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Distributed Monotonicity Restoration

Michael SaksRutgers University

C. SeshadhriPrinceton University

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Overview Introduce a new class of algorithmic

problems:Distributed Property Restoration

A solution for the case of theMonotonicity Property

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Data Sets

Data set = function f : Γ V

Γ = finite index setV = value set

In this talk,Γ = [n]d = {1,…,n}d

f is a d-dimensional array

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Data Sets: Examples

Directed Graphsboolean valued matrix

This talk: Nonnegative integer-valued arrays

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For f,g with common domain Γ:

dist(f,g) = fraction of domain where f(x) ≠ g(x)

= relative Hamming distance

Distance between two data sets

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Properties of data setsExamples

Graphs: planarity, Hamiltonicity, etc.

For multidimensional arrays Linear mod n Distinct entries Monotone: nondecreasing along every line

(Order preserving) When d=1, monotone = sorted

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Some Algorithmic problems for PGiven data set f (as input): Recognition: Does f satisfy P? Testing:

(Define ε(f) = min{ dist(f,g) : g satisfies P})Decide either ε(f) > 0: f does not satisfy P ε(f) ≤ δ: f is close to P

(If 0 < ε(f) ≤ δ then can decide either)

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Property RestorationSetting:

Given f We expect f to satisfy P

(e.g. we run algorithms on f that assume P) but f may not satisfy P

Restoration problem for P: Given data set f, produce data set g that satisfies P is close to f: d(f,g) is not much bigger than ε(f)

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What does it mean to produce g? Offline computation

Input: function table for f

Output: function table for g

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Distributed monotonicity restoration For each domain value x in Γ,

Processor Px computes g(x) works independently of other processors may access f(y) for (not too many) y has access to a short random string s

(common to all processors)

and is otherwise deterministic.

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Distributed Property RestorationGoal:

WHP (with probability close to 1) (over choices of random string s):

g has property P d(g,f) = O( ε(f) ) Each Px runs quickly

in particular only reads f(y) for a small number of y.

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Distributed Property Restoration

Precursor to this work: Online Data Reconstruction Model

(Ailon, Chazelle, Liu, Seshadhri)

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Example: Error Correcting Codes Data set f = boolean string of length n

Property = Code word of a given error correcting code C

Recognition: Does f belong to C?Restoration = Decoding to a close code wordDistributed restoration = Local decoding

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Monotonicity Restoration: d=1 f is a linear array of length n

First attempt at distributed restoration: Px looks at f(x) and f(x-1)

If f(x) ≥ f(x-1),then g(x) = f(x)

Otherwise, we have a non-monotonicity

g(x) = max { f(x) , f(x-1) }

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Monotonicity Restoration: d=1 Second attempt

Set g(x) = max{ f(1), f(2),…, f(x) }

g is monotone but

Px requires time Ω(x) dist(g,f) may be much larger than ε(f)

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Our results (for general d )A distributed monotonicity restoration algorithm

for general dimension d such that:

Time to compute g(x) is (log n)O(d)

dist(f,g) = C1(D) f) Shared random string s has size (d log n)O(1)

(Builds on similar results of Ailon, et al for Online Monotonicity reconstruction.)

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Which array values should be changed?A subset S of Γ is f-monotone

if f restricted to S is monotone.

For each x in Γ, Px must: Decide whether g(x) = f(x) If not , then determine g(x)

Preserved = { x : g(x) = f(x) }Corrected = { x : g(x) ≠ f(x) }

In particular, Preserved must be f-monotone

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Identifying Preserved

The partition (Preserved, Corrected)must satisfy:

Preserved is f-monotone |Corrected|/|Γ| = O(ε(f))

Preliminary algorithmic problem:

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Classification problem

Classify each y in Γ as Green or Red Green is f - monotone Red has size O(ε(f)|Γ|)

Need subroutine Classify(y).

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A sufficient condition for f-monotonicityA pair (x,y) in Γ × Γ is a violation if

x < y and f(x) > f(y)

To guarantee that Green is f - monotone:

Red should hit all violations:

For every violation (x,y) at least one of x,y is Red

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Classify: 1-dimensional case

d=1: Γ={1,…,n} f is a linear array.

For x in Γ, and subinterval J of Γ:violations(x,J)=|{y in J : (x,y) is a violation}|

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Constructing a large f-monotone setThe set Bad:

x in Bad if for some interval J containing x|violations(x,J)|≥|J|/2

Lemma. Good=Γ \ Bad is f-monotone |Bad| ≤ 4 ε(f)|Γ| .

So we’d like to take:Green=Good Red = Bad

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How do we compute Good?

To test whether y in Good:For each interval J containing y,

check violations(y,J)< |J|/2Difficulties

There are (n) intervals J containing y For each J, computing violations(y,J)

takes time (|J|) .

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Speeding up the computation

Estimate violations(y,J) by random sampling sample size polylog(n) is sufficient

violations* (y,J) denotes the estimate

Compute violations* (y,J) only for a carefully chosen set of test intervals

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Set of test intervals

Want set T of intervals of [n] with:

Each x is in few ( O(log n) ) intervals of T

If S is any subset of T, then S has a subfamily C such that:

each x in US belongs to 1 or 2 sets of C.

For any interval I, there is a J in T containing I,of length at most 4|I|.

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The Test Set T

Assume n=|Γ|=2k

k layers of intervalsLayer j consists of 2k-j+1-1 intervals of size 2j

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Subroutine classify

To classify y If for each J in T containing y

violations*(y,J) < .1 |J|then y is Greenelse y is Red

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Where are we?We have a subroutine Classify On input x,

Classify outputs Green or Red Runs in time polylog(n)

WHP Green is f-monotone |Red| ≤ 20ε(f)|Γ|

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Defining g(x) for Red x

The natural way to define g(x) is: Green(x) = { y : y ≤x and y Green}

g(x) = max{f(y) : y in Green(x))} = f(max{Green(x)})

In particular, this givesg(x) = f(x) for Green x

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Computing m(x)

Can search back from x to find first Green

Inefficient if x is preceded by a long Red stretch

xm(x)

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Approximating m(x)?

x

Pick random Sample(x) of points less than x Density inversely proportional to distance from x Size is polylog(n)

Green* (x) = { y: y in Sample(x) , y Green}m*(x) = max {y in Green* (x)}

m*(x)

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Is m*(x) good enough?

xm*(x) y

Suppose y is Green and m*(x) ≤ y ≤ x Since y is Green:

g(y) = f(y) and

g(x) = f(m*(x)) < f(y) = g(y)

g is not monotone

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Is m*(x) good enough?

To ensure monotonicity we need:x < y implies m*(x) < m*(y)

Requires relaxing the requirement: for all Green y, m*(y) = y

xm*(x) y

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Thinning out Green* (x)

Plan: Eliminate certain unsafe points from Green*(x)

Roughly, y is unsafe for x if for some z > x

(There is a non-trivial chance that Sample(z)has no Green points ≥ y.)

Some interval beginning with y and containing x has a high density of Reds.

xy z

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Thinning out Green* (x)

Green* (x) = { y: y in Sample(x) , y Green}m*(x) = max {y in Green* (x)}

Green^(x) = { y: y in Green* (x) , y safe for x}m^(x) = max {y in Green^ (x)}

(Hiding: Efficient implementation of Green^(x))

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Redefining Green^(x)

WHP

if x ≤ y, then m^(x) ≤ m^(y)

{x: m^(x) ≠ x} is O(ε(f) |Γ|).

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Summary of 1-dimensional case Classify points as Green and Red

Few Red points f restricted to Green is f-monotone

For each x, choose Sample(x) size polylog(n) All points less than x Density inversely proportional to distance from x

Green^ (x) from Sample(x) that are safe for x m^(x) is the maximum of Green^(x)

Output g(x)=f(m^(x))

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Dimension greater than 1

For x < y, want g(x) < g(y)

x

y

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Red/Green Classification

Extend the Red/Green classification to higher dimensions: f restricted to Green is Monotone Red is small

Straightforward (mostly) extension of 1-dimensional case

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Given Red/Green classificationIn the one-dimensional case,

Green^ (x) = sampled Green points safe for x

g(x) = f(max {y : y in Green^ (x) }

.

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The Green points below x

Set of Green maxima could be very large Sparse Random Sampling will only roughly capture the frontier Identifying the unsafe points is much harder than in the one

dimensional case

01

x

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Further work The g produced by our algorithm has

d(g,f) ≤ C(d)ε(f)|Γ| Our C(d) is exp(d2) . What should C(d) be? (Guess: C(d) = exp(d) )

Monotonicity restoration for general posets

Distributed restoration for other properties(Graph Properties?)