Bounding Variance and Expectation of Longest Path Lengths in DAGs
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Bounding Variance and Expectation of Longest Path Lengths in DAGs Jeff Edmonds, York University Supratik Chakraborty, IIT Bombay. Motivation. Statistical timing analysis of circuits Mean and std deviation of component delays provided by manufacturers

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Bounding Variance and Expectation of Longest Path Lengths in DAGs Jeff Edmonds, York University

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Bounding variance and expectation of longest path lengths in dags jeff edmonds york university

Bounding Variance and Expectation of Longest Path Lengths in DAGs

Jeff Edmonds, York University

Supratik Chakraborty, IIT Bombay


Motivation

Motivation

Statistical timing analysis of circuits

Mean and std deviation of component delays provided by manufacturers

Joint distributions of component delays difficult to obtain in practice


The longest path problem

The Longest Path Problem

Input:

st-DAG G gives job precedence.

For each edge i, xi is the time to complete job i

Output:

Time for all jobs to complete in parallel = length of longest st-path = Maxp i p xi = XG

s

x1

x3

x2

t

Easy with Dynamic Programming


The longest path problem1

The Longest Path Problem

Input:

st-DAG G gives job precedence.

For each edge i, xi is the time to complete job i

Output:

Time for all jobs to complete in parallel = length of longest st-path = Maxp i p xi = XG

s

Inter-dependent random variables

x1

x3

x2

t

Understand random variable XG


The longest path problem2

The Longest Path Problem

Input:

st-DAG G gives job precedence.

For each edge i, xi is the time to complete job i

Output:

Time for all jobs to complete in parallel = length of longest st-path = Maxp i p xi = XG

s

Exp[Xi] & Var[Xi]

x1

x3

x2

t

Bound Exp[XG] & Var[XG]


The longest path problem3

The Longest Path Problem

Input:

XG = Max( x1+x2, x3 )

4

5

0

1

s

  • Possible distributions :

x1

x3

x2

t

  • Another possibility :


The longest path problem4

The Longest Path Problem

Input:

XG = Max( x1+x2, x3 )

Upper & Lower bounds

s

x1

x3

x2

t


Contributions

Contributions

Upper bounds of Exp[XG] and Var[XG]

A spring “algorithm” for computing bounds

Proof no distributions give higher values (skip)

Cake distributions that achieve bounds

Lower bounds of Exp[XG] and Var[XG]

Continuum of values for Exp[XG] and Var[XG]

Cake distributions that achieve any Exp[XG] and Var[XG] within range

Special results for series-parallel graphs


Series graphs

Series Graphs

If G is a series graph,

XG = ∑i xi

Exp[xG] = ∑i Exp[xi]

0 ≤ Var[xG] ≤ (∑i √Var[xi] )2

s

t


Series graphs1

Series Graphs

If G is a parallel graph,

XG = Maxi xi

Maxi Exp[xi] ≤ Exp[xG] ≤ ?

0 ≤ Var[xG] ≤ ?

s

t


Representing random variables

Representing Random Variables

X

5

0

r

0

0.5

1

X : Two-valued random variable, prob 0.5 for each value


Representing random variables1

Representing Random Variables

X

5

Z

0

r

0

0.5

1

X, Z : Two equivalent independent random variables.


Representing random variables2

Representing Random Variables

X

Y

5

0

r

0

0.5

1

X, Y : Two-valued random variables, prob 0.5 for each value

X, Y have perfect negative correlation

Exp( Max(x,y) ) = Exp(x) + Exp(y)

Var( Max(x,y) ) = 0


Series graphs2

Series Graphs

If G is a parallel graph,

XG = Maxi xi

Maxi Exp[xi]

≤ Exp[xG]

≤ Min(

∑i Exp[xi],

Maxi Exp[xi] + √∑i Var[xi] )

0 ≤ Var[xG] ≤ ∑i Var[xi]

s

t


Series parallel graphs

Series Parallel Graphs

Theorem

In a series-parallel graph,

Rules for maximum variance applied recursively to obtain Max Var[XG].

Not so Max Exp[XG]


Maximizing var x g

Maximizing Var [ XG ]

There are no distributions xi for which

Var[xi] = vi and Exp[xi] = mi

Var[XG] >

Proof uses lots of calculus.

Theorem


Cakes maximizing var x g

Cakes Maximizing Var [ XG ]

There exists “cake” distributions xi such that

Var[xi] = vi and Exp[xi] = mi

Var[XG] =

Theorem


Cake distribution

Cake Distribution

s

t

Find a cake distribution for each edgewith correct Exp[xi] & Var[xi]

to maximize Var[xG]


Cake distribution1

Cake Distribution

s

Exp[xi]

t


Cake distribution2

Cake Distribution

s

t

Var[xi] = ∑c (ε hc)2


Cake distribution3

Cake Distribution

s

t

  • Series graphs G:

  • XG ≈ x1 + x2

  • Candle heights add

  • Want candle heights to be in same location


Cake distribution4

Cake Distribution

s

t

  • Parallel graphs G:

  • XG ≈ Max( x1 , x2 )

  • Candle heights max

  • Want candle heights to be in different location


Cake distribution5

Cake Distribution

s

t

A candle location

for each st-path in G

but in the end

# candles ≈ # edges


Cake distribution6

Cake Distribution

s

t

If edge i not in path p,

candle for xi at location p has height 0

If candle is selected,

then corresponding path pis the longest path


Cake distribution7

Cake Distribution

s

t

“Springs” give

give candle heights.


Cakes maximizing var x g1

Cakes Maximizing Var [ XG ]

There exists “cake” distributions xi such that

Var[xi] = vi and Exp[xi] = mi

Var[XG] =

Theorem

Proved


Lower bound of var x g

Lower Bound of Var[ XG ]

TheoremVar[xG] ≥ 0

Continuum Results

Theorem

Every Var[XG] in this range achievable.


Bounding variance and expectation of longest path lengths in dags jeff edmonds york university

Lower bound of Exp [ XG ]


Upper bound of exp x g

Upper bound of Exp [ XG ]

XG

r

tp

0

1

For st-path p, tp is interval for which p is the longest path.

 p P tp = 1


Upper bound of exp x g1

Upper bound of Exp [ XG ]

XG

r

tp

0

1

ti

For edge i, ti is interval for which i is in the longest path.

ti =  p itp


Upper bound of exp x g2

Upper bound of Exp [ XG ]

Xi

r

tp

0

1

ti

If it can edge i contributes all of its mi =Exp[Xi] to Exp[XG]


Upper bound of exp x g3

Upper bound of Exp [ XG ]

Xi

r

tp

0

1

ti

But if vi = Var[Xi] is too small,

it can only contribute


Upper bound of exp x g4

Upper bound of Exp [ XG ]

XG

r

tp

0

1

ti


Conclusion future work

Conclusion & Future Work

Tight analysis for upper bounds was achieved

Cake distributions particularly important for achieving tight bounds

A related question is that of finding tight bounds of mean and expectation of difference in longest paths to two given nodes in a DAG

Spring algorithm involves solving non-linear constraints iteratively. Can an alternative algorithm be obtained?


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