Maverick Woo 2003-10-29 TidbitsTidbits Boston, 2003 FOCS

2003-10-29

Maverick Woo

TidbitsTidbitsTidbitsTidbits

Boston, 2003

FOCS

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Game PlanGame PlanGame PlanGame Plan

There is no plan!

Chip in whenever you feel like it

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GoalGoalGoalGoal

Manuel told us that a PhD should

Know something about everythingKnow everything about something

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Put my name on your paper!Put my name on your paper!Put my name on your paper!Put my name on your paper!

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#authors

#papers

#papers 7 24 21 5 2 1 0 1

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TutorialsTutorialsTutorialsTutorials

1. Machine Learning: my favorite results, directions and open problemAvrim Blum

We need a speaker next week…2. Mixing

Dana Randall3. Performance Analysis of Dynamic

Network ProcessesEli Upfal

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Yan Can Cook, So Can YouYan Can Cook, So Can YouYan Can Cook, So Can YouYan Can Cook, So Can You

monomer-dimer coverings

dimer coveringshard core lattice gas

modelground states of

Potts modelpartition function

ferromagnetismBethe lattice

mean-field

matchingsperfect matchingsindependent setsvertex coloringsnormalizing

constantpositive correlationcomplete regular

treeKn

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Heads UpHeads UpHeads UpHeads 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

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Traveling RepairmanTraveling RepairmanTraveling RepairmanTraveling 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)

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Traveling RepairmanTraveling RepairmanTraveling RepairmanTraveling Repairman

TSP: 9 TRP: 1+2+3+4+5+6+7+8=36

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.

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Back To Traveling SalesmanBack To Traveling SalesmanBack To Traveling SalesmanBack To Traveling SalesmanApproximation Algorithms for Asymmetric TSP

by Decomposing Directed Regular MultigraphsH. Kaplan, M. Lewsenstein, N. Shafrir, M.

Sviridenko

0.842 log n-approximation for minimum asymmetric TSP2/3-approximation for maximum TSP (from 5/8)5/2-approximation for shortest superstring2/3-approximation for maximum 3-cover (from 3/5)10/13-approximation for maximum ATSP with ¢

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A Brief History of Min-ATSPA Brief History of Min-ATSPA Brief History of Min-ATSPA 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.

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Pseudorandom Object Pseudorandom Object GeneratorGeneratorPseudorandom Object Pseudorandom Object GeneratorGenerator

On the Implementation of Huge Random Objects

O. Goldreich, S. Goldwasser, A. Nussboim

It’s hard to look random…(But they show it’s doable.)

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Random GraphsRandom GraphsRandom GraphsRandom GraphsYou want to run some simulations on HUGE random graphs.

Having read Bollobas’s 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.(Don’t worry about the details.)

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Wait a second!Wait a second!Wait a second!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?

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List DecodingList DecodingList DecodingList Decoding

List-Decoding Using the XOR LemmaL. Trevisan

What is List Decoding?

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Coding ProblemCoding ProblemCoding ProblemCoding Problem

Noisy Channel100 110 100

000 001

010 011

101100

110 111

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DecodingDecodingDecodingDecoding

Classical DecodingOutput the unique closest codeword

Output = Original

List DecodingOutput a list of codewords that are within Hamming distance e

Output List 3 Original

Success Criteria

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List DecodingList DecodingList DecodingList DecodingWe 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 listis short (polynomial in the length of the

codeword), andeach codeword in the list can be

computed efficiently?

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Adversarial Queuing ModelAdversarial Queuing ModelAdversarial Queuing ModelAdversarial Queuing Model

Instability of FIFO at Arbitrarily Low Rates in the Adversarial Queuing Model

R. Bhattacharjee, A. Goel

Adversarial?

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Stochastic Arrival is Stochastic Arrival is UnrealisticUnrealisticStochastic Arrival is Stochastic Arrival is UnrealisticUnrealistic

Complexity of network traffic has grown over the years

(Was a Poisson stream ever realistic in a network?)

Poisson Poisson???

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Adversarial Queuing ModelAdversarial Queuing ModelAdversarial Queuing ModelAdversarial Queuing ModelWe allow a capable-but-constrained

adversary toinject packets such that

over any window of T time units, there can be at most w+rT packets traversing each edge

This is called a (w, r)-adversary of burst rate w and

injection rate r < 1.

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Network StabilityNetwork StabilityNetwork StabilityNetwork StabilityA 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.

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““I Invented the Internet”I Invented the Internet”““I Invented the Internet”I Invented the Internet”On Certain Connectivity Properties of the

Internet TopologyM. Mihail, C. Papadimitriou, A. Saber

Model: Growth w/ Preferential Attachment

Result: Almost all scale-free graphs have constant conductance

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VCG OverpaymentVCG OverpaymentVCG OverpaymentVCG OverpaymentFor 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?

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Good Turing HuntingGood Turing HuntingGood Turing HuntingGood Turing Hunting

Always Good Turing: Asymptotically Optimal Probability EstimationA. Orlitsky, N. P. Santhanam, J. Zhang

I.J. Good and A.M. Turing

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SafariSafariSafariSafariIn preparation for your next safari, you observe a random sample of African animals. Youfind 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 EstimatorA Naïve EstimatorA Naïve EstimatorA 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?

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LaplaceLaplaceLaplaceLaplaceTo address this unseen-elements problem, Laplace proposed to add 1 to count of each species and an “unseen” species, ie,P(giraffes) = (3+1)/10,P(zebras) = (1+1)/10, P(elephants) = (2+1)/10,P(unseen) = (0+1)/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 One-tenth for all other species?species?One-tenth for all other One-tenth for all other species?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

goodin fact, many other existing estimators are

much worsehow to construct an asymptotically optimal

estimator

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SortingSortingSortingSorting

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

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Sorting LowerboundsSorting LowerboundsSorting LowerboundsSorting Lowerbounds

Comparisonslog n! ¸ n log n – 1.443n

Moves1.5n(think selection sort, which actually does 2n-1 moves)

SpaceIn-place, ie, constant auxiliary storage(think insertion sort, but not quicksort)

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Matrix Multiplication AgainMatrix Multiplication AgainMatrix Multiplication AgainMatrix Multiplication Again

A Group-theoretic Approach to Fast Matrix Multiplication

H. Cohn, C. Umans

“It is widely believed that = 2.”Anyone knows why? (They didn’t say.)

Current best is still2.376 by Coppersmith and Winograd, 1990

Current best is still2.376 by Coppersmith and Winograd, 1990

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PreconditionersPreconditionersPreconditionersPreconditioners

Solving Sparse, Symmetric, Diagonally-Dominant Linear Systems in Time O(m1.31)

D.A. Spielman, S. Teng

… (what am I supposed to say? :P)

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Smoothed CompetitivenessSmoothed CompetitivenessSmoothed CompetitivenessSmoothed Competitiveness

Average Case and Smoothed Competitive Analysis of the Multi-level Feedback Algorithm

L. Becchetti, S. Leonardi, A. Marchetti-Spaccamela, G. Schafer, T. Vredeveld

c = supI

E I 2 f N (I )

£ A(I )OP T(I )

¤

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EmbeddingEmbeddingEmbeddingEmbedding

On The Impossibility of Dimension Reduction in

B. Brinkman, M. Charikar

No Johnson-Lindenstrauss in

1̀

1̀

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Johnson-LindenstraussJohnson-LindenstraussJohnson-LindenstraussJohnson-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-L’s)

2̀

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Diamond GraphsDiamond GraphsDiamond GraphsDiamond Graphs

1

1/21/4

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Embedding AgainEmbedding AgainEmbedding AgainEmbedding Again

Bounded-geometries, fractals, and low-distortion embeddings

A. Gupta, R. Krauthgamer, J.R. Lee

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Déjà vuDéjà vuDéjà vuDéjà vu

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From The Polynomial Time From The Polynomial Time DeptDeptFrom The Polynomial Time From The Polynomial Time DeptDept

A Polynomial Algorithm for Recognizing Perfect Graphs

G. Cornuejols, X. Liu, K. Vuskovic

O(V 10)

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From The Polynomial Time From The Polynomial Time DeptDeptFrom The Polynomial Time From The Polynomial Time DeptDept

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

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LogconcaveLogconcaveLogconcaveLogconcave

Logconcave Functions:Geometry and Efficient Sampling Algorithms

L. Lovasz, S. Vempala

A function is logconcave if it satisfies

for every and 0 · · 1.

x;y 2 Rn

f (®x + (1¡ ®)y) ¸ f (x)®f (y)1¡ ®

f : Rn ! R+

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From the Constants From the Constants DepartmentDepartmentFrom the Constants From the Constants DepartmentDepartment

Clustering with Qualitative InformationM. Charikar, V. Guruswami, A.

Wirth

4-approximation for MinDisagree on complete graphs(from 442)

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Qualitative InformationQualitative InformationQualitative InformationQualitative Information

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MinDisagreeMinDisagreeMinDisagreeMinDisagree

Minimize #disagree in a cluster and #agree across clusters

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That Reminds MeThat Reminds MeThat Reminds MeThat 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…