Locally Self-Adjusting Tree Networks Chen Avin (BGU) Bernhard - - PowerPoint PPT Presentation

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Locally Self-Adjusting Tree Networks Chen Avin (BGU) Bernhard - - PowerPoint PPT Presentation

Nov 04, 2022 184 likes •468 views

Locally Self-Adjusting Tree Networks Chen Avin (BGU) Bernhard Hupler (MIT) Zvi Lotker (BGU) Christian Scheideler (U. Paderborn) Stefan Schmid (T-Labs) 1 From Optimal Networks to Self -Adjusting Networks Networks become more and

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SLIDE 1

Locally Self-Adjusting Tree Networks

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Chen Avin (BGU) Bernhard Häupler (MIT) Zvi Lotker (BGU) Christian Scheideler (U. Paderborn) Stefan Schmid (T-Labs)

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SLIDE 2

From “Optimal” Networks to Self-Adjusting Networks

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  • Networks become more and more dynamic (e.g., flexible SDN control)
  • Vision: go beyond classic “optimal” static networks
  • Example (of this paper): Peer-to-peer

Chord, Pastry, SHELL Koorde, ... Pancake

  • Hypercubic
  • Log diameter
  • Log degree
  • Log routing
  • Constant degree
  • Log routing
  • Log/loglog degree and

log/loglog routing

Stefan Schmid (T-Labs)

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SLIDE 3

From “Optimal” Networks to Self-Adjusting Networks

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  • Networks become more and more dynamic (e.g., flexible SDN control)
  • Vision: go beyond classic “optimal” static networks
  • Example: Peer-to-peer

Chord, Pastry, SHELL Koorde, ... Pancake

  • Hypercubic
  • Log diameter
  • Log degree
  • Log routing
  • Constant degree
  • Log routing
  • Log/loglog degree and

log/loglog routing

Wh What at if if ne networks

  • rks coul
  • uld

d sel elf-adjust adjust de depe pendin nding g

  • n
  • n comm
  • mmunic

unication ation pat attern? ern?

Stefan Schmid (T-Labs)

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SLIDE 4

An Old Concept: Move-to-front, Splay Trees, …

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  • Classic data structures: lists, trees
  • Linked list: move frequently accessed elements to front!
  • Trees: move frequently accessed elements closer to root

Stefan Schmid (T-Labs)

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SLIDE 5

An Old Concept: Move-to-front, Splay Trees, …

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  • Classic data structures: lists, trees
  • Linked list: move frequently accessed elements to front!
  • Trees: move frequently accessed elements closer to root

Stefan Schmid (T-Labs)

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SLIDE 6

An Old Concept: Move-to-front, Splay Trees, …

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  • Classic data structures: lists, trees
  • Linked list: move frequently accessed elements to front!
  • Trees: move frequently accessed elements closer to root

Stefan Schmid (T-Labs)

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SLIDE 7

An Old Concept: Move-to-front, Splay Trees, …

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  • Classic data structures: lists, trees
  • Linked list: move frequently accessed elements to front!
  • Trees: move frequently accessed elements closer to root

Stefan Schmid (T-Labs)

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SLIDE 8

An Old Concept: Move-to-front, Splay Trees, …

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  • Classic data structures: lists, trees
  • Linked list: move frequently accessed elements to front!
  • Trees: move frequently accessed elements closer to root

Stefan Schmid (T-Labs)

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SLIDE 9

An Old Concept: Move-to-front, Splay Trees, …

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  • Classic data structures: lists, trees
  • Linked list: move frequently accessed elements to front!
  • Trees: move frequently accessed elements closer to root

Splay Trees!

Stefan Schmid (T-Labs)

Splay Trees!

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SLIDE 10

The Vision: Splay Networks (“Distributed Splay Trees”)

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  • Most simple self-adjusting tree network: Binary Search Tree (BST)

Stefan Schmid (T-Labs)

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SLIDE 11

The Vision: Splay Networks (“Distributed Splay Trees”)

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  • Most simple self-adjusting tree network: Binary Search Tree (BST)

Stefan Schmid (T-Labs)

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SLIDE 12

The Vision: Splay Networks (“Distributed Splay Trees”)

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  • Most simple self-adjusting tree network: Binary Search Tree (BST)

Stefan Schmid (T-Labs)

Communication between peer pairs! (Not only lookups from root…)

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SLIDE 13

The Vision: Splay Networks (“Distributed Splay Trees”)

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  • Most simple self-adjusting tree network: Binary Search Tree (BST)

Stefan Schmid (T-Labs)

Why BST?!

  • Most simple generalization of

classic data structure

  • Allows for local routing!
  • Allows for algebraic gossip
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SLIDE 14

Model: Self-Adjusting SplayNets

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Input:

  • communication pattern:

(static or dynamic) graph

Stefan Schmid (T-Labs)

Output:

  • sequence of network adjustments

Cost metric:

  • expected path length
  • # (local) network updates

“Host Graph” “Guest Graph”

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SLIDE 15

Our Contribution

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SplayNets

  • “Online algorithm” for

self-adjusting distributed trees

  • Optimal offline algorithm

(polynomial time, for large class

  • f graphs!)

Stefan Schmid (T-Labs)

Performance evaluation:

  • General bounds on amortized costs
  • Lower bounds (empirical entropy)
  • Analysis of convergence times

for important static comm. patterns

  • Optimality of online algorithm for

special patterns (e.g., matchings)

  • Simulation study (Facebook data)
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SLIDE 16

The Optimal Offline Solution

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Stefan Schmid (T-Labs)

Dynamic program

  • Binary search:

decouple left from right!

  • Polynomial time

(unlike MLA!)

  • So: solved M”BST”A

See also:

  • Related problem of

phylogenetic trees

OPT OPT OPT

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SLIDE 17

The Online SplayNets Algorithm

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Stefan Schmid (T-Labs)

From Splay tree to SplayNet:

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SLIDE 18

The Online SplayNets Algorithm

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Stefan Schmid (T-Labs)

From Splay tree to SplayNet:

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SLIDE 19

The Online SplayNets Algorithm

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Stefan Schmid (T-Labs)

From Splay tree to SplayNet:

Least Common Ancestor Local rotations!

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SLIDE 20

Analysis: Basic Lower and Upper Bounds

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Stefan Schmid (T-Labs)

Adaption of Tarjan&Sleator

A-Cost < H(X) + H(Y)

Upper Bound

where H(X) and H(Y) are empirical entropies of sources

  • resp. destinations

A-Cost > H(X|Y) + H(Y|X)

Lower Bound

where H( | ) are conditional entropies.

Assuming that each node is the root for “its tree” Therefore, our algorithm is optimal, e.g., if communication pattern describes a product distribution!

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Properties: Convergence

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Stefan Schmid (T-Labs)

Nodes communicate within local clusters only!

Over time, nodes will form clusters in BST! No paths “outside”.

Cluster scenario:

IDs

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Properties: Optimal Solutions

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Stefan Schmid (T-Labs)

Will converge to optimum: Amortized costs 1.

Laminated scenario:

IDs Will converge to optimum: Amortized costs 1.

Non-crossing matching (= “no polygamy”) scenario:

IDs

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SLIDE 23

Properties: Optimal Solutions

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Stefan Schmid (T-Labs)

Multicast scenario (BST): Example Invariant over “stable” subtrees (from right):

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SLIDE 24

Improved Lower Bounds (and More Optimality)

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Stefan Schmid (T-Labs)

Cut of interval: entropy yields amortized costs!

Via interval cuts or conductance entropy:

IDs

Grid:

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SLIDE 25

Simulation Results

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Stefan Schmid (T-Labs)

  • Facebook component with 63k nodes and 800k edges
  • SplayNet exploit random walk locality, to less extent also matching
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SLIDE 26

Conclusion

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Stefan Schmid (T-Labs)

  • Vision: self-adjusting networks
  • Interesting generalization of Splay trees
  • SplayNets
  • Formal analysis reveals nice properties
  • Amortized costs good: but tight?
  • Competitive ratio remains open
  • Future work? Yes 
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SLIDE 27

Thank you! Questions?

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Stefan Schmid (T-Labs)

“Host Graph” “Guest Graph”