Algorithms and Data Structures Group

Gerth Stølting Brodal

Faculty Meeting, Department of Computer Science, Aarhus University, October 3, 2014

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

Programmer Svend Christian Svendsen

Administration Trine Ji Holmgaard Jensen, Katrine Østerlund Rasmussen, Ellen Kjemtrup

Weekly visitors

Peder Klith Bøcher Brody Steven Sandel (Biodiversity Group)

Events Workshop on Massive Data Algorithmis (2009 -)

Symposium on Computational Geometry (2009)

European Symposium on Algorithms / ALGO 2016

MADALGO Summer Schools 2007, 2008, 2010 -

MADALGO retreat (October)

Research – Efficiency

O(n3)

O(n2)

O(nlog2 n)

O(nlog n)

O(nlog n

loglog n)

quartet distance between two trees

?

O(log n)

O(log n

loglog n)

O(loglog n)

O( loglog n)

O(1)

integer sorting, cost per element

?

Less practical ?

Deeper insights

Improved

asymptotics

More complicated ?

lower bound ?

algorithm engineering

?

Algorithms Research

Theory

Algorithm engineering

Applications

Peyman

Gerth

Lars

EcoInformatics BioInformatics

COWI, DES, EIVA, …

Kasper

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

NEW

Integer Sorting Results

Bucket sort O(n+2w)

Radix sort; Hollerith 1887 O nw

log n

van Emde Boas 1975 Willard 1983

O n logw superlinear space expected

Kirkpatrick and Reicsh 1983 O n logw

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. 1998 O(n) expected, w ≥ Ω log2+εn

Belazzougui, Brodal, Nielsen 2014 O(n) expected, w ≥ Ω log2n ∙ log log n

Time per element

w log2 n·loglog n log2+ε n

O log log n

O(1)

log n

1

2

3

1

2

3

? (n words of w bits)

Threesomes, Degenerates, and Love Triangles

Allan Grønlund, Seth Pettie FOCS 2014

3SUM problem : Given real numbers x1,..., xn, does there exist xi + xj + xk = 0 ?

Conjecture : 3SUM requires time Θ(n2)

Theorem : 3SUM can be solved in time O(n2 / (log n/loglog n)2/3)

Theorem’ : 3SUM has decision tree complexity O(n3/2 log n)

3 2 -9 11 -7 5 9 10 -4 6

Planar Orthogonal Skyline Queries

Space (words)

Query

Reporting O(n)

O(nlglg n) O(nlgε n)

O(klgε n) O(klglg n + lg n/lglg n)

O(k + lg n/lglg n)

Counting O(n)

O(nlgO(1) n) O(lg n/lglg n) (lg n/lglg n)

Gerth Brodal, Kasper Larsen SWAT 2014

Half-Space Range Reporting

Peyman Afshani JCGA 2013

Query time Q(n) Space n

Q(n)

d

Memory Hierarchies

Processor L1 L2 A

R

M L3 Disk

bottleneck

increasing access times and memory sizes

CPU

IO Model

I/O model...but algorithms do not know B and M

Assume optimal cache replacement strategy

Optimal on all levels (under some assumptions)

Memory, size M

Block size B

Alok Aggarwal and Jeff Vitter. The Input/Output Complexity of Sorting and Related Problems. Communications of the ACM 31(9) 1988.

Matteo Frigo, Charles E. Leiserson, Harald Prokop, Sridhar Ramachandran. Cache-Oblivious Algorithms. ACM Transactions on Algorithms, 8(1), Article No. 4, 2012.

Disk

Cache-Oblivious Model

Cost = # block transfers

Scanning O(N/B) IOs, Sorting O(N/BlogM/B N/B) IOs

Computing Multiresolution Rasters

Input

n x n n2 x n2

n3 x n3

n4 x n4 n x n

Output

IO Model - O(Sort(n2)) IOs Lars Arge, Herman Haverkort and Constantinos Tsirogiannis. ACM SIGSPATIAL 2012. Cache Oblivious Model - O(Scan(n2)) IOs Lars Arge, Gerth Stølting Brodal, Jakob Truelsen, and Constantinos Tsirogiannis. ESA 2013.

x

n x n

average n2

2=

n

=1O(n2) Output size =

... 1x1

RAM algorithm O(n2) time

Terrain Research

3 2 4

7 5 8

7 1 9

3 2 4

7 5 8

7 1 9

3 2 4

7 5 8

7 1 9

3 2 4

7 5 8

7 1 9

Height matrix

Terrain model

Areas Flooded by 100 mm Rain

SoCG 2010 MADALGO prototype SCALGO prodct

Lars Arge, Morten Revsbæk, and Norbert Zeh SoCG 2010

Algorithms Research

Theory

Algorithm engineering

Applications

Peyman

Gerth

Lars

EcoInformatics BioInformatics

COWI, DES, EIVA, …

Kasper