Distributed (Local)
Monotonicity Reconstruction
Michael Saks
Rutgers University
C. Seshadhri
Princeton University
(Now IBM Almaden)
1
Overview
Introduce a new class of algorithmic
problems:
Distributed Property Reconstruction
(extending framework of
program self-correction,
robust property testing
locally decodable codes)

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A solution for the property Monotonicity
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Data Sets
Data set = function f : Γ  V
Γ = finite index set
V = value set
In this talk,
Γ = [n]d = {1,…,n}d
V = nonnegative integers
f = d-dimensional array of nonnegative integers
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Distance between two data sets
For f,g with common domain Γ:
dist(f,g) = fraction of domain where f(x) ≠ g(x)
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Properties of data sets
Focus of this talk:

Monotone: nondecreasing along every line
(Order preserving)
When d=1, monotone = sorted
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Some Algorithmic problems for P
Given data set f (as input):
 Recognition: Does f satisfy P?
 RobustTesting:
(Define ε(f) = min{ dist(f,g) : g satisfies P})
For some 0 ≤ ε1 < ε2 < 1, output either
 ε(f) > ε1 :f is far from P
 ε(f) < ε2: f is close to P
(If ε1 < ε(f) ≤ ε2 then can decide either)
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Property Reconstruction
Setting:
Given f
 We expect f to satisfy P
(e.g. we run algorithms on f that assume P)
 but f may not satisfy P
Reconstruction problem for P:
Given data set f, produce data set g that
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
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 reconstruction
Want algorithm A that on input x, computes
g(x)

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may query f(y) for any y
has access to a short random string s
and is otherwise deterministic.
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Distributed Property Reconstruction
Goal:
WHP (with probability close to 1)
(over choices of random string s):
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
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g has property P
d(g,f) = O( ε(f) )
Each A(x) runs quickly
in particular only reads f(y) for a small number of y.
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Distributed Property Reconstruction
Precursors:
 Online Data Reconstruction Model
(Ailon-Chazelle-Liu-Seshadhri)[ACCL]

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Locally Decodable Codes and Program selfcorrection (Blum-Luby-Rubinfeld; Rubinfeld-Sudan; etc )
Graph Coloring (Goldreich-Goldwasser-Ron)
Monotonicity Testing (Dodis-Goldreich- LehmanRaskhodnikova-Ron-Samorodnitsky; Goldreich-GoldwasserLehman-Ron-Samorodnitsky;Fischer;Fischer-Lehman-NewmanRaskhodnikova-Rubinfeld-Samorodnitsky;Ergun-Kannan-KumarRubinfeld-Vishwanathan; etc)

Tolerant Property Testing (Parnas, Ron, Rubinfeld)
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Example: Local Decoding of Codes
Data set f = boolean string of length n
Property = is a Code word of a
given error correcting code C
Reconstruction = Decoding to a close code word
Distributed reconstruction = Local decoding
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Key issue: making answers consistent

For error correcting code, can assume
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
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input f decodes to a unique g.
The set of positions that need to be corrected is
determined by f.
For general property,

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many different g (even exponentially many) that
are close to f may have the property
We want to ensure that A produces one of them.
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An example

Monotonicity with input array:
1,….,100, 111,…,120,101,…,110,121,…,200.
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Monotonicity Reconstruction: d=1
f is a linear array of length n
First attempt at distributed reconstruction:
A(x) 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 Reconstruction: d=1
Second attempt
Set g(x) = max{ f(1), f(2),…, f(x) }
g is monotone but
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A(x) 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 reconstruction
algorithm for general dimension d such that:
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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 prior results on monotonicity testing
and 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, A(x) 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:

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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

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Green is f - monotone
Red has size O(ε(f)|Γ|)
Need subroutine Classify(y).
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A sufficient condition for f-monotonicity
A 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 set
The set Bad:
x in Bad if for some interval J containing x
|violations(x,J)|≥|J|/2
Lemma.
1)Good=Γ - Bad is f-monotone
2)|Bad| ≤ 4 ε(f)|Γ| .
Proof:
1)
If x,y are a violation then one of them is Bad for the
interval [x,y].
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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?
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To test whether y in Good:
For each interval J containing y,
check violations(y,J)< |J|/2
Difficulties

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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
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Compute violations* (y,J) only for a
carefully chosen set of test intervals
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The Test Set T
Assume n=|Γ|=2k
k layers of intervals
Layer 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 Green
else y is Red
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Where are we?
We have a subroutine Classify
 On input x,
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Classify outputs Green or Red
Runs in time polylog(n)
WHP
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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)})
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In particular, this gives
g(x) = f(x) for Green x
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Computing m(x)
m(x)
x
Can search back from x to find first Green
Inefficient if x is preceded by a long Red stretch
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Approximating m(x)?
m*(x)
x
Pick random Sample(x) of points less than x
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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)}
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Is m*(x) good enough?
m*(x)
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y
x
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?
m*(x)
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y
x
To ensure monotonicity we need:
x < y implies m*(x) < m*(y)
Requires relaxing the requirement:
for all Green z, m*(z) = z
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Thinning out Green* (x)
y
x
z
Plan: Eliminate certain unsafe points from
Green*(x)
Roughly, y is unsafe for x if for some
z>x
Some interval beginning with y and containing x has a
high density of Reds.
(There is a non-trivial chance that Sample(z)
has no Green points ≥ y.)
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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)
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{x: m^(x) ≠ x} is O(ε(f) |Γ|).
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Summary of 1-dimensional case
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Classify points as Green and Red
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For each x, choose Sample(x)
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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
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Few Red points
f restricted to Green is f-monotone
m^(x) is the maximum of Green^(x)
Output g(x)=f(m^(x))
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Dimension greater than 1
y
x
For x < y, want g(x) < g(y)
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Red/Green Classification

Extend the Red/Green classification to higher
dimensions:
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f restricted to Green is Monotone
Red is small
Straightforward (mostly) extension of
1-dimensional case
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Given Red/Green classification
In 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
0
x
1
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Set of Green maxima could be very large
Sparse Random Sampling will only roughly capture the frontier
Finding an appropriate definition of unsafe points is much harder
than in the one dimensional case
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Further work

The g produced by our algorithm has
d(g,f) ≤ C(d)ε(f)|Γ|
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Our C(d) is exp(d2) .
What should C(d) be? (Guess: C(d) = exp(d) )
Distributed reconstruction for other interesting
properties?

(Reconstructing expanders, Kale,Peres,
Seshadhri, FOCS 08)
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