Tidbits

1
Tidbits

• Boston, 2003

2
Game Plan

• There is no plan!
• Chip in whenever you feel like it

3
Goal

• Manuel told us that a PhD should
• Know something about everything
• Know everything about something

4
Put my name on your paper!
5
Tutorials

1. Machine Learning my favorite results, directions
   and open problemAvrim Blum We need a speaker
   next week
2. MixingDana Randall
3. Performance Analysis of Dynamic Network
   ProcessesEli Upfal

6
Yan Can Cook, So Can You

• monomer-dimer coverings
• dimer coverings
• hard core lattice gas model
• ground states of Potts model
• partition function
• ferromagnetism
• Bethe lattice
• mean-field

• matchings
• perfect matchings
• independent sets
• vertex colorings
• normalizing constant
• positive correlation
• complete regular tree
• Kn

7
Heads Up

• Approximation Algorithms for Orienteering and
  Discounted-Reward TSP
• A. Blum, S. Chawla, D.R. Karger, T. Lane, A.
  Meyerson, M. Minkoff
• Coming with a free lunch for you on Nov 19

8
Traveling Repairman

• Paths, Trees, and Minimum Latency Tours
• K. Chaudhuri, B. Godfrey, S. Rao, K. Talwar
• 3.59-Approximation (probably the best we can do
  if we keep trying to stitch tours with
  geometrically increasing costs)

9
Traveling Repairman

• TSP 9 TRP 1234567836
• Since edges in the earlier part of the tour will
  be counted many times, it makes senseto find
  sub-tours of geometrically increasing costs and
  stitchthem together.

10
Back To Traveling Salesman

• Approximation Algorithms for Asymmetric TSP by
  Decomposing Directed Regular Multigraphs
• H. Kaplan, M. Lewsenstein, N. Shafrir, M.
  Sviridenko
• 0.842 log n-approximation for minimum asymmetric
  TSP
• 2/3-approximation for maximum TSP (from 5/8)
• 5/2-approximation for shortest superstring
• 2/3-approximation for maximum 3-cover (from 3/5)
• 10/13-approximation for maximum ATSP with

11
A Brief History of Min-ATSP

• 2003 this paper 0.842 log n
• 2002 Blaser 0.999 log n
• 1982 Frieze, Galbiati, Maffioli log n
• Thanks to Abie for telling me about this 20 year
  gap.

12
Pseudorandom Object Generator

• On the Implementation of Huge Random Objects
• O. Goldreich, S. Goldwasser, A. Nussboim
• Its hard to look random(But they show its
  doable.)

13
Random Graphs

• You want to run some simulations on HUGE random
  graphs.
• Having read Bollobass book, you are willing
  toassume all random graphs are Hamiltonian.
• Being limited in memory, you plan to
  usepseudorandom functions in order to
  efficiently generate and store representations of
  your graphs.(Dont worry about the details.)

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Wait a second!

• Why should the graphs you get be Hamiltonian?
• Like every other proof in crypto, we show a
  reduction.
• Being Hamiltonian is a global property that
  requires checking an exponential number of
  adjacencies (unless)
• So its violation cannot be translated to a
  contradiction of the pseudorandomness of the
  function you used.
• Reduction argument will fail, see?

15
List Decoding

• List-Decoding Using the XOR Lemma
• L. Trevisan
• What is List Decoding?

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Coding Problem
Noisy Channel
100
110 ? 100
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Decoding

• Classical Decoding
• Output the unique closest codeword
• Output Original

• List Decoding
• Output a list of codewords that are within
  Hamming distance e
• Output List 3 Original

Success Criteria
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List Decoding

• We can now allow a higher noise level that
  corrupts a codeword so that the received code is
  closer to another codeword, as long as the
  original codeword is also in the list.
• How to design codes such that the list
• is short (polynomial in the length of the
  codeword), and
• each codeword in the list can be computed
  efficiently?

19
Adversarial Queuing Model

• Instability of FIFO at Arbitrarily Low Rates in
  the Adversarial Queuing Model
• R. Bhattacharjee, A. Goel
• Adversarial?

20
Stochastic Arrival is Unrealistic

• Complexity of network traffic has grown over the
  years
• (Was a Poisson stream ever realistic in a
  network?)

Poisson
Poisson???
21
Adversarial Queuing Model

• We allow a capable-but-constrained adversary to
• inject packets such that
• over any window of T time units, there can be at
  most wrT packets traversing each edge
• This is called a (w, r)-adversary of burst rate w
  and
• injection rate r lt 1.

22
Network Stability

• A packet forwarding protocol is stable against a
  given adversary and for a given network if
• the maximum queue size, and
• the maximum delay experienced by a packet
• remain bounded.
• This paper showed there is a truly ugly network
  where FIFO leads to instability.

23
I Invented the Internet

• On Certain Connectivity Properties of the
  Internet Topology
• M. Mihail, C. Papadimitriou, A. Saber
• Model Growth w/ Preferential Attachment
• Result Almost all scale-free graphs have
  constant conductance

24
VCG Overpayment

• For any graph G and vertices s and t, consider
  the shortest path P from s to t.
• For each edge e, define the Vickrey-Clarke-Groves
  overpayment of e w.r.t. s and t denoted v(e,s,t),
  to be the increase in the length of the shortest
  path from s to t if e were deleted.
• How should we define v(e,s,t) if e is a bridge?

25
Good Turing Hunting

• Always Good Turing Asymptotically Optimal
  Probability Estimation
• A. Orlitsky, N. P. Santhanam, J. Zhang
• I.J. Good and A.M. Turing

26
Safari

• In preparation for your next safari, you observe
  a random sample of African animals. You
• find 3 giraffes, 1 zebra and 2 elephants. How
  would you estimate the probability distributions
  of the various species you may encounter on your
  trip?

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A Naïve Estimator

• We have seen 6 animals in total,
  henceP(giraffes) 1/2,P(zebras) 1/6,
  P(elephants) 1/3.
• Wait, but what about the lions?

28
Laplace

• To address this unseen-elements problem, Laplace
  proposed to add 1 to count of each species and an
  unseen species, ie,
• P(giraffes) (31)/10,P(zebras) (11)/10,
  P(elephants) (21)/10,P(unseen) (01)/10.
• Other add-constant methods have been analyzed
  under the condition of fixed-species and
  increasing sample size.

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One-tenth for all other species?

• When the number of possible species is large
  compared to the sample size, add-constant is
  still an excessive overestimate.
• This paper shows
• that the Good-Turing estimator is reasonably good
• in fact, many other existing estimators are much
  worse
• how to construct an asymptotically optimal
  estimator

30
Sorting

• An In-Place Sorting with O(n log n) Comparisons
  and O(n) Moves
• G. Franceschini, V. Geffert
• Basically optimal in all computational resources
  in the comparison-based model, but their
  algorithm is not stable

31
Sorting Lowerbounds

• Comparisons
• log n! n log n 1.443n
• Moves
• 1.5n(think selection sort, which actually does
  2n-1 moves)
• Space
• In-place, ie, constant auxiliary storage(think
  insertion sort, but not quicksort)

32
Matrix Multiplication Again

• A Group-theoretic Approach to Fast Matrix
  Multiplication
• H. Cohn, C. Umans
• It is widely believed that ? 2.
• Anyone knows why? (They didnt say.)

33
Preconditioners

• Solving Sparse, Symmetric, Diagonally-Dominant
  Linear Systems in Time O(m1.31)
• D.A. Spielman, S. Teng
• (what am I supposed to say? P)

34
Smoothed Competitiveness

• Average Case and Smoothed Competitive Analysis of
  the Multi-level Feedback Algorithm
• L. Becchetti, S. Leonardi, A. Marchetti-Spaccamela
  , G. Schafer, T. Vredeveld

35
Embedding

• On The Impossibility of Dimension Reduction in
• B. Brinkman, M. Charikar
• No Johnson-Lindenstrauss in

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Johnson-Lindenstrauss

• Any n points in Euclidean space (with distances
  measured by the norm) may be mapped down to
  O((log n)/?2) dimensions such that no pairwise
  distance is distorted by more than a (1?)
  factor.
• Many simpler proofs are known (compare to J-Ls)

37
Diamond Graphs
1/4
1/2
1
38
Embedding Again

• Bounded-geometries, fractals, and low-distortion
  embeddings
• A. Gupta, R. Krauthgamer, J.R. Lee

39
Déjà vu
40
From The Polynomial Time Dept

• A Polynomial Algorithm for Recognizing Perfect
  Graphs
• G. Cornuejols, X. Liu, K. Vuskovic
• O(V 10)

41
From The Polynomial Time Dept

• Simulated Annealing in Convex Bodies and an
  O(n4) Volume Algorithm
• L. Lovasz, S. Vempala
• Back in 1991, it was about 23

video clip from FOCS
42
Logconcave

• Logconcave FunctionsGeometry and Efficient
  Sampling Algorithms
• L. Lovasz, S. Vempala
• A function is
  logconcave if it satisfies
• for every and 0 ? 1.

43
From the Constants Department

• Clustering with Qualitative Information
• M. Charikar, V. Guruswami, A. Wirth
• 4-approximation for MinDisagree on complete
  graphs(from 442)

44
Qualitative Information
45
MinDisagree

• Minimize disagree in a cluster and agree across
  clusters

46
That Reminds Me

• Their algorithm was combinatorial in contrast
  our algorithm is based on a natural linear
  programming relaxation and rounding the
  fractional solution using the region-growing
  approach.
• Once upon a time, in a room far far away in MIT,
  Bruce was a graduate student in his q-exam