Overlay Network Construction in Highly Decentralized Networks

Overlay Network Construction in Highly Decentralized Networks Melih Onus PhD Thesis Defense Committee: Andrea W. Richa (Chair) Goran Konjevod Rida Bazzi Christian Scheideler April14, 2009, Arizona State University

Publish/Subscribe (Pub/Sub) {A,B,C,E,} Subscription(N1)={B,C,D} N2 {A,D} N1 N3 M1 Message Bus M1 {A,X} Publish(M1, A) N5 M1 N4 {A,B,X}

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 Relay {A,B,C,E} {B,C,D} (M1, A) N2 {A,D} N1 N3 (M1, A) (M1, A) (M1, A) • SCRIBE • Corona • Feedtree • Sub-2-Sub • TERA • ... N5 (M1, A) {A,X} N4 {A,B,X}

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

Topics B,C,X,E are connected Topics A and D are disconnected Topic-Connectivity {A,B,C,E} {B,C,D} N2 {A,D} N1 N3 N5 {A,X} N4 {A,B,X}

Node degree grows linearly with the subscription size • Roughly twice as big as the subscription size for rings/trees Topic-Connectivity: Simple Solution {A,B,C,E} {B,C,D} N2 {A,D} N1 N3 N5 {A,X} N4 {A,B,X}

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?

Outline • Minimum Maximum Degree Publish-Subscribe Overlay Network Design • Parameterized Maximum and Average Degrees in Publish-Subscribe Overlay Network Design • Constant Diameter Publish-Subscribe Overlay Network Design

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 MinMax-TCO Lemma:MinMax-TCO(V,T,Interest,k)NP Proof: Topic connectivity is verifyable in polynomial time Lemma:MinMax-TCO(V,T,Interest,k) is NP-hard Proof: • 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 • Set Cover is polynomially reducible to SN-TCO • SN-TCO is polynomially reducible to TCO Theorem: MinMax-TCO is NP-complete

Approximating MinMax-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

Overlay Design Algorithm {B,C,D} {A,B,C,E} N1 N2 {A,D} N3 N5 {A,X} N4 {A,B,X}

Maximum degree of 2 vs. almost 4 for ring-per-topic! Overlay Design Algorithm {B,C,D} {A,B,C,E} N1 N2 {A,D} N3 N5 {A,X} N4 {A,B,X}

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 connectedcomponents 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 greedyalgorithm 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: 10 Uniform distribution

Experimental Results II Average Node Degree #topics: 100 #subscriptions: 10 Uniform distribution

Experimental Results III Maximum Node Degree #topics: 100 #nodes: 100 Uniform distribution

v3 v2 v2 v1 v3 vn v1 vn … … vn-1 vn-1 v2 v1 v3 vn … vn-1 ODA Algorithm • The ODA algorithm can have average degree of (n), when constant average degree overlay network exists.

ODA and GM Algorithms • GM Algorithm: Choose edge with maximum benefit • Average Degree: O(log nt) approximation • Maximum Degree: O(n) approximation • ODA Algorithm: Choose edge with maximum benefit among the ones that increases maximum degree minimally • Average Degree: O(n) approximation • Maximum Degree: O(log nt) approximation How to approximate both average and maximum degree?

Parameterized Algorithm • e1: Edge with maximum benefit • e2: Edge with maximum benefit among the ones that increases maximum degree minimally • If w(e2) > w(e1) / k, choose e2 • Otherwise, choose e1 1 < k < n

Algorithms • GM Algorithm: • Average Degree: O(log nt) approximation • Maximum Degree: O(n) approximation • ODA Algorithm: • Average Degree: O(n) approximation • Maximum Degree: O(log nt) approximation • P-ODA Algorithm: • Average Degree: O(k * log nt) approximation • Maximum Degree: O((n/k)*log nt) approximation

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 • Constant Diameter Overlay Design Algorithm: • At each iteration: • Find number of neighbors for each node • Add a starwhich 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:

Constant Diameter Overlay Algorithm I • Constant Diameter Overlay Design Algorithm I: • At each iteration: • Find weight for each node • Add a starwhich connects the node with maximum weight, • Remove topics which are connected by the star • Stop, once there is a single connected component for each topic Weight of node u:

Constant Diameter Overlay Algorithm II • Constant Diameter Overlay Design Algorithm II: • At each iteration: • Find number of neighbors for each node • Add a starwhich connects the node with maximum density, • Remove topics which are connected by the star • Stop, once there is a single connected component for each topic Density of node u:

Experimental Results I Average Node Degree Varying #nodes #topics: 100 #subscription: 10 Uniform distribution Only 2.3 times more edge

Experimental Results II Average Node Degree Varying #topics #nodes: 100 #subscription: 20 Uniform distribution Only 1.9 times more edge

Experimental Results III Average Node Degree Varying #subscription #nodes: 100 #topics: 100 Uniform distribution Only 1.8 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 • Parameterized algorithm with low maximum and average degree • 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 • Proving diameter results theoretically

Publications • Parameterized Maximum and Average Degrees in Topic-based Publish-Subscribe Overlay Network Design, M. Onus and A. W. Richa,Submitted to 21st Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), August 2009. • Minimum Maximum Degree Publish-Subscribe Overlay Network Design, M. Onus and A. W. Richa, 28th Annual IEEE Conference on Computer Communications (INFOCOM), April 2009, Rio De Janeiro, Brazil. • Distributed Coloring with O(log n) bits, K. Kothapalli, M. Onus, C. Scheideler and C. Schindelhauer, To appear in Journal of Parallel and Distributed Computing (JPDC), 2008. • Linearization: Locally Self Stabilizing Sorting in Graphs, M. Onus, A. W. Richa, C. Scheideler, Workshop on Algorithm Engineering & Experiments (ALENEX), January 2007, New Orleans, Louisiana. • A Scalable Multilevel Algorithm for Community Structure Detection, H. Djidjev and M. Onus, 4th Workshop on Algorithms and Models for the Web-Graph (WAW), November 2006, Banff, Alberta. • Heuristics for Minimum Brauer Chain Problem, F.Gelgi and M.Onus, 21st International Symposium on Computer and Information Sciences (ISCIS), Springer LNCS 4263, November 2006, Istanbul, Turkey. • Distributed Coloring with O(log n) bits, K. Kothapalli, C. Scheideler, M. Onus and C. Schindelhauer, 20th IEEE Parallel & Distributed Processing Symposium (IPDPS), April 2006, Rhodes Island, Greece. • Efficient Broadcasting and Gathering in Wireless Ad-Hoc Networks, M. Onus, A. W. Richa, K. Kothapalli and C. Scheideler.International Symposium on Parallel Architectures, Algorithms and Networks (I-SPAN), December 2005, Las Vegas, Nevada. • Constant Density Spanners for Wireless Ad-Hoc Networks, K. Kothapalli, C. Scheideler, M. Onus and A. W. Richa. 17th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), July 2005, Las Vegas, Nevada.

Thank You!

Overlay Network Construction in Highly Decentralized Networks