structures Group Gerth Stølting Brodal Faculty Meeting, Department of Computer Science, Aarhus University, October 3, 2014

2 VIP Lars Arge (Professor) Gerth Stølting Brodal (Lektor) Peyman Afshani (Adjunkt) Kasper Green Larsen (Adjunkt) PostDocs Allan Grønlund Jørgensen (HTF) Wanbin Son Zengfeng Huang Constantinos Tsirogiannis (AUFF) PhD students Bryan Wilkinson Edvin Berglin Ingo van Duijn Jakob Truelsen Jesper Asbjørn Sindahl Nielsen Jungwoo Yang Morten Revsbæk Sarfraz Raza Mathias Rav Konstantinos Mampentzidis MSc students Claus Jespersen Mikkel Engelbrecht Hougaard Bo Mortensen Weekly visitors Peder Klith Bøcher Brody Steven Sandel (Biodiversity Group) Programmer Svend Christian Svendsen Administration Trine Ji Holmgaard Jensen, Katrine Østerlund Rasmussen, Ellen Kjemtrup

3 Events Workshop on Massive Data Algorithmis (2009 -) Symposium on Computational Geometry (2009) European Symposium on Algorithms / ALGO 2016 MADALGO Summer Schools 2007, 2008, MADALGO retreat (October)

4 Research Efficiency O(n 3 ) O(n 2 ) O(n log 2 n) O(n log n)? log n O(n loglog n ) quartet distance between two trees Less practical? Deeper insights Improved asymptotics More complicated? algorithm engineering? O(log n) log n O( loglog n ) O(loglog n) O( loglog n)? O(1) lower bound? integer sorting, cost per element

5 Algorithms Research Theory Gerth Kasper Peyman Lars EcoInformatics BioInformatics COWI, DES, EIVA, Algorithm engineering Applications

6 Research Models of Computation RAM model memory access and other operations O(1), sometimes * is (1) Pointer model disallow arrays, memory is a graph with O(1) out-degree Functional model pointer model with no side-effects, implies persistence Comparison model/decision trees simple lower bounds Cell-probe model strong lower bounds, applies to RAM model Bit-probe model fundamental lower bounds, special case of cell-probe Implicit model O(1) working space, store information as input permutation IO model focus on number of memory-disk transfers Cache-oblivious model abstract model to model multiple memory layers Streaming model limited working space, single or multiple scans of input

7 Integer Sorting Time per element Results (n words of w bits) O log log n O(1) 1 log n? 2 NEW 3 log 2 n loglog n log 2+ε n w Bucket sort O(n+2 w ) Radix sort; Hollerith 1887 O n w log n van Emde Boas 1975 O n log w Willard 1983 superlinear space expected Kirkpatrick and Reicsh 1983 O n log w log n Merge sort: von Neumann 1945 O n log n comparison based optimal Thorup and Han 2002 O n log (w/log n) O n log log n expected Andersson et al O(n) expected, w Ω log 2+ε n Belazzougui, Brodal, Nielsen 2014 O(n) expected, w Ω log 2 n log log n

8 Threesomes, Degenerates, and Love Triangles Allan Grønlund, Seth Pettie FOCS SUM problem : Given real numbers x 1,..., x n, does there exist x i + x j + x k = 0? Conjecture : 3SUM requires time Θ(n 2 ) Theorem : 3SUM can be solved in time O(n 2 / (log n/loglog n) 2/3 ) Theorem : 3SUM has decision tree complexity O(n 3/2 log n)

9 Gerth Brodal, Kasper Larsen SWAT 2014 Planar Orthogonal Skyline Queries Space (words) Query Reporting O(n) O(n lglg n) O(n lg ε n) O(k lg ε n) O(k lglg n + lg n/lglg n) O(k + lg n/lglg n) Counting O(n) O(n lg O(1) n) O(lg n/lglg n) (lg n/lglg n)

10 Peyman Afshani JCGA 2013 Half-Space Range Reporting Query time Q(n) Space n Q(n) d

11 Memory Hierarchies CPU Processor R L1 L2 L3 A M Disk bottleneck increasing access times and memory sizes

12 IO Model Cost = # block transfers Cache-Oblivious Model Memory, size M Disk Block size B I/O model...but algorithms do not know B and M Assume optimal cache replacement strategy Optimal on all levels (under some assumptions) Scanning O(N/B) IOs, Sorting O(N/B log M/B N/B) IOs Alok Aggarwal and Jeff Vitter. The Input/Output Complexity of Sorting and Related Problems. Communications of the ACM 31(9) Matteo Frigo, Charles E. Leiserson, Harald Prokop, Sridhar Ramachandran. Cache-Oblivious Algorithms. ACM Transactions on Algorithms, 8(1), Article No. 4, 2012.

13 Computing Multiresolution Rasters average Output size = n =1 n 2 2 = O(n 2 ) Input x Output RAM algorithm O(n 2 ) time n x n n x n n 2 x n 2 n 3 x n 3 n 4 x n 4 n x n... 1x1 IO Model - O(Sort(n 2 )) IOs Lars Arge, Herman Haverkort and Constantinos Tsirogiannis. ACM SIGSPATIAL Cache Oblivious Model - O(Scan(n 2 )) IOs Lars Arge, Gerth Stølting Brodal, Jakob Truelsen, and Constantinos Tsirogiannis. ESA 2013.

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15 Terrain Research Height matrix Terrain model

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17 Lars Arge, Morten Revsbæk, and Norbert Zeh SoCG 2010 Areas Flooded by 100 mm Rain SoCG 2010 MADALGO prototype SCALGO prodct

18 Algorithms Research Theory Kasper Peyman Gerth Lars EcoInformatics BioInformatics COWI, DES, EIVA, Algorithm engineering Applications