Syllabus: This seminar will cover recent algorithmic
topics that involve geometric concepts and techniques, including
algorithms for nearest neighbor search, dimensionality reduction,
traveling salesman problem, metric embeddings, metric
decompositions, and their applications to combinatorial algorithms
such as approximation algorithms for graph partitioning.
Prerequisites: Basic
knowledge of algorithms, at an undergraduate level, is required,
and familiarity with Euclidean geometry, probability theory, and
linear algebra may be necessary.
2014-04-23: Polynomial Time Approximation Schemes for
Euclidean Traveling Salesman
[paper presented by Ami Mor + Dima Kogan]
2014-04-30: no class (cancelled)
2014-05-07: Low-distortion embeddings and probabilistic
embedding into dominating trees
[homework: none] [reading: Fakcharoenphol-Rao-Talwar]
2014-05-14: Probabilistic embedding into dominating trees
(cont'd)
[homework: problem set #3] [reading: Fakcharoenphol-Rao-Talwar]
2014-05-21: Approximating Multicut -- LP Rounding via Tree
Embeddings;
2014-05-28: no class (cancelled)
2014-06-04: no class (SHAVUOT)
2014-06-11: Turning big data into tiny data: Constant-size
coresets for k-means, PCA and projective clustering
[paper presented by Matan Karklinsky + Noam Arkind]
2014-06-18: Near-Optimal Hashing Algorithms for Approximate
Nearest Neighbor in High Dimensions
[paper presented by Idan Himlich]
2014-06-25 (morning): Excluded minors, network decomposition,
and multicommodity flow
[paper presented by Lior Kamma]
Suggested papers for presentations (in crude categorization):
Low-distortion embeddings:
[EE] Jittat
Fakcharoenphol, Satish Rao, Kunal Talwar: A tight bound on
approximating arbitrary metrics by tree metrics. J. Comput.
Syst. Sci. 69(3): 485-497 (2004)
[link]
N. Linial, A. Magen, Assaf Naor: Girth and Euclidean Distortion,
Geometric and Functional Analysis (GAFA)12 (2002), no. 2,
380-394
[EE] Satish Rao: Small Distortion and Volume
Preserving Embeddings for Planar and Euclidean Metrics.
Symposium on Computational Geometry 1999: 300-306
[EE] Anupam Gupta, Ilan Newman, Yuri Rabinovich,
Alistair Sinclair: Cuts, Trees and l1-Embeddings of
Graphs. Combinatorica 24(2): 233-269 (2004)
See also:
[EE] Ilan Newman, Yuri Rabinovich: A lower
bound on the distortion of embedding planar metrics into
Euclidean space. Symposium on Computational Geometry 2002:
94-96
[EE] Piotr Indyk, Anastasios Sidiropoulos:
Probabilistic embeddings of bounded genus graphs into planar
graphs. Symposium on Computational Geometry 2007: 204-209
[EE]
James R. Lee, Anastasios Sidiropoulos: Pathwidth, trees, and
random embeddings. Combinatorica 33(3): 349-374 (2013)
[EE]
James R. Lee, Arnaud de Mesmay, Mohammad Moharrami: Dimension
Reduction for Finite Trees in ℓ 1. Discrete & Computational
Geometry 50(4): 977-1032 (2013)
[EE]
James R. Lee, Prasad Raghavendra: Coarse Differentiation and
Multi-flows in Planar Graphs. Discrete & Computational
Geometry 43(2): 346-362 (2010)
Embedding a graph into spanning trees:
[EE] Michael Elkin, Yuval Emek, Daniel A.
Spielman, Shang-Hua Teng: Lower-Stretch Spanning Trees. SIAM
J. Comput. 38(2): 608-628 (2008)
[EE]
Ittai Abraham, Ofer Neiman: Using petal-decompositions to
build a low stretch spanning tree. STOC 2012: 395-406
[EE] Rafail Ostrovsky, Yuval Rabani: Low
distortion embeddings for edit distance. J. ACM 54(5): (2007)
Nearest Neighbor Search (NNS):
[EE]
Sariel Har-Peled, Piotr Indyk, Rajeev Motwani: Approximate
Nearest Neighbor: Towards Removing the Curse of Dimensionality.
Theory of Computing 8(1): 321-350 (2012)
See also:
[EE]
Piotr Indyk, Rajeev Motwani: Approximate Nearest Neighbors:
Towards Removing the Curse of Dimensionality. STOC 1998:
604-613
[EE]
Eyal Kushilevitz, Rafail Ostrovsky, Yuval Rabani: Efficient
Search for Approximate Nearest Neighbor in High Dimensional
Spaces. SIAM J. Comput. 30(2): 457-474 (2000)
[EE]
Aristides Gionis, Piotr Indyk, Rajeev Motwani: Similarity
Search in High Dimensions via Hashing. VLDB 1999: 518-529
[link]
Sariel Har-Peled: A Replacement for Voronoi Diagrams of Near
Linear Size. FOCS 2001: 94-103
[EE] Sanjoy Dasgupta, Yoav Freund: Random
projection trees and low dimensional manifolds. STOC 2008:
537-546
[EE]
Rajeev Motwani, Assaf Naor, Rina Panigrahy: Lower Bounds on
Locality Sensitive Hashing. SIAM J. Discrete Math. 21(4):
930-935 (2007)
See also:
[EE]
Ryan O'Donnell, Yi Wu, Yuan Zhou: Optimal lower bounds for
locality sensitive hashing (except when q is tiny). ICS 2011:
275-283
[EE] Piotr Indyk: On Approximate Nearest
Neighbors under linfinity Norm. J. Comput. Syst. Sci.
63(4): 627-638 (2001)
[EE] Piotr Indyk: Approximate nearest neighbor
algorithms for Frechet distance via product metrics. Symposium
on Computational Geometry 2002: 102-106
[EE] Sariel
Har-Peled, Nirman Kumar: Approximate Nearest Neighbor Search for
Low-Dimensional Queries. SIAM J. Comput. 42(1): 138-159 (2013)
[EE]
Kenneth Clarkson: Nearest-neighbor searching and metric space
dimensions. In G. Shakhnarovich, T. Darrell, and P. Indyk,
editors, Nearest-Neighbor Methods for Learning and Vision:
Theory and Practice, pages 15-59. MIT Press, 2006.
[EE] Piotr
Indyk,
Assaf
Naor: Nearest-neighbor-preserving embeddings. ACM Transactions
on Algorithms 3(3): (2007)
(taken) [EE] Alexandr
Andoni, Piotr Indyk: Near-Optimal Hashing Algorithms for
Approximate Nearest Neighbor in High Dimensions. FOCS 2006:
459-468
Algorithms for processing
metric spaces (including low-dimensional, doubling, or
graphical):
(taken) [EE]
Sanjeev Arora: Polynomial Time Approximation Schemes for
Euclidean Traveling Salesman and other Geometric Problems. J.
ACM 45(5): 753-782 (1998)
See also:
[EE] Kunal
Talwar:
Bypassing
the embedding: algorithms for low dimensional metrics. STOC
2004: 281-290
[EE]
Yair Bartal, Lee-Ad Gottlieb, Robert Krauthgamer: The
traveling salesman problem: low-dimensionality implies a
polynomial time approximation scheme. STOC 2012: 663-672
[EE]
Yair Bartal, Lee-Ad Gottlieb: A Linear Time Approximation
Scheme for Euclidean TSP. FOCS 2013: 698-706
[EE] Mikkel Thorup, Uri Zwick: Approximate
distance oracles. J. ACM 52(1): 1-24 (2005)
[EE] Mikkel Thorup: Compact oracles for
reachability and approximate distances in planar digraphs. J.
ACM 51(6): 993-1024 (2004)
[EE]
Sariel Har-Peled, Piotr Indyk, Anastasios Sidiropoulos:
Euclidean spanners in high dimensions. SODA 2013: 804-809
[EE]
Claire Mathieu, Hang Zhou: Graph Reconstruction via Distance
Oracles. ICALP (1) 2013: 733-744
[EE]
Ittai Abraham, Amos Fiat, Andrew V. Goldberg, Renato Fonseca F.
Werneck: Highway Dimension, Shortest Paths, and Provably
Efficient Algorithms. SODA 2010: 782-793
[EE]
Alexandr Andoni, Huy L. Nguyen: Near-Optimal Sublinear Time
Algorithms for Ulam Distance. SODA 2010: 76-86
[EE]
Artur Czumaj, Christiane Lammersen, Morteza Monemizadeh,
Christian Sohler: (1+ Є)-approximation for facility location in
data streams. SODA 2013: 1710-1728
(taken) [EE]
Dan Feldman, Melanie Schmidt, Christian Sohler: Turning big data
into tiny data: Constant-size coresets for k-means, PCA and
projective clustering. SODA 2013: 1434-1453
Graph partitioning algorithms:
[link, link2] Richard
J. Lipton and Robert Endre Tarjan: A Separator Theorem for
Planar Graphs. SIAM Journal on Applied Mathematics, Vol. 36, No.
2 (Apr., 1979), pp. 177-189
See also: [link]
Alon, Noga; Seymour, Paul; Thomas, Robin: Planar
separators.SIAM J. Discrete Math. 7
(1994), no. 2, 184--193
[link]
Daniel A. Spielman, Shang-Hua Teng: Spectral Partitioning Works:
Planar Graphs and Finite Element Meshes. FOCS 1996:96-105
[EE]
Punyashloka Biswal, James R. Lee, Satish Rao: Eigenvalue
Bounds, Spectral Partitioning, and Metrical Deformations via
Flows. FOCS 2008: 751-760
[link]
Naveen Garg, Vijay V. Vazirani, Mihalis Yannakakis: Approximate
Max-Flow Min-(Multi)Cut Theorems and Their Applications. SIAM J.
Comput. 25(2): 235-251 (1996)
[link]
Guy Even, Joseph Naor, Satish Rao, Baruch Schieber: Fast
Approximate Graph Partitioning Algorithms. SIAM J. Comput.
28(6): 2187-2214 (1999)
(taken) [EE] Philip N. Klein, Serge A. Plotkin, Satish
Rao: Excluded minors, network decomposition, and multicommodity
flow. STOC 1993: 682-690
See also: [EE] Jittat Fakcharoenphol, Kunal Talwar: An
Improved Decomposition Theorem for Graphs Excluding a Fixed
Minor. RANDOM-APPROX 2003: 36-46
[EE] Uriel Feige, MohammadTaghi Hajiaghayi, James
R. Lee: Improved Approximation Algorithms for Minimum Weight
Vertex Separators. SIAM J. Comput. 38(2): 629-657 (2008)
[EE]
James R. Lee, Manor Mendel, Mohammad Moharrami: A
node-capacitated Okamura-Seymour theorem. STOC 2013: 495-504
[EE] Noga Alon, Assaf Naor: Approximating the
Cut-Norm via Grothendieck's Inequality. SIAM J. Comput. 35(4):
787-803 (2006)
[EE] Jon
M.
Kleinberg,
Éva
Tardos: Approximation algorithms for classification problems
with pairwise relationships: metric labeling and Markov random
fields. J. ACM 49(5): 616-639 (2002)
[EE] Harald Räcke: Optimal hierarchical
decompositions for congestion minimization in networks. STOC
2008: 255-264