Theory of Parallel and Distributed Systems (WS2016/17)

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Kapitel 6 Routing II

Walter Unger

Lehrstuhl für Informatik 1

11:54 Uhr, den 30. Januar 2017

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Inhalt I

1 Introduction Situation The Model Lower Bound Proof Application

2 Consistent Hashing Introduction Statements

3 Chord Network Introduction Statements

4 Randomized Oblivious Routing Introduction

5 Path Selection

Path Selection on the Hypercube: Valiant’s Trick

Analyzing a random routing problem

6 Packet Scheduling for the Hypercube The algorithm

Proof

7 Packet scheduling for general networks The algorithm

Proof

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Situation

Current Situation:

every permutation could be routed on permutation network and mesches number of steps proportional to the diameter algorithm was centralized

needed global knowledge about the sources and destinations of all packets

We now want to deviselocal-controlalgorithms.

Each nodes decides on the next step by some local information.

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Situation

Current Situation:

every permutation could be routed on permutation network and mesches number of steps proportional to the diameter algorithm wascentralized

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Situation

Current Situation:

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Situation

Current Situation:

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Situation

Current Situation:

every permutation could be routed on permutation network and mesches number of steps proportional to the diameter algorithm was centralized

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Situation

Current Situation:

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Situation

Current Situation:

We now want to devise local-control algorithms.

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Situation

Current Situation:

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The Model

The network is modelled by a graphG = (V,E).

Arouting problemRonG is defined by a finite set of packets each of which comes with a source and a destination node.

We assume that time proceeds in synchronous steps:

Before the first step, each packet is placed at its source.

In each step, each edge can forward at most one packet in each direction.

The routing is completed as soon as all packets have reached their destination.

The number of stepsT taken by an algorithm to deliver all packets is referred to as routing time.

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The Model

A routing problemRonG is defined by a finite set of packets each of which comes with a source and a destination node.

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The Model

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The Model

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The Model

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The Model

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The Model

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Oblivious routing

Here: algorithms that belong to the class of oblivious routing algorithms:

the path of each packet depends only on the source and the destination of this packet but not on the sources and destinations of other packets.

One specifies a path systemW with a pathPu,v fromutov, for every possible source-destination pair(u,v)∈V²,u6=v.

Every packet with sourceuand destinationv is sent along the path Pu,v.

Example: bit-fixing paths on the hypercube

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Oblivious routing

Here: algorithms that belong to the class ofoblivious routing algorithms:

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Oblivious routing

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Oblivious routing

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Oblivious routing

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Lower Bound by Borodin and Hopcroft

Theorem

LetG= (V,E)be any graph and

W be any path system with a pathPu,v fromutov, for every (u,v)∈V²,u6=v.

Letndenote the number of nodes and∆the maximum degree ofG. There exists a permutationπ:V →V and an edgee^∗∈E such that at least

pn/(2∆²) = Ω(√ n/∆)

of the paths selected byπfromW contain the edgee^∗∈E.

This is very bad newsabout deterministic oblivious routing:

The time complexity for permutation routing under this paradigm is lower bounded byΩ(√

n/∆), which is polynomial inn.

Even a small diameter, say logarithmic inn, does not help.

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Lower Bound by Borodin and Hopcroft

Theorem

pn/(2∆²) = Ω(√ n/∆)

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Lower Bound by Borodin and Hopcroft

Theorem

Letndenote the number of nodes and∆the maximum degree ofG.

There exists a permutationπ:V →V and an edgee^∗∈E such that at least pn/(2∆²) = Ω(√

n/∆)

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Lower Bound by Borodin and Hopcroft

Theorem

pn/(2∆²) = Ω(√ n/∆)

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Lower Bound by Borodin and Hopcroft

Theorem

pn/(2∆²) = Ω(√ n/∆)

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Lower Bound by Borodin and Hopcroft

Theorem

pn/(2∆²) = Ω(√ n/∆)

This is very bad news about deterministic oblivious routing:

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Lower Bound by Borodin and Hopcroft

Theorem

pn/(2∆²) = Ω(√ n/∆)

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Lower Bound by Borodin and Hopcroft

Theorem

pn/(2∆²) = Ω(√ n/∆)

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Proof of the lower bound by Borodin and Hopcroft

Definition

Forv∈V, letWv ={Pv,u|u∈V}.

For a positive numbert, a nodev ∈V, and an edgee∈E, we say thate ist-popular forv if at leastt paths fromWv containe.

Outline of the proof:

First, we prove a lemma showing that, for any given nodev ∈V, there are “many” edges that are “quite popular” forv.

Then we use the lemma to show that there is an edgee^∗ that is “quite popular” for “many” nodes, that is,e^∗ist-popular fort different nodes, fort= Ω(√

n/∆).

Given this, we will be able to construct a permutationπsuch thattof the paths selected byπcontaine^∗, which proves the lower bound.

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Proof of the lower bound by Borodin and Hopcroft

Definition

n/∆).

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Proof of the lower bound by Borodin and Hopcroft

Definition

n/∆).

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Proof of the lower bound by Borodin and Hopcroft

Definition

n/∆).

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Proof of the lower bound by Borodin and Hopcroft

Definition

n/∆).

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Proof of the lower bound by Borodin and Hopcroft

Definition

n/∆).

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Proof of the lower bound by Borodin and Hopcroft

Definition

Fort>0, we define a 0-1 matrixA(t):

The matrix hasnrows and|E|columns.

Forv∈V, ande∈E, define Av,e(t) =

1 ife ist-popular forv, and 0 otherwise,

Forv∈V, letAv(t) =P

e∈EAv,e(t)denote the row sum ofv. Fore∈E, letAe(t) =P

v∈VAv,e(t)denote the column sum ofe.

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Proof of the lower bound by Borodin and Hopcroft

Definition

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Proof of the lower bound by Borodin and Hopcroft

Definition

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Proof of the lower bound by Borodin and Hopcroft

Definition

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Proof of the lower bound by Borodin and Hopcroft

Definition

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One Lemma for the Proof of the lower bound

Lemma

∀v ∈V andt6(n−1)/∆ :Av(t)> _2∆tⁿ . Proof of lemma:

LetQ⊆V be the set of nodes from which there is a path tov that contains only edges that aret-popular forv.

LetL=V −QandB=E∩(L×Q), that is,Bis the set of those edges connecting a node inLwith a node inQ.

It holds

|B| ·(t−1)>|L|because, for each nodeu∈L, the pathPv,uleads through at least one edge inB and these edges are nott-popular so that each of them can be contained in at mostt−1 paths fromWv.

|B|6∆|Q|as each node inQ has at most∆incident edges.

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One Lemma for the Proof of the lower bound

Lemma

It holds

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One Lemma for the Proof of the lower bound

Lemma

It holds

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One Lemma for the Proof of the lower bound

Lemma

It holds

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One Lemma for the Proof of the lower bound

Lemma

It holds

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One Lemma for the Proof of the lower bound

Lemma

It holds

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Proof of the lemma

|B| ·(t−1)>|L|and|B|6∆|Q|

Combining the two equations, we obtain

∆|Q|(t−1) > |L| = n− |Q| , which implies

∆|Q|t > n .

and, hence,

|Q| > n

∆t .

Next we will show|Q|62A_v(t)which completes the proof of the lemma as it implies

Av(t) > |Q|

2 ≥ n

2∆t .

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Proof of the lemma

|B| ·(t−1)>|L|and|B|6∆|Q|

∆|Q|t > n .

and, hence,

|Q| > n

∆t .

Av(t) > |Q|

2 ≥ n

2∆t .

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Proof of the lemma

|B| ·(t−1)>|L|and|B|6∆|Q|

∆|Q|t > n .

and, hence,

|Q| > n

∆t .

Av(t) > |Q|

2 ≥ n

2∆t .

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Proof of the lemma

|B| ·(t−1)>|L|and|B|6∆|Q|

∆|Q|t > n .

and, hence,

|Q| > n

∆t .

Av(t) > |Q|

2 ≥ n

2∆t .

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Proof of the lemma

Show:|Q|62Av(t)

LetE⁰ denote the set of edges that aret-popular forv. To complete the proof of the lemma, we have to show|Q|62|E⁰|=2Av(t).

At first, we obersve thatt6(n−1)/∆implies thatE⁰6=∅.

This is because

v has at most∆incident edges, and Wv containsn−1 paths

such that at least one of the edges incident tov is contained in at least (n−1)/∆>z paths fromWv.

Therefore, there is at least one edge that ist-popular forv.

Given thatE⁰ is non-empty, each node inQ is incident to an edge inE⁰. Consequently,|Q|62|E⁰|as each of the edges inE⁰ is incident to at most two nodes fromQ.

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Proof of the lemma

Show:|Q|62Av(t)

This is because

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Proof of the lemma

Show:|Q|62Av(t)

This is because

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Proof of the lemma

Show:|Q|62Av(t)

This is because

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Proof of the lemma

Show:|Q|62Av(t)

This is because

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Proof of the lemma

Show:|Q|62Av(t)

This is because

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Proof of the lemma

Show:|Q|62Av(t)

This is because

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Proof of the lemma

Show:|Q|62Av(t)

This is because

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Proof of the lower bound by Borodin and Hopcroft

Show:∃e^∗:e^∗ist-popular fortdifferent nodes, fort= Ω(√ n/∆).

Our next goal is to show that there exists an edgee^∗that ist-popular for t nodes wheret= Ω(√

n/∆).

We observe that X

e∈E

Ae(t) =X

e∈E

X

v∈V

Ae,v(t) =X

v∈V

X

e∈E

Ae,v(t) =X

v∈V

Av(t)> n² 2∆t , where the inequality follows from the lemma.

Because of the “pigeonhole principle”, there has to exist an edgee^∗∈E such that

Ae^∗(t)>

n²

|E| ·2∆t

>l n

2∆²t m

, where the last step follows from|E|6∆n.

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Proof of the lower bound by Borodin and Hopcroft

n/∆).

We observe that X

e∈E

Ae(t) =X

e∈E

X

v∈V

Ae,v(t) =X

v∈V

X

e∈E

Ae,v(t) =X

v∈V

Ae^∗(t)>

n²

|E| ·2∆t

>l n

2∆²t m

, where the last step follows from|E|6∆n.

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Proof of the lower bound by Borodin and Hopcroft

n/∆).

We observe that X

e∈E

Ae(t) =X

e∈E

X

v∈V

Ae,v(t) =X

v∈V

X

e∈E

Ae,v(t) =X

v∈V

Ae^∗(t)>

n²

|E| ·2∆t

>l n 2∆²t

m , where the last step follows from|E|6∆n.

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Proof of the lower bound by Borodin and Hopcroft

Show:∃e^∗:e^∗ist-popular fortdifferent nodes, fort= Ω(√

n/∆). We haveA_e∗(t)>d ⁿ 2∆2te.

Next we chooset such thatAe^∗(t) = _2∆ⁿ2t, that is, we set t=√

n/(√ 2∆).

Observe thatt=√ n/(√

2∆)implies t6(n−1)/∆, for anyn>2, so that the assumption aboutt that we made in the lemma is satisfied.

For this choice oft, our analysis gives Ae^∗(t)>d ⁿ

2∆²√ n/(√

2∆)e=dⁿ

√2∆

2∆²√ ne=d

√n

√ 2∆e.

Ae^∗(t)>dte, that is,

e^∗isdte-popular fordtenodes, wheret=√ n/(√

2∆).

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Proof of the lower bound by Borodin and Hopcroft

n/(√ 2∆).

2∆²√ n/(√

2∆)e=dⁿ

√2∆

2∆²√ ne=d

√n

√ 2∆e.

2∆).

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Proof of the lower bound by Borodin and Hopcroft

n/(√ 2∆).

2∆²√ n/(√

2∆)e=dⁿ

√2∆

2∆²√ ne=d

√n

√ 2∆e.

2∆).

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Proof of the lower bound by Borodin and Hopcroft

n/(√ 2∆).

2∆²√ n/(√

2∆)e=dⁿ

√2∆

2∆²√ ne=d

√n

√ 2∆e.

2∆).

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Proof of the lower bound by Borodin and Hopcroft

n/(√ 2∆).

2∆²√ n/(√

2∆)e=dⁿ

√2∆

2∆²√ ne=d

√n

√ 2∆e.

2∆).

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Proof of the lower bound by Borodin and Hopcroft

n/(√ 2∆).

2∆²√ n/(√

2∆)e=dⁿ

√2∆

2∆²√ ne=d

√n

√ 2∆e.

2∆).

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Proof of the lower bound by Borodin and Hopcroft

n/(√ 2∆).

2∆²√ n/(√

2∆)e=dⁿ

√2∆

2∆²√ ne=d

√n

√ 2∆e.

2∆).

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Proof of the lower bound by Borodin and Hopcroft

Cconstruct a permutationπsuch thattof the paths selected byπcontaine^∗.

Finally, we construct a permutationπsuch thatdteof the paths selected byπ containe^∗:

LetV⁰ denote a set ofdtenodes for whiche^∗isdte-popular.

W.l.o.g.,V⁰={1, . . . ,dte}.

For everyv ∈V⁰, there exists a subsetUv ⊆V of cardinalitydtesuch that, for everyu∈Uv, the pathPv,ucontainse^∗.

Forv=1 todte, setπ(v) =uwhereuis chosen arbitrarily from Uv\ {π(1), . . . , π(v−1)}.

Forv=dte+1 ton, setπ(v) =u whereuis chosen arbitrarily from V \ {π(1), . . . , π(v−1)}.

By our construction,πande^∗satisfy the properties described in the theorem.

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Proof of the lower bound by Borodin and Hopcroft

W.l.o.g.,V⁰={1, . . . ,dte}.

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Proof of the lower bound by Borodin and Hopcroft

W.l.o.g.,V⁰={1, . . . ,dte}.

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Proof of the lower bound by Borodin and Hopcroft

W.l.o.g.,V⁰={1, . . . ,dte}.

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Proof of the lower bound by Borodin and Hopcroft

W.l.o.g.,V⁰={1, . . . ,dte}.

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Proof of the lower bound by Borodin and Hopcroft

W.l.o.g.,V⁰={1, . . . ,dte}.

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Proof of the lower bound by Borodin and Hopcroft

W.l.o.g.,V⁰={1, . . . ,dte}.

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Application to the hypercube and Goal

For thed-dimensional hypercube withn=2^d nodes, the lower bound of Borodin and Hopcroft implies a lower bound ofΩ(√

n/logn)for permutation routing.

There is a permutationπsuch thatΩ(√

n)paths contain the same edge when using bit-fixing paths on the hypercube.

Consequently, when using bit-fixing paths the time complexity for permutation routing isΩ(√

n).

Our goal is to devise a distributed permutation routing algorithm with time complexityO(logn).

This will take some time.

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Application to the hypercube and Goal

n).

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Application to the hypercube and Goal

n).

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Application to the hypercube and Goal

n).

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Application to the hypercube and Goal

n).

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Introduction Con.Hash. Ch.NW. Rand.Obl.Routing Path Selection Hypercube General NW.

6:14 Introduction 1/5 Walter Unger 30.1.2017 11:54 WS2016/17 Z

Outline of the approach

We build a dynamic system of storage devices supporting the addition and removal of storage devices using dynamic hashing:

devices are mapped i.u.r. to the ring[0,1), that is, each devicei gets assigned a random adressa(i)∈[0,1)

data objects are mapped to the ring using a random hash function h:U→[0,1), that is, objectx is mapped to positionh(x) data objectx is stored on the device found next toh(x)in clock-wise direction on the ring

We assume an idealistic hash function, that is, the hash values are real numbers chosen i.u.r. from[0,1).

1independently, uniformly at random

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Outline of the approach

devices are mapped i.u.r.¹to the ring[0,1), that is, each devicei gets assigned a random adressa(i)∈[0,1)

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Outline of the approach

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