A Distributed and Oblivious Heap

Christian Scheideler and Stefan SchmidDept. of Computer Science

University of Paderborn

MotivationP2P systems: significant amount of attention.

Holy grail: P2P systems that are highly scalable and robust

Internet

Motivation

P2P systems face many threats:stop/fail attacks, join/leave attacks, Eclipse attacks, Sybil attacks, Denial-of-Service attacks, content pollution,…

Internet

Sybil Attacks

Goal: flood network with fake peers

Effect: disconnect good peers, drop messages, drop or manipulate data,…

Internet

Sybil Attacks

Previous solutions:• Douceur 02: Use trusted authority to limit generation of

identities• Bazzi et al. 05/06: Limit identities created at a particular

location (e.g., within an ISP) of the Internet• Danezis et al. 05, Yu et al. 06: Network based on social

contacts• Margolin & Levine 07: Game theoryGeneral approach: add constraintsOur approach: working P2P network despite Sybil attacks

Our Approach

Connect to the older peers!

Our Approach

Similar to preferential attachment graphs.

Hope: peers already in the system when Sybil attack starts are not affected by it.

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Network Entry Point

How to connect to older peers?

Naive solution:

Connection to d older peers chosen uniformly at random:

• How to join and leave efficiently?• Not a self-routing topology!Better: distributed heap based on de Bruijn graph

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Dynamic de Bruijn Network

Classical network theory: family of graphs

Example: d-dim. de Bruijn graph G=(V,E)• V = {0,1}d

• E = { {x,y} | x=(x1,…,xd), y=(b,x1,…,xd-1), for some b{0,1} }

We need: dynamic form of de Bruijn graph (nodes join and leave) 00

01

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Dynamic de Bruijn Network

Example: d-dim. de Bruijn graph G=(V,E)• V = {0,1}d

• E = { {x,y} | x=(x1,…,xd), y=(b,x1,…,xd-1), for some b{0,1} }

• Represent (x1,…,xd) as 0.x1 x2…xd [0,1), e.g. 0.101 = 1(1/2) + 0(1/4) + 1(1/8)

• Let d

Dynamic de Bruijn Network

Example: d-dim. de Bruijn graph G=(V,E)• V = {0,1}d

• E = { {x,y} | x=(x1,…,xd), y=(b,x1,…,xd-1), for some b{0,1} }

Result for d:• V = [0,1)• E = { {x,y}[0,1)2 | y=x/2 or y=(1+x)/2 }

Dynamic de Bruijn Network

Continuous de Bruijn graph:• V = [0,1)• E = { {x,y}[0,1)2 | y=x/2 or y=(1+x)/2 }

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For any two regions R, R´: {x,y}E: (x,y)RxR´ or (x,y)R´xR: each peer in R to all older peers in R´

• cut [0,1) into regions of same size • assign peers to random points in [0,1)

RR´

Dynamic de Bruijn Network

Continuous de Bruijn graph:• V = [0,1)• E = { {x,y}[0,1)2 | y=x/2 or y=(1+x)/2 }

0 1

Dilemma: for any fixed region size EITHER : no neighbors OR : too many neighbors

SHELL

SHELL: Hierarchy of de Bruijn networks

SHELL

SHELL: distributed heap based on age

SHELL

For every node v:• v has forward edges to older peers in its home region and

de Bruijn regions• v has backward edges to all peers that have forward edges

to it• Level of v is the lowest s.t. v´s home region contains at

least c log nv older nodes in this and higher levels(nv: total # nodes older than v) (How does v know?)

Forward degree: O(log n), backward degree: O(log2 n)(w.h.p., based on random positions of peers)

SHELL

Routing between two peers v and w:1. use deBruijn routing along forward edges till region of w

reached2. descend along backward edges till w reached

v

w

1.

2.

SHELL

Routing between two nodes v and w:1. use deBruijn routing along forward edges till region of w

reached2. descend along backward edges till w reached

v1.

forward edgeyoungest peer

SHELL

Routing between two nodes v and w:1. use deBruijn routing along forward edges till region

of w reached2. descend along backward edges till w reached

Nice properties:• Only nodes visited that are at least as old as v and w• When every node v sends packet to random w,

congestion at most O(log2 n) w.h.p.

SHELLJoin(v):1. find right region in right level prescribed by order nv and

random point assigned to vO(log n) time w.h.p. (use routing)

2. integrate v into SHELL and adapt regions of neighbors (if needed)O(1) time, O(log2 n) work w.h.p. (for general heap)

v

w Expected O(1) lifts per levelO(log n) work per lift w.h.p.

Join/leave verkehrt!!

SHELL

Leave(v):• Remove v from SHELL and adapt regions of

neighbors (if needed)O(1) time, O(log2 n) work w.h.p.

vw

Expected O(1) drops per levelO(log n) work per drop w.h.p.

SHELL

Problem: node v needs to estimate its order nv

(i.e. # nodes older than v)Solution:• B(j): number of nodes in home region of v in

level j that are older than v• v finds largest j so that j < B(j)/c – log B(j)

Chosen level deviates by 1 from ideal level w.h.p.

Distributed Heaps

Concurrent heaps:• Shavit & Zemach 99: skip queue• Shavit & Zemach 00: funnel treenot applicable in our context

Distributed heaps:• Bhargava et al. 04: PAGODA

join/leave: O(log2 n) time and work

Main Results

Theorem: For any Sybil attack after step t0, all peers that were in the SHELL system before t0 can still perform reliable and efficient routing (and based on that, also data management).

SHELL can also be applied as scalable, low-congestion network for peers with arbitrary bandwidths.

Conclusion

SHELL: distributed heap with essentially optimal runtime for routing, joining and leaving

Future tasks:• Better congestion (O(log2 n) down to O(log n))• Locality• Realization?