Lecture 2: Introduction to sublinear algorithms

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Lecture 2: Introduction to sublinear algorithms

Themis Gouleakis

April 20, 2021

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Sublinear-time algorithms (examples)

Problem: Compute the diameter of a point set m points in metric spaceX.

Distances given by:

D=







0 d₁₂ . . . d_1m d₂₁ 0 ...

... . ..

d_m1 . . . 0







– Symmetric: d_ij=d_ji

– Triangle inequality:d_ij<d_ik+d_kj

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Theorem

Diameter-Estimator returns a2-approximation to the actual diameter.

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Sublinear-time algorithms (examples)

Problem: Number of connected components Input:G= (V,E),|V|=n

Goal: Estimatec=]connected components ofG.

Letn_v :]of nodes in the connected component ofv. We need to estimate:c =P

v∈V 1 nv

Lemma

For all v ∈V , it holds that _n_ˆ¹

v −_n¹

v ≤/2, wherenˆ_v = min{n_v,2/}

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Algorithm

Lemma

It holds that: Pr[|ˆc−˜c|> n/2]≤1/4

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Finishing the proof

Theorem

Let c be the number of connected components of G and let˜c be the output of Algorithm3. Then,Pr[|c−˜c| ≤n]≥3/4.

Proof:

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Property testing definitions

Computational problems (exact)

Search problems

x :R(x) ={y : (x,y)∈R}

v :{0,1}^∗→R(value)

Goal: Findy^∗ = max_y∈R{v(y)}

Decision problems

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Property testing definitions

Computational problems (approximate) Search problems

x :R(x) ={y : (x,y)∈R}

v :{0,1}^∗→R(value) Goal: Find

y^∗:v(y^∗)>C· max

y∈R(x){v(y)}

Decision problems

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Definitions for property testing

Definition

LetΠnbe a set of functionsf : [n]→Rn,n∈N. The union Π =∪n∈NΠ_nπof these sets will be called aproperty.

Oracle access: Queryi →f(i) Distance: Letδ(f,g) = ^|{i∈[n]:f(i)6=g(i)}|

n = Pri∈U_[n][f(i)6=g(i)]

Distance from propertyΠ =∪_n∈_NΠn: – δΠ(f) =δ(f,Π) = ming∈Π_n{δ(f,g)}

– δΠ(f) =∞ifΠ_n=∅.

Query complexity:q :N×(0,1]→N

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Definitions for property testing

Definition

Atesterfor a propertyπ is a probabilistic oracle machine that outputs a binary verdict that satisfies the following:

1. IfS ∈Π, then the tester accepts with probability at least 2/3.

2. IfS is-far fromΠ, then the tester accepts with probability at most 1/3

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Examples

Problem: Testing convex position

Definition

A point setP is inconvex positionif every point inPbelongs to the convex hull ofP.

Definition

A setP ofnpoints is-farfromconvex positionif no setQof size (at most)nexists such thatP\Qis in convex position.

Goal: Design a tester that can distinguish the above in sub-linear time.

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Tester

Completeness case: Clearly, ifP is in convex position, then any S⊆Pwill also be in convex position.

Soundness case:We need to show thatCONVEX TESTER rejects every point set that is-far from convex position with probability at least 2/3.