max |L -L | << max |L -L | Gradient Clock Synchronization

(1)

Gradient Clock Synchronization

max _{v,w}ϵE |L _v -L _w | << max _v,wϵV |L _v -L _w |

(2)

Today: Lower Bound on Local Skew

Theorem Suppose

dH/dt ≤ dL/dt ≤ (1+μ)dH/dt

and (θ-1)d ≤ u/8, then the local skew is Ω(u log _σ D),

where

σ = μ/(θ-1).

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Recall: We Can “Hide” HW Clock Skew...

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H _v

H _w H _w

We can do this on each edge of a shortest path!

=> u dist(v,w)/2 HW clock skew between v and w

(4)

Recall: We Can “Hide” HW Clock Skew...

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H _v

H _w H _w

This yields skew Ω(u dist(v,w)) between v and w:

L _v ^f (t) – L _w ^f (t) = L _v ^s (t + u dist(v,w)/2) – L _w ^s (t)

≥ L _v ^s (t) + u dist(v,w)/2 – L _w ^s (t)

(5)

Recall: ...but at most u per edge!

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Message delay known to be between d-u and d

=> (local) reception times in indistinguishable

executions differ by at most u!

(6)

Racing Against Time

- in order to build up local skew ω(u), we need to

“reveal” the skew and the algorithm can react

dist(v,w) = D

average skew: u/2

(7)

Racing Against Time

- in order to build up local skew ω(u), we need to

“reveal” the skew and the algorithm can react - but it cannot remove all of the skew at once

dist(v,w) = D average skew: u/2

dist(v,w) = D

average skew:* u/4

*assuming dH/dt = 1 u dist(v,w)/(4μ)

time passes

(8)

Racing Against Time

- in order to build up local skew ω(u), we need to

“reveal” the skew and the algorithm can react - but it cannot remove all of the skew at once

=> can add up to Θ(u dist(v,w)/σ) HW skew

dist(v,w) = D average skew: u/2

dist(v,w) = D

average skew:* u/4

*assuming dH/dt = 1 u dist(v,w)/(4μ)

time passes

(9)

Whack-a-Mole and Pigdeon Hole

- the algorithm has choices: it can locally reduce skew anywhere, but not everywhere

?

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Whack-a-Mole and Pigdeon Hole

- the algorithm has choices: it can locally reduce skew anywhere, but not everywhere

=> if we subdivide the line into k segments, at least one still has average skew u/4

?

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Why Subdivide?

...because we‘re on a clock: the rate at which we

introduce HW skew is factor σ smaller than the rate of removal!

- note also: can only “hide” u/2 HW skew per hop

=> choose k so dist(v,w)/dist(v’,w’) ≈ σ

?

u/2 u/4

(12)

Why Subdivide?

...because we‘re on a clock: the rate at which we

introduce HW skew is factor σ smaller than the rate of removal!

- note also: can only “hide” u/2 HW skew per hop

=> choose k so dist(v,w)/dist(v’,w’) ≈ σ

u/2 u/4

v’ w’

(13)

Why Subdivide?

...because we‘re on a clock: the rate at which we

introduce HW skew is factor σ smaller than the rate of removal!

- note also: can only “hide” u/2 HW skew per hop

=> choose k so dist(v,w)/dist(v’,w’) ≈ σ

- dL/dt ≥ dH/dt => HW skew adds logical clock skew

? +HW

skew u/4 + u/2

u/2 u/4

v’ w’ v’ w’

(14)

Lower Bound on Local Skew

Theorem Suppose

dH/dt ≤ dL/dt ≤ (1+μ)dH/dt

and (θ-1)d ≤ u/8, then the local skew is Ω(u log _σ D),

where

σ = μ/(θ-1).

(15)

Proving the Theorem

- build up u D/2 between nodes v and w in distance D

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Proving the Theorem

- build up u D/2 between nodes v and w in distance D - until dist(v,w) ≤ σ :

- run algorithm for u dist(v,w)/(4μ) time

=> average skew between v and w decreases by at

at most u/4

(17)

Proving the Theorem

- build up u D/2 between nodes v and w in distance D - until dist(v,w) ≤ σ :

- run algorithm for u dist(v,w)/(4μ) time

=> average skew between v and w decreases by at at most u/4

- find v’ and w’ in distance dist(v,w)/σ with same

average skew as between v and w

(18)

Proving the Theorem

- build up u D/2 between nodes v and w in distance D - until dist(v,w) ≤ σ :

- run algorithm for u dist(v,w)/(4μ) time

=> average skew between v and w decreases by at at most u/4

- find v’ and w’ in distance dist(v,w)/σ with same average skew as between v and w

- add average skew u/2 by inserting hardware skew

(which translates to logical skew)

(19)

Proving the Theorem

- build up u D/2 between nodes v and w in distance D - until dist(v,w) ≤ σ :

- run algorithm for u dist(v,w)/(4μ) time

=> average skew between v and w decreases by at at most u/4

- find v’ and w’ in distance dist(v,w)/σ with same average skew as between v and w

- add average skew u/2 by inserting hardware skew (which translates to logical skew)

- set v := v‘ and w := w‘

(20)

Proving the Theorem

- build up u D/2 between nodes v and w in distance D - until dist(v,w) ≤ σ :

- run algorithm for u dist(v,w)/(4μ) time

=> average skew between v and w decreases by at at most u/4

- find v’ and w’ in distance dist(v,w)/σ with same average skew as between v and w

- add average skew u/2 by inserting hardware skew (which translates to logical skew)

- set v := v‘ and w := w‘

Overall, we get a skew of u/4 * log _σ D this way.

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Proving the Theorem*

- build up u D/2 between nodes v and w in distance D - until dist(v,w) ≤ σ :

- run algorithm for u dist(v,w)/(4μ) time

=> average skew between v and w decreases by at at most u/4

- find v’ and w’ in distance dist(v,w)/σ with same average skew as between v and w

- add average skew u/2 by inserting hardware skew (which translates to logical skew)

max |L -L | << max |L -L | Gradient Clock Synchronization

Gradient Clock Synchronization

max {v,w}ϵE |L v -L w | << max v,wϵV |L v -L w |

Today: Lower Bound on Local Skew

Theorem Suppose

dH/dt ≤ dL/dt ≤ (1+μ)dH/dt

and (θ-1)d ≤ u/8, then the local skew is Ω(u log σ D),

where

σ = μ/(θ-1).

Recall: We Can “Hide” HW Clock Skew...

w

v 7

7

8

8

9

9

10

10

11

11

w

v 7

7

8

8

9

9

10

10

11

11

H v

H v

H w H w

We can do this on each edge of a shortest path!

=> u dist(v,w)/2 HW clock skew between v and w

Recall: We Can “Hide” HW Clock Skew...

w

v 7

7

8

8

9

9

10

10

11

11

w

v 7

7

8

8

9

9

10

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11

11

H v

H v

H w H w

This yields skew Ω(u dist(v,w)) between v and w:

L v f (t) – L w f (t) = L v s (t + u dist(v,w)/2) – L w s (t)

≥ L v s (t) + u dist(v,w)/2 – L w s (t)

Recall: ...but at most u per edge!

w

v 7

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v 7

max _{v,w}ϵE |L _v -L _w | << max _v,wϵV |L _v -L _w |

and (θ-1)d ≤ u/8, then the local skew is Ω(u log _σ D),

v ⁷

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H _v

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L _v ^f (t) – L _w ^f (t) = L _v ^s (t + u dist(v,w)/2) – L _w ^s (t)

≥ L _v ^s (t) + u dist(v,w)/2 – L _w ^s (t)

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H _v

H _v

H _w H _w