Minimum Maximum Degree Publish-Subscribe Overlay Network Design

Melih Onus

TOBB Ekonomi ve Teknoloji Üniversitesi, 28 Mayıs 2009

Publish/Subscribe (Pub/Sub)

N1

Subscription(N1)={B,C,D}N2

{A,B,C,E,}

N3

{A,D}

N4

{A,B,X}

N5

{A,X}Message BusMessage Bus

Publish(M1, A)

M1

M1

M1

Scalability of Pub/Sub

Most traditional pub/sub systems are geared towards small scale deployment– E.g., Isis MDS, TIB, MQSeries, Gryphon

New generation of applications…– Large data centers: Amazon, Google, Yahoo, EBay,…– RSS, feed/news readers, on-line stock trading and banking– Web 2.0, Second Life

…drive dramatic growth in scale– 10,000s of nodes, 1000s of topics, Internet-wide distribution

Emerging systems address this trend using P2P techniques

Overlay-Based Pub/Sub

N1

{B,C,D}N2

{A,B,C,E}

N3

{A,D}

{A,B,X}

N5

{A,X}

N4

(M1,

A)

(M1, A)

(M1, A)

(M1, A

)(M1, A)

•SCRIBE•Corona •Feedtree •Sub-2-Sub •TERA•...

Relay

Overlay Topologies for Pub/Sub

“Good” overlay will allow for efficient and simple publication routing– Small routing tables, low load on relays, – low latency

Ideally, overlay is topic-connected: i.e., one connected component for each topic-induced sub-graph– Most existing implementations construct topic-connected

overlays

Topic-Connectivity

Topics B,C,X,E are connected

Topics A and D are disconnected

N1

{B,C,D}N2

{A,B,C,E}

N3

{A,D}

{A,B,X}

N5

{A,X}

N4

Topic-Connectivity: Simple Solution

N1

{B,C,D}N2

{A,B,C,E}

N3

{A,D}

{A,B,X}

N5

{A,X}

N4

Node degree grows linearly with the subscription size Roughly twice as big as the subscription size for

rings/trees

Scalability of the Simple Solution

Negative impact on performance due to– CPU load: neighbor monitoring, message processing– Connection maintenance and header overhead– Memory overhead: per-link state associated with routing

and/or compression schemes being used, etc.

Scalability barrier for large systems offering a wide range of subscription choices

Can we do better?Can we do better?

The MinMax-TCO Problem

Minimum Maximum Degree Topic-Connected Overlay (MinMax-TCO) problem:– For a set of nodes V, set of topics T, and Interest: V T

{true, false}– Construct a topic-connected overlay G with the minimum

possible maximum degree TCO (decision version):

– Decide whether there is a topic-connected overlay with maximum degree k (for a given k)

GM Algorithm

The GM algorithm can have maximum degree of (n), when constant maximum degree overlay network exists.

Complexity of TCO

Lemma: TCO(V,T,Interest,k)NPProof: Topic connectivity is verifyable in polynomial time

Lemma: TCO(V,T,Interest,k) is NP-hardProof: 1. Define an auxiliary problem Single Node TCO (SN-TCO)

which is to decide if there is a topic-connected overlay in which the degree of single given node d

2. Set Cover is polynomially reducible to SN-TCO3. SN-TCO is polynomially reducible to TCO

Theorem: TCO is NP-complete

Approximating Min-TCO

The idea: exploiting subscription overlaps– Connecting the nodes with overlapping interests improves

connectivity of several topics at once Overlay Design Algorithm (ODA):

– Start from a singleton connected component for each (v, t) V T

– At each iteration: add an edge that reduces the number of connected components for the biggest number of topics among the ones which increase maximum degree minimally

– Stop, once there is a single connected component for each topic

Greedy Merge

N1

{B,C,D}

N2

{A,B,C,E}

N3

{A,D}

{A,B,X}

N5

{A,X}

N4

Topic # of conn. comps

A 4

B 3

C 2

D 2

X 2

E 1

Greedy Merge

N1

{B,C,D}

N2

{A,B,C,E}

N3

{A,D}

{A,B,X}

N5

{A,X}

N4

Topic # of conn. comps

A 3

B 2

C 2

D 2

X 2

E 1

Greedy Merge

N1

{B,C,D}

N2

{A,B,C,E}

N3

{A,D}

{A,B,X}

N5

{A,X}

N4

Topic # of conn. comps

A 3

B 2

C 2

D 1

X 2

E 1

Greedy Merge

N1

{B,C,D}

N2

{A,B,C,E}

N3

{A,D}

{A,B,X}

N5

{A,X}

N4

Topic # of conn. comps

A 3

B 1

C 1

D 1

X 2

E 1

Greedy Merge

N1

{B,C,D}

N2

{A,B,C,E}

N3

{A,D}

{A,B,X}

N5

{A,X}

N4

Topic # of conn. comps

A 2

B 1

C 1

D 1

X 1

E 1

Greedy Merge

N1

{B,C,D}

N2

{A,B,C,E}

N3

{A,D}

{A,B,X}

N5

{A,X}

N4

Topic # of conn. comps

A 1

B 1

C 1

D 1

X 1

E 1

Maximum degree of 2 vs. almost 4 for ring-per-topic!

ODA Running Time

O(|V|4|T|)– At most |V|2 iterations – At most |V|2 edges inspected at each iteration– At most |T| steps to inspect an edge

Can be optimized to run in O(|V|2 |T|)– For each e V V, weight(e) = the number of connected

components merged by e– At each iteration, output the heaviest edge and adjust the other

edge weights accordingly– Stop once there are no more edges with weight > 0

Approximability Results

Lemma: The number of edges in the overlay constructed by GM log(|V||T|) OPT

Proof: Similar to that of the approximation ratio of the greedy algorithm for Set Cover

Uses Maximum Weighted Matching Uses Edge Coloring

Theorem: No algorithm can approximate MinMax-TCO within a constant factor (unless P=NP)

Proof: Existence of such an algorithm would imply existence of the constant factor approximation for Set Cover which is known to be impossible (unless P=NP)

Experimental Results I

Maximum Node Degree#topics: 100#subscriptions: 10Uniform distribution

Experimental Results II

Average Node Degree#topics: 100#subscriptions: 10Uniform distribution

Experimental Results III

Maximum Node Degree#topics: 100#nodes: 100Uniform distribution

Constant Diameter Overlays

Constant Diameter Topic-Connected Overlay (CD-TCO) problem:– For a set of nodes V, set of topics T, and Interest: V T

{true, false}– Construct a topic-connected, constant diameter overlay G

with the minimum possible average degree

The GM algorithm can have diameter of (n), where n is number of nodes in the pub/sub system.

Constant Diameter Overlay Algorithm

The idea: adding stars– Make topics connected with star structures

Constant Diameter Overlay Design Algorithm:– Start from a singleton connected component for each (v, t)

V T– At each iteration:

• Add a star which connects maximum number of nodes, • Remove topics which are connected by the star

– Stop, once there is a single connected component for each topic

Number of neighbors of node u:

Experimental Results

Average Node Degree#topics: 100#nodes: 100Uniform distribution

Only 2.3 times more edge

Conclusions

Formal study of the problem of designing efficient and scalable overlay topologies for pub/sub

Defined the problem (MinMax-TCO) capturing the cost of constructing topic-connected overlays– NP-Completeness, polynomial approximation,

inapproximability results

Empirical evaluation showed effectiveness of our approximation algorithm on practical inputs

Defined the problem (CD-TCO), empirical results

Future Directions

Study dynamic case Investigate other overlay design problems Study distributed case

– Partial knowledge of other node interest– Dynamically changing interest assignments

Thank You!