600.770 Selected Topics in Algorithms for Metric Spaces, Spring 2014

Instructor: Michael Dinitz

Time: M 11:00-12:00

Place: Shaffer 204E

Course description: 

This course will focus on algorithms that use, simplify, or exploit metric spaces. Examples of topics we will address include metric embeddings and their applications, algorithms to exploit low dimensionality, graph spanners and sparsifiers, and data structures such as distance oracles and compact routing schemes that allow us to efficiently find distances and paths. This course will mostly be in the form of a reading group, and students will present a paper and lead a discussion. Prereq: 600.463 or permission of the instructor.

Papers

Here is a small selection of papers related to the topics of this course. All students will need to choose and present one paper, either from this list or your own choice (subject to approval).

• Metric Embeddings

  1. J. Bourgain. On {Lipschitz} embeddings of finite metric spaces in {Hilbert} space. Israel J. of Math. 1985.
         How to embed any n-point metric into any l_p space with distortion O(log n)
  2. N. Linial, E. London and Yu. Rabinovich, The geometry of graphs and some of its algorithmic applications, Combinatorica (1995) 15, pp. 215-245.
         Algorithmic version of Bourgain's embedding, many other embeddings results.
  3. J. Matousek, On embedding expanders into $l\sb p$ spaces, Israel Journal of Math, 1997.
         Expanders suffer (log n)/p distortion when embedded into Lp spaces
  4. N. Alon, R. Karp, D. Peleg, D. B. West, A graph-theoretic game and its application to the k-server problem, SICOMP 1995.
         The paper that introduced the idea of embedding into distributions of trees.
  5. Y. Bartal, Probabilistic Approximations of Metric Spaces and its Algorithmic Applications, FOCS 1996.
         Gives a O(log² n) approximation for metrics by tree distributions, shows that graph decompositions give embeddings
  6. J. Fakcharoenphol, S. Rao and K. Talwar, A tight bound on approximating arbitrary metrics by tree metrics. STOC 2003.
         A simple algorithm that gets the tight O(log n) result for all graphs.


• Dimension Reduction
  
  
  1. W. Johnson, J. Lindenstauss. Extensions of Lipschitz maps into a Hilbert space. Contemporary Mathematics 1984.
         Any n-point Euclidean space embeds into O(log n/x²) space with (1+ x) 
         distortion. In fact, a random such subspace will do.
  2. S. Dasgupta, A. Gupta. An elementary proof of a theorem of Johnson and Lindenstrauss, Random Structures Algorithms, 2002.
         Chernoff-type proof of [JL84]
  3. D. Achlioptas. Database Friendly Random Projections, PoDS 2000.
         How to do [JL84] with unbiased coins. 
  
  
• Dimensionality of Metric Spaces:
  
  
  1. A. Gupta, R. Krauthgamer and J.R. Lee. Bounded geometries, fractals,and low-distortion embeddings. FOCS 2003
         Properties of metrics which cannot grow very fast.
  2. K. Talwar. Bypassing the Embedding: Approximation schemes and Compact Representations for Low Dimensional Metrics. STOC '04.
         Fast and good approximations for optimization problems in bounded dimension metrics
  3. H. Chan, A. Gupta, B. Maggs, and S. Zhou. On Hierarchical Routing in Doubling Metrics. SODA '05.
         Good spanners and routing schemes for doubling metrics
  4. Y. Bartal, L. Gottlieb, and R. Krauthgamer. The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme. STOC '12.
         PTAS for TSP in doubling metrics.
  5. Y. Bartal, L. Gottlieb, and O. Neiman. On the Impossibility of Dimension Reduction for Doubling Subsets of ℓp, p > 2. SoCG '14.
         Shows instances with small doubling dimension but where any embedding needs large dimension.


• Spanners:
  
  
  1. I. Althofer, G. Das, D. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. D&CG 1993.
         Finding sparse subgraphs that maintain all pair distances. The fundamental tradeoff for sparse spanners
  2. S. Khuller, B. Raghavachari, and N.E. Young. Balancing minimum spanning and shortest path trees, Algorithmica, 1995.
         Finds trees that are simultaneously near-MST and near-shortest path trees
  3. M. Thorup, U. Zwick. Approximate distance oracles. STOC 2001.
         Foundational distance oracle paper. Gives tree covers: a small set of subtrees such that each shortest-path is maintained by one of them.
         Used, e.g., in compact routing paper [TZ02]
  4. T. Chan, M. Dinitz, and A. Gupta. Spanners with Slack. ESA '06.
         Shows spanners and distance oracles with constant average stretch
  5. M. Dinitz and R. Krauthgamer. Fault-Tolerant Spanners: Better and Simpler. PODC '11.
         Simple construction of fault-tolerant spanners.
  6. P. Berman, A. Bhattacharyya, K. Makarychev, S. Raskhodnikova, and G. Yaroslavtsev. Improved Approximation for the Directed Spanner Problem. ICALP '11.
         Best known approximation to the Directed k-Spanner Problem.
  7. E. Chlamtac, M. Dinitz, and R. Krauthgamer. Everywhere-Sparse Spanners via Dense Subgraphs. FOCS '12.
         Approximation algorithm for the Lowest Degree 2-Spanner problem, using the Sherali-Adams LP hierarchy.
  8. S. Pettie. Low Distortion Spanners. ICALP '07.
         Spanners with additive and combined multiplicative-additive stretch. Modular construction.