Transiently Consistent SDN Updates: Being Greedy is Hard Saeed - - PowerPoint PPT Presentation

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Transiently Consistent SDN Updates: Being Greedy is Hard Saeed - - PowerPoint PPT Presentation

Feb 04, 2024 185 likes •545 views

Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases Transiently Consistent SDN Updates: Being Greedy is Hard Saeed Akhoondian Amiri 1 Arne Ludwig 1 Jan Marcinkowski 2 Stefan Schmid 3 1 TU Berlin 2 University of Wroclaw 3

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Transiently Consistent SDN Updates: Being Greedy is Hard

Saeed Akhoondian Amiri1 Arne Ludwig1 Jan Marcinkowski2 Stefan Schmid3

1TU Berlin 2University of Wroclaw 3Aalborg University Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Table of contents

1 Introduction 2 Greedy is Hard 3 Polynomial Time Algorithm for Special Cases

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Plan

1 Introduction 2 Greedy is Hard 3 Polynomial Time Algorithm for Special Cases

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Software Defined Networks

1 Centralised Controller, 2 Asynchronous Updates, 3 Consistent Updates

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Packet Routing

1 Initial routing from source to destination 2 New routing policy by controller 3 Consistent updates in few rounds.

Saeed Amiri SDN Updates: Being Greedy is Hard

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SLIDE 6

Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Relaxed Loop Freedom

Consistency:

1 Always there is a path from the source to the destination. 2 There cannot be any loop at any moment in a path from

source to destination.

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Relaxed Loop Freedom

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Relaxed Loop Freedom

The source and the destination are not connected via an active path( solid edges).

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Strong Loop Freedom

1 Always there is a path from the source to the destination. 2 There cannot be any loop at any moment w.r.t. active edges.

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Relaxed (Strong) Loop Freedom

Goal: Minimise the number of rounds.

1 Optimal solution? 2 Greedy algorithm: In the round i, find the maximum number

  • f vertices that we can update.

Greedy Algorithm (Ludwing,Marcinkowski,Schmid PODC15):

1 It is at least as hard as Feedback Arc Set (FAS) problem in

the update graph.

2 FAS is hard in general graphs but easy for some graph classes. 3 Network update graph is very simple:

Out degrees at most two They must constitute a legal path Is greedy algorithm in P ?!

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Plan

1 Introduction 2 Greedy is Hard 3 Polynomial Time Algorithm for Special Cases

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Greedy is Hard

Theorem 1 Finding the greedy solution for each round is NP-hard even in update graph. Reduction from Hitting Set Problem: Input: A set S = {S1, . . . , Sm} of subsets of U = {a1, . . . , an}. Output: Set S′ ⊆ U of minimum size such that S′ ∩ Si = ∅, For i ∈ {1, . . . , m}

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Greedy is Hard: Proof’s General Idea

Construction

1 Directed rooted tree with 2n + 1 branches: 1 One S − D branch 2 For each element ai ∈ U two branches: ain

i − D (input branch)

and aout

i

− D (output branch)

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Greedy is Hard: Proof’s General Idea

Construction

1 Directed rooted tree with 2n + 1 branches 2 Source Connectors: Thick dashed blue edges

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Greedy is Hard: Proof’s General Idea

Construction

1 Directed rooted tree with 2n + 1 branches 2 Source Connectors: Thick dashed blue edges 3 Element Selectors: From each ain i

an edge to aout

i

, thin blue edges.

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Greedy is Hard: Proof’s General Idea

Construction

1 Directed rooted tree with 2n + 1 branches 2 Source Connectors: Thick dashed blue edges 3 Element Selectors: From each ain i

an edge to aout

i

, thin blue edges

4 Set Selectors: n + 1 edges from element aout i

to ain

j if ai, aj

appear after each other in sequence St, Thick red edges.

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Greedy is Hard: Proof’s General Idea

S = {S1, S2, S3, S4} = {{1, 2, 3}, {1, 4}, {2, 5}, {3, 2, 5}}.

1 Each set Si corresponds to a cycle, 2 Deleting n element selector edge destroys all cycles, 3 Minimum number of not updatable edges = Minimum hitting

set.

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Greedy is Hard: Proof’s General Idea

S = {S1, S2, S3, S4} = {{1, 2, 3}, {1, 4}, {2, 5}, {3, 2, 5}}.

1 Each set Si corresponds to a cycle, 2 Deleting n element selector edge destroys all cycles, 3 Minimum number of not updatable edges = Minimum hitting

set.

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Greedy is Hard: Proof’s General Idea

S = {S1, S2, S3, S4} = {{1, 2, 3}, {1, 4}, {2, 5}, {3, 2, 5}}.

1 Each set Si corresponds to a cycle, 2 Deleting n element selector edge destroys all cycles, 3 Minimum number of not updatable edges = Minimum hitting

set.

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Greedy is Hard: Special Update Graph?

Is it possible to reach that special update graph from some initial architecture? Long Story Short: Yes it is possible, in one greedy step we are there

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Greedy is Hard: Forward Edges

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Plan

1 Introduction 2 Greedy is Hard 3 Polynomial Time Algorithm for Special Cases

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Two Branches

Update graph: Active Edges: Rooted directed tree with two branches Update edges: Rest of edges (Dashed Edges)

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Two Branches

Cycle:

1 If two update edges (e1, e2) cross each other, and 2 Head of ei is below tail of ei′(i, i′ ∈ {0, 1}).

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Two Branches: Strong LF (SLF)

1 Update Edge: Assign a Vertex, 2 Connect two vertices if they correspond to a cycle.

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Two Branches: SLF

Minimum Vertex Cover ⇔ Minimum Non Updatable Edges Graph is Bipartite ⇒ Polynomial Time Solvable.

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Two Branches: Relaxed LF

Simulate update edges in the second branch to reduce it to SLF

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Two Branches: Relaxed LF

Simulate update edges in the second branch to reduce it to SLF

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Two Branches: Relaxed LF

Simulate update edges in the second branch to reduce it to SLF

Saeed Amiri SDN Updates: Being Greedy is Hard

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SLIDE 31

Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Two Branches: Relaxed LF

Simulate update edges in the second branch to reduce it to SLF

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

Three Branches

Main Edges: They form three branches Update edges: Connection between three branches Construct a similar model as before? Vertex cover in cubic graphs is NP-Hard (Brooks’ Theorem)

Saeed Amiri SDN Updates: Being Greedy is Hard

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Introduction Greedy is Hard Polynomial Time Algorithm for Special Cases

More General Graphs?

1 What if the graph has bounded directed tree-width

Claim: It is NP-hard.

2 Is greedy hard even in update graph with 3 branches? 3 What if Feedback Arc Set is small?

Saeed Amiri SDN Updates: Being Greedy is Hard

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Thank you