CS 598 TMC, Fall 2019
Geometric Data Structures
Instructor
Timothy Chan
(Siebel 3230, tmc "at" illinois.edu,
office hours: Wed 2:303:30 or by appointment)
Meeting Time and Place
Wed & Fri 11:0011:15, Siebel 1103
Course Overview
This course is about
data structures in computational geometry.
We will discuss fundamental techniques and recent theoretical developments for
a variety of basic geometric data structuring problems in a variety of models.
Possible topics include:
 Orthogonal range searching:
from basic methods (such as range trees) to current best results (such as
my paper with Larsen and Patrascu)
 Point location:
from standard solutions (segment trees, fractional cascading,
persistence, planar separators, ...),
to more recent sublogarithmic methods on the word RAM
 Nonorthogonal range searching:
partition trees, cutting trees, random sampling, iterative reweighting,
shallow cuttings
 Dynamic data structures:
dynamic convex hull, dynamic point location, general dynamization techniques
 Approximate nearest neighbor searching:
quadtrees, shifting, zordering, locality sensitive hashing
 Big data:
externalmemory geometric data structures, geometric streaming algorithms, ...
Prerequisite: a solid background in algorithm
design and analysis (e.g., at the level of CS 374) is assumed.
Course Work
 4 assignments (problem sets), worth 40%
 presentation, worth 20%
 a project (reading some papers and writing a report,
or implementing some data structures,
or doing some original research), worth 40%
(Assignments and projects may be done individually or in groups of 2.)
See guidelines for presentation and project.
References
There is no required textbook, but you may find the following general references useful:

[BCKO] M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars,
Computational
Geometry: Algorithms and Applications (3rd ed.),
SpringerVerlag, 2008

[GRT] J. Goodman, J. O'Rourke, and C. D. Toth (eds.),
Handbook of Discrete and Computational Geometry (3rd ed.),
CRC Press, 2017
Lectures
(I'll provide copies of my handwritten notes, and links to relevant papers, below...)
Aug 28, 30, Sep 4: Introduction. Orthogonal range searching.
 kd trees (O(n) space, O(n^{11/d} + k) time;
see Sec 5.2 of BCKO)
 Priority search trees and Cartesian trees for 2D 3sided emptiness/reporting (O(n) space, O(log n + k) time; see Sec 10.2 of BCKO
and wikipedia page)
 Range trees (O(n log^{d1} n) space, O(log^{d1} n) time; see Sec 5.35.6
of BCKO)
 my notes
Sep 6, 11:
 Improving space by bit packing: Chazelle'88 for 2D counting (O(n) space,
O(log n) time;
see Sec 3.1 of
paper)
 Improving time: warmup with 1D predecessor search

van Emde Boas trees (O(n) space, O(loglog U) time; see wikipedia, or the Cormen et al. textbook, 3rd ed.)

fusion trees (O(n) space, O(log_w n) time, but rough sketch only;
see Fredman and Willard's paper)
 my notes
Sep 11, 13, 18:
 Improving time in 2D (O(n log n) space, O(loglog U) time (for emptiness))
 Improving time and space in 2D:
Alstrup, Brodal, and Rauhe's gridbased method (O(n loglog U)
space, O((loglog U)^2) time; see Sec 3 of
paper)
 Improving time and space in 2D: C.LarsenPatrascu'11 (O(n loglog n) space, O(loglog U) time;
see Sec 2 of paper)
 my notes
Sep 18, 20:
Planar point location.
 The naive slab method (O(n^2) space, O(log n) time)
 The segment tree method (O(n log n) space, O(log^2 n) time; see Sec 10.3 of
BCKO)
 ... with fractional cascading (O(n log n) space, O(log n) time;
see Chazelle and Guibas' two
papers)
 Preparata's trapezoid tree method (also O(n log n) space, O(log n) time;
see paper or Preparata and Shamos' old textbook)
 Persistent search tree method
(O(n log n) space, O(log n) time by path copying;
see Sarnak and Tarjan's
paper,
which also reduced space to O(n))
 my notes
Sep 25:
 Planar separator method
(O(n) space, O(log n) time;
see Lipton and Tarjan's
paper)
 Kirkpatrick's hierarchical triangulation method
(see paper, or Preparata and Shamos'
book, or Iacono's video)
 my notes
Sep 27:
Oct 2:
 Point location in sublogarithmic time (O(n) space, O(loglog U) time;
see my paper from '11)
 my notes
Oct 4:
Oct 9, 11: Nonorthogonal range searching.
 Willard's method
(O(n) space, O(n^{0.793}) time in 2D, via the hamsandwich cut theorem;
see paper)
 A dual method (O(n^{4.82}) space, O(log n) time in 2D;
for duality, see Sec 8.2 of
BCKO)

Clarkson's cutting tree (O(n^{2+eps}) space, O(log n) time in 2D)
via the Cutting Lemma (proof by random sampling;
see
Clarkson's paper)
 my notes
Oct 16, 18:

Matousek's partition tree (O(n) space, O(n^{1/2+eps}) time in 2D)
via the Partition Theorem (proof by multiplicative weights update;
see
Matousek's paper)

Time/space tradeoffs

"Applications" to combinatorial geometry (SzemerediTrotter theorem
and Erdos' unit distance problem;
see Ch4 of
Matousek's book)
 Nonlinear range searching (via the linearization technique)
 Multilevel versions of partition trees
(see Sec 16.2 of BCKO)
 my notes
Oct 23:
 Halfspace range reporting in 2D and 3D
(O(n log n) or O(n loglog n) space, O(log n + k) time; see
my paper) via
the Shallow Cutting Lemma
(see Matousek's paper)
 my notes
Oct 25: Dynamic geometric data structures. Dynamic 2D convex hull.
Oct 30: General dynamization techniques.
 Static to insertonly:
The "logarithmic method" (Bentley and Saxe '80)
 Static to fully dynamic:
The "square root method" (same paper)
 Static to offline dynamic, via segment tree
 my notes
Nov 1: Dynamic 2D point location.
 Dynamic segment tree (O(log^2 n) query and update time)
 ... with dynamic fractional cascading (O(log n loglog n) query and O(log^2 n) update time)
 Sketch of a version of the latest method by
C. and Nekrich'15 (O(log n (loglog n)^2) query and O(log n loglog n) update time)
 my notes
Nov 6, 8: Approximate nearest neighbor search. In constant dimensions.
 Grid method (O(n) space and O(1) time, for fixed radius)
 (Compressed) quadtree method (O(n) space and O(log U) time for decision problem;
see Sariel's book)
 ... with shifting (O(n) space and O(log U) time; see
Bern'93
and Lemma 3.3 in C.'98)
 Balanced quadtree (O(n) space and O(log n) time; see again the above
papers)
 Zordering (same result but simplified; see
C.'02
and wikipedia,
and also a recent paper by C., HarPeled, and Jones)
 my notes
Nov 13, 15: In high dimensions.
 Dimension reduction via the JohnsonLindenstrauss Lemma
(n^{O((1/eps^2)log(1/eps))} space, O(poly(d)) time for Euclidean distances;
see Indyk and Motwani's
original paper and also Matousek's survey on metric embedding)
 Localitysensitive hashing
(near O(n^{1+1/c}) space, O(poly(d) n^{1/c}) time for approximation factor c,
for Hamming, L_1, and Euclidean distances;
see again Indyk and Motwani's original paper and Sec 2.9 of Matousek's
notes)
 my notes
Nov 20:
 The L_infinity case: Indyk's search tree method
(near O(n^t) space, O(d log n) time for approximation factor O(log_t log d);
see paper)
 my notes
Nov 22:
 Offline case: the "polynomial method"
(near n^{2sqrt{eps}} time via matrix multiplication and Chebyshev polynomials;
this paper also got near n^{2eps^{1/3}})
 my notes
Dec 4: Wrapup. Orthogonal range searching in moderate dimensions.
Dec 7, 11: Student presentation.