Divide and Conquer Algorithms for Pub/Sub Overlay Design

1
Divide and Conquer Algorithms for Pub/Sub Overlay
Design

• Chen Chen 1
• joint work with Hans-Arno Jacobsen 1,2, Roman
  Vitenberg 3
• 1 Department of Electrical and Computer
  Engineering
• 2 Department of Computer Science
• University of Toronto
• 3 Department of Informatics
• University of Oslo

2
Example Pub/Sub
Interests boy
boy
Interests boy
girl
Interests girl
3
Pub/Sub

• A communication paradigm
• Subscribers express their interests
• Publishers disseminate messages
• Many applications and industry standards
• Application integration, financial data
  dissemination,
• RSS feed distribution, business process
  management
• WS Notifications, WS Eventing,
• OMGs Real-time Data Dissemination Service
• Topic-based pub/sub
• TIBCO RV
• Googles GooPS

4
Two componentsin pub/sub implementation

• Design of routing protocols

• Construction of overlay

• The design of protocols so that publications and
  subscriptions are sent most efficiently across
  the overlay network.
• G. Li et al., ICDCS08
• M. Castro et al., JSAC02

• The construction of the overlay topology such
  that network traffic is minimized.
• Chockler et al., PODC07
• Onus et al., INFOCOM09

5
Desirable properties for overlays

• Low average node degree
• Low fan-out of a node
• Low diameter
• Topic-connectivity
• Efficiency to construct
• Adaptability to churn
• Ease of distributed implementation

6
Our contributions
Previous algorithm GM
High running time cost
Full knowledge requirement
Centralized operation (difficult to decentralize)
No support for dynamic changes
Constructing from scratch only (No support for incremental addition)
Our algorithms
Low running time cost
Partial knowledge requirement
Centralized operation (easy to decentralize)
No direct support for dynamic changes
Constructing both from scratch and incrementally
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Topic-connectivity
b,c,d
b,c,d
V1
V1
a,c
a
a,c
a
V5
V2
V5
V2
V4
V3
V4
V4
V3
b,d
a,b
a,b
b,d
a,b
Suboverlay Ga is topic-connected
Suboverlay Gb is NOT topic-connected
An overlay G
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MinAvg-TCO problem
b,c,d
b,c,d
V1
V1
a,c
a,c
a
a
V5
V2
V5
V2
V4
V3
V4
V3
b,d
a,b
b,d
a,b
TCO1 has 5 edges
TCO2 has 10 edges
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MinAvg-TCO problem
b,c,d
V1

• A high-quality overlay
• Topic-connectivity
• Total number of edges
• Input
• a set of nodes V,
• a set of topics T,
• the interest function Int
• MinAvg-TCO(V,T,Int) (optimization version)
• Construct a TCO(V,T,Int,E) such that E is
  minimum.
• Avg-TCO(V,T,Int,k) (decision version)
• Is there a TCO(V,T,Int,E) such that Ek?
• Theorem MinAvg-TCO is NP-complete

a
V2
a,c
V5
a,b
V3
b,d
V4
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Greedy-Merge (GM) algorithm

• Greedy
• always making the choice that looks best at the
  moment
• GM for MinAvg-TCO
• always adding an edge with maximum link
  contribution
• Running Time O(V2?T)
• Approximation Ratio O(log(V?T))

11
Our contributions
Previous algorithm GM
High running time cost
Full knowledge requirement
Centralized operation (difficult to decentralize)
No support for dynamic changes
Construction from scratch only (No support for incremental addition)
Our algorithms
Low running time cost
Partial knowledge requirement
Centralized operation (easy to decentralize)
No direct support for dynamic changes
Construction both from scratch and incrementally
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TCO join problem

• Given p TCOs TCOd (Vd,Td,Intd,Ed), d1,..,p
• MinAvg-TCO-Join(V,T,Int,p) (optimization version)
• Construct a TCO(V,T,Int,E) such that E is
  minimum
• Avg-TCO-Join(V,T,Int,p,k) (decision version)
• Is there a TCO(V,T,Int,E) such that Ek?
• MinAvg-TCO is a special case of MinAvg-TCO-Join
• Theorem MinAvg-TCO-Join is NP-complete

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Solving MinAvg-TCO-Join

• MinAvg-TCO-Join could be solved by GM,
• but NOT practical
• Tear down all existing links
• Rebuild the overlay from scratch using GM
• It is better to preserve all existing edges and
  only add edges incrementally.

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Bad case for incremental addition of edges
Vall interested in all topics in T
Constructing incrementally
Constructing from scratch
Vall
Vall
V1
V1
V1
V2
Vn
V2
Vn
V2
Vn
Vi
Vn-1
Vi
Vn-1
Vi
Vn-1
TCO0
TCO2
TCO1
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Naive Merge (NM) algorithm
NM is based on the same greedy heuristic as GM.

• GM algorithm

• NM algorithm

• Input (V,T,Int)
• Output one TCO
• Algorithm
• - Start with an empty edge set
• - Always add an edge with maximum link
  contribution.
• Running time

• Input (Vd,Td,Intd,Ed), d1,...,p
• Output one TCO
• Algorithm
• - Start with existing internal-TCO links
• - Always add a cross-TCO edge with maximum link
  contribution.
• Running time

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Example of NM
c
a
V0
V1
c
a,c,d
V4
d
V12
V3
a,b,c
V13
V7
c
V9
V6
V10
d
a,b,c
c
Still a prohibitively high running time!!!
a,b,c
V2
V11
b,c,d
V8
a,b,d
V14
V5
a
a,b,d
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Star set
Given a TCO (V,T,Int,E) A Star set S is a subset
of V that covers all Vs topics.
b,c,d
b,c,d
b,c,d
V1
V1
V1
a
a
a
V5
V2
V5
V2
V5
V2
a,c
a,c
a,c
V4
V3
V4
V3
V4
V3
a,b
b,d
a,b
b,d
b,d
a,b
v3, v5 is a star set which covers all topics
a,b,c,d
v2, v3, v4 is not a star set it only covers
a,b,d
A topic-connected overlay
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Star set

• Star set nodes
• Represents the interests of all the nodes
• Can function as bridges to determine cross-TCO
  links
• Observation minimal star sets tend to be
  substantially smaller than the total number of
  nodes.
• How to find a minimum star set S for (V,T,Int)?
• Equal to classic set cover problem NP-complete
• Could be approximated with a log approximation
  ratio

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Star Merge (SM) algorithm

• NM algorithm

• SM algorithm

• Input (Vd,Td,Intd,Ed), d1,..,p
• Output one TCO
• Algorithm
• - Start with existing internal-TCO links
• - // Do nothing
• - Always add a cross-TCO edge with maximum link
  contribution.

• Input (Vd,Td,Intd,Ed), d1,..,p
• Output one TCO
• Algorithm
• - Start with existing internal-TCO links
• - Find a star set for each sub-TCO
• - Always add a cross-Star edge with maximum link
  contribution.

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Example of SM
c
a
V0
V1
c
a,c,d
V4
d
V12
V6
a,b,c
V13
V7
c
V9
a,b,c
V10
V3
d
c
Running time largely improved because stars ltlt
nodes for most cases.
a,b,c
V2
V11
b,c,d
V8
a,b,d
V14
V5
a
a,b,d
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Divide and Conquer (DC) for MinAvg-TCO

• The number of nodes is a dominant factor for the
  running time of the GM algorithm.
• Divide-and-conquer
• Divide the MinAvg-TCO problem into several
  sub-overlay construction problems
• Conquer the sub-MinAvg-TCO problems independently
  and build sub-overlays into sub-TCOs
• Combine these sub-TCOs to one TCO

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Design of DC algorithm

• How to divide the node set V
• Node clustering vs. random partitioning
• The number of partitions p
• The balance between conquer and combine
• p 1 (single partition) conquer only GM
• p V (each node is a partition) combine only
  GM
• How to decentralize DC
• Note the DC algorithm as presented is fully
  centralized.
• However, it is possible to decentralize it.
• Theoretical analysis not straightforward.

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Example of DC
c
a
V0
V1
c
a,c,d
V4
d
V12
V6
a,b,c
V13
V7
c
V9
a,b,c
V10
V3
d
c
- Divide overlay based on V - Conquer each
sub-TCO by GM - Combine TCO into one by SM
a,b,c
V2
V11
b,c,d
V8
a,b,d
V14
V5
a
a,b,d
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Experiment setting

• The number of nodes
• V 1000 ranging from 1000 to 8000
• The number of topics
• T 100 ranging from 100 to 1000
• The number of topics that subscribed by a node
• NodeIntSize20 ranging from 10 to 100
• Topic distribution uniform, zipf, exponential

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Experiment design

• Evaluation average node degree, running time
• Star Merge for MinAvg-TCO-Join
• DC for MinAvg-TCO
• Random node partitioning
• The effects of the number of nodes
• The effects of the number of topics
• The effects of average subscription size of a
  node
• Comparison with RingPT
• RingPT is an algorithm that mimics the common
  practice of building separate overlay for each
  topic.

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Star MergeSM vs NM vs GM
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Divide-and-conquerThe effect of the number of
nodes
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Divide-and-conquerDC vs GM vs RingPT
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Algorithm summary
Running time Quality of overlay edges (avg node degree) Required information Potential to Decentralize
RingPT good poor full knowledge good
GM poor O(V2?T) good O(log(V?T)) full knowledge poor
NM poor 75 of GM good full knowledge good
SM good 1.0 of GM good 0.15 compared to GM partial knowledge good
DC good 1.7 of GM good 2.12 compared to GM partial knowledge good
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Minimal Number of Links

• A typical pub/sub system combines a number of
  protocols, many of which maintaining per-link
  state
• A node must constantly monitor the availability
  of each of its neighbors (heartbeats and
  keep-alive state)
• If the links are maintained using TCP, there is
  the cost of connection state for each link
• The more links there are, the fewer topics can be
  routed over each individual link, thereby
  diminishing cross-topic aggregation benefits
• If sequential-diff-based compression scheme is
  used, there is an extra cost associated with a
  history table