Sensor Networks Costas Busch Division of Computer Science and Eng. - - PowerPoint PPT Presentation

Universal Steiner Trees for Efficient Data Aggregation in Sensor Networks Costas Busch

Division of Computer Science and Eng. Louisiana State University

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Talk Overview

Oblivious Network Design

• Distr. Transactional Memory

Single-Sink Network Design

Task: For each terminal, choose a path to sink. : terminals sink

Single-Sink Network Design

Task: For each terminal, choose a path to sink. : terminals sink Goal: Minimize 

E e e

w

edge weights

Communication Cost

Oblivious Network Design

Task: pre-compute paths to sink. : terminals sink Goal: Minimize 

E e e

w

For every set of terminals Communication Cost

: terminals sink sink Tree 1 Tree 2 Different Steiner trees for different sources

• n top of same universal spanning tree

Applicability

Sensor Networks

Distributed sensor nodes send information to central node(s)

Content-based publish-subscribe systems

Users “publish” or “subscribe” to information

VLSI, Transportation, Power Grids, Pipelines

Problem Hardness

Steiner Tree Problem is NP-complete For universal Steiner trees:

stretch log log log           n n

Our Results

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Busch, Dutta, Radhakrishnan, Rajaraman and Srinivasagopalan “Split and Join: Strong Partitions and Universal Steiner Trees for Graphs” IEEE Symposium on Foundations of Computer Science (FOCS 2012) Srinivasagopalan, Busch, and Iyengar “An Oblivious Spanning Tree for Single-Sink Buy-at-Bulk in Low Doubling-Dimension Graphs” IEEE Transactions on Computers, 2012

stretch 2

) log (



n O

General graphs

stretch log poly  n

Planar graphs

stretch log poly  n

Doubling dimension

Partitions

10

partition ) , , (    

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partition ) , , (    

      Cluster strong diameter

   

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partition ) , , (    

Cluster strong diameter

   

      #clusters in radius



  

Partition Hierarchy

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) , , , ( hierarchy partition ) , , (

1 d

P P P      ) / ( log d a DG





partition ) , , ( :   b a P

……

Partition Hierarchy

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) / ( log d a DG





……

partition ) , , ( :

1 1

  b a P

……

partition ) , , ( :   b a P

) , , , ( hierarchy partition ) , , (

1 d

P P P     

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) / ( log d a DG





……

partition ) , , ( :

1 1

  b a P

……

partition ) , , ( :

1 1



  i i

b a P 

……

partition ) , , ( : 

i i

b a P 

…… ……

partition ) , , ( :   b a P

) , , , ( hierarchy partition ) , , (

1 d

P P P     

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) / ( log d a DG





……

partition ) , , ( :

1 1

  b a P

……

partition ) , , ( :

1 1



  i i

b a P 

……

partition ) , , ( : 

i i

b a P 

……

partition ) , , ( : 

d d

b a P 

Whole graph …… ……

partition ) , , ( :   b a P

) , , , ( hierarchy partition ) , , (

1 d

P P P     

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Root padding

partition ) , , ( :

1 1



  i i

b a P  partition ) , , ( : 

i i

b a P  i



1  i

 r

r

Universal Tree

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partition ) , , ( :   b a P

Universal Tree

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partition ) , , ( :   b a P partition ) , , ( :

1 1

  b a P

…… Combine sub-trees of clusters by organizing them on a virtual shortest path tree

Universal Tree

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partition ) , , ( :   b a P partition ) , , ( :

1 1

  b a P

……

partition ) , , ( :

1 1



  i i

b a P 

……

partition ) , , ( : 

i i

b a P 

…… ……

Universal Tree

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partition ) , , ( :   b a P partition ) , , ( :

1 1

  b a P

……

partition ) , , ( :

1 1



  i i

b a P 

……

partition ) , , ( : 

i i

b a P 

……

partition ) , , ( : 

d d

b a P 

Whole graph

root

…… ……

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Approximation:

) , , , ( hierarchy partition ) , , (

1 d

P P P     

  stretch

n O

n

 log ) (

2 log 





Bad! For small (constant) stretch is some root of n

   , ,

Top Down Approach

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d

P

1  d

P

1  d

P

1  d

P

1  d

P

Top Down Approach

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d

P

1  d

P

1  d

P

1  d

P

1  d

P

Find shortest paths from lower level leader to the root

Top Down Approach

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d

P

1  d

P

1  d

P

1  d

P

1  d

P

Find shortest paths from lower level leader to the higher level

Top Down Approach

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d

P

1  d

P

1  d

P

1  d

P

1  d

P

Find shortest paths from lower level leader to the higher level

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Approximation:

) , , , ( hierarchy partition ) , , (

1 d

P P P     

  stretch

n O  log

2 2

  

Better!

Why Partitions?

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Many nearby nodes Many long paths BAD CASE

Why Partitions?

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Many nearby nodes Many long paths Impossible with partitions

Partition Algorithm for General Graphs

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partition

1  

P

……

Individual nodes

Recursive construction Basis of Recursion:

partition ) , , ( :

1 1



  i i

b a P 

……

partition ) , , ( : 

i i

b a P 

…… Assuming We will build

) , , , ( hierarchy partition ) , , (

1 d

P P P     

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Recursive step works in stages

partition ) , , ( :

1 1



  i i

b a P 

…… Stage 0

rank 0 rank 0 rank 0 rank 0

The rank of a cluster is the stage that it has been created

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Recursive step works in stages

partition ) , , ( :

1 1



  i i

b a P 

…… Stage 0

rank 0 rank 0 rank 0 rank 0

Stage 1

rank 1 rank 1

Merge Merge Problematic nodes:

k

n

1



clusters of rank 0 in

• d

neighborho

i 



i



i



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A stage has 2 phases Phase 1:

rank 1 rank 1

Problematic nodes with rank 1:

k

n

1



clusters of rank 0 in

• d

neighborho

i 



……

rank 0

Phase 2:

i



rank 1

Merge

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In stage 2 we merge clusters of rank 1

• r less if there are problematic nodes

Final stage has no more problematic nodes Algorithm repeats to higher stages

Analysis

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Stage j, Phase 1:

rank j rank j-1 rank j-1 rank j-1

Merge

Problematic node

i



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Stage j, Phase 2:

rank j

Problematic node

rank j-1 rank j-1 rank j-1 i



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Stage j, Phase 2:

rank j rank j-1 rank j-1 rank j-1 rank j

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Stage j, Phase 2:

rank j rank j-1 rank j-1 rank j-1 rank j i



rank j-1 rank j-1 rank j-1

New problematic node IMPOSSIBLE!

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Stage j, Phase 2:

rank j rank j-1 rank j-1 rank j-1 rank j i



rank j-1 rank j-1 rank j-1

The only possibility for problematic nodes

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Stage j, Phase 2:

rank j rank j-1 rank j-1 rank j-1 rank j rank j-1 rank j-1 rank j-1

core Problematic nodes can only be at core

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Stage j, Phase 2:

rank j rank j-1 rank j-1 rank j-1 rank j rank j-1 rank j-1 rank j-1

core Problematic nodes can only be at core This helps to bound the diameter

1 1 1

6 2 4

  

  

j j j j

D D D D

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Minimum size of cluster

1

k

n

1 k k k

n n n

2 1 1

 

Stage 0: Stage 1: Stage 2:

k j

n

Stage j: How many stages?

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Minimum size of cluster

k j

n

Stage j: Therefore, maximum stage: k Maximum diameter:

6 D D

k k 

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     

 

) , , , ( hierarchy partition 2 , 2 , 2

1 log log log d n O n O n O

P P P  

Take

6 log log n k 

1 log

2





i n i

D D

) , , , ( hierarchy partition ) , , (

1 d

P P P     

n

a

log

2 

 

n O k

n

log 1

2   

level i level i-1

6 D D

k k 

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     

 

hierarchy partition 2 , 2 , 2

log log log



n O n O n O

hierarchy partition ) , , (    

  stretch

n O  log

2 2

  

stretch

n O



) log (

2

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For planar graphs:

 

hierarchy partition log , log , log

3 4 3

 n n n

By recursively splitting the graph

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Talk Overview

Oblivious Network Design

• Distr. Transactional Memory

Distributed Transactional Memory

• Transactions run on network nodes
• They ask for shared objects distributed over the network

for either read or write

• They appear to execute atomically
• The reads and writes on shared objects are supported

through three operations:

 Publish  Lookup  Move

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Predecessor node Suppose the object ξ is at node and is a requesting node ξ Requesting node

Suppose transactions are immobile and the objects are mobile

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Read-only copy Main copy Lookup operation ξ ξ

Replicates the object to the requesting node

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Read-only copy Main copy Lookup operation ξ ξ

Replicates the object to the requesting nodes

Read-only copy ξ

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Main copy Invalidated Move operation ξ ξ

Relocates the object explicitly to the requesting node

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Invalidated Move operation ξ

Relocates the object explicitly to the requesting node

Main copy ξ Invalidated ξ

Need a distributed directory protocol

 To provide objects to the requesting nodes efficiently implementing Publish, Lookup, and Move

• perations

 To maintain consistency among the shared object copies

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Previous Approaches

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Protocol Stretch Network Kind Runs on Arrow [DISC’98] O(SST)=O(D) General Spanning tree Relay [OPODIS’09] O(SST)=O(D) General Spanning tree Combine [SSS’10] O(SOT)=O(D) General Overlay tree Ballistic [DISC’05] O(log D) Constant- doubling dimension Hierarchical directory with independent sets

➢ D is the diameter of the network kind ➢ S is the stretch of the tree used

Our Results

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Sharma, Busch, and Srinivasagopalan “Distributed Transactional Memory for General Networks” IEEE International Parallel and Distributed Processing Symposium (IPDPS 2012) Sharma and Busch “Towards Load Balanced Distributed Transactional Memory” International European Conference on Parallel and Distributed Computing (EUROPAR 2012)

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Problem of Spanning Tree Approaches These two nodes are far in tree

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Spiral directory protocol for general networks O(log2n . log D) stretch avoids race conditions

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Hierarchical clustering Spiral Approach: Network graph

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Hierarchical clustering Spiral Approach:

Alternative representation as a hierarchy tree with leader nodes

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At the lowest level (level 0) every node is a cluster

Directories at each level cluster, downward pointer if object locality known

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Owner node

root

A Publish operation

➢ Assume that is the creator of which invokes the Publish operation ➢ Nodes know their parent in the hierarchy

ξ ξ

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root

Send request to the leader

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root

Continue up phase Sets downward pointer while going up

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root

Continue up phase Sets downward pointer while going up

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root

Root node found, stop up phase

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root

A successful Publish operation Predecessor node ξ

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Requesting node Predecessor node

root

Supporting a Move operation

➢ Initially, nodes point downward to object owner (predecessor node) due to Publish operation ➢ Nodes know their parent in the hierarchy

ξ

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Send request to leader node of the cluster upward in hierarchy

root

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Continue up phase until downward pointer found

root

Sets downward path while going up

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Continue up phase

root

Sets downward path while going up

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Continue up phase

root

Sets downward path while going up

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Downward pointer found, start down phase

root

Discards path while going down

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Continue down phase

root

Discards path while going down

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Continue down phase

root

Discards path while going down

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Predecessor reached, object is moved from node to node

root

Lookup is similar without change in the directory structure and only a read-only copy of the object is sent

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Distributed Queue

root

u u tail head

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Distributed Queue

root

u u tail head v v

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root

u v w

Distributed Queue

u tail head v w

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root

u v w

Distributed Queue

tail head v w

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root

u v w

Distributed Queue

tail head w

Race Conditions for Ballistic

Locking is required

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A

• bject

parent(A) lookup parent(A) move parent(A)

B

Probes left to right

C

Level k Level k-1 Level k+1

Ballistic configuration at time t

From root Lookup from C is probing parent(B) at t

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(Our protocol) Spiral avoids Race condition

➢ Label all the parents in each level and visit them in the

• rder of the labels.

2 1

A

• bject

parent(A) lookup parent(A) move parent(A)

B

3

C

Level k Level k-1 Level k+1 From root parent(B)

Spiral Hierarchy

(O(log n), O(log n))-labeled sparse cover hierarchy constructed from O(log n) hierarchical partitions from Gupta, Hajiaghayi, Racke SODA 2006 (adapts FRT scheme)

 Level 0, each node belongs to exactly one cluster  Level h, all the nodes belong to one cluster with root r  Level 0 < i < h, each node belongs to exactly O(log n) clusters which are labeled different

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Spiral Hierarchy

• How to find a predecessor node?

 Via spiral paths for each leaf node u by visiting leaders of all the clusters that contain u from level 0 to the root level

The hierarchy guarantees: (1) For any two nodes u,v, their

spiral paths p(u) and p(v) meet at level min{h, log(dist(u,v))+2} (2) length(pi(u)) is at most O(2i log2n)

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root

u

p(u) p(v)

v

(Canonical) downward Paths

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root

u

p(u)

root

u

p(u)

v

p(v) p(v) is a (canonical) downward path

Analysis: lookup Stretch

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v w vi

x

Level k Level i O(2k log2n) O(2i log2n) O(2k log n) 2i If there is no Move, a Lookup r from w finds downward path to v in level log(dist(u,v))+2 = O(i) When there are Moves, it can be shown that r finds downward path to v in level k = O(i + log log2n) p(w) p(v)

C(r)/C*(r) = O(2k log2n)+O(2k log n)+O(2i log2n) / 2i-1 = O(log4n)

Canonical path spiral path

Analysis: move Stretch

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Level Assume a sequential execution R of l+1 Move requests, where r0 is an initial Publish request.

C*(R) ≥ max1≤k≤h (Sk-1) 2k-1 C(R) ≥ σ

k=1

ℎ (Sk−1) O(2k log2n)

C(R)/C*(R) = σ

k=1

ℎ (Sk−1) O(2k log2n) / max1≤k≤h (Sk-1) 2k-1

= O(log2n. h) max1≤k≤h (Sk-1) 2k-1 / max1≤k≤h (Sk-1) 2k-1 = O(log2n. log D)

h . . . k . . . 2 1

request x

r0 . . . r0 . . . r0 r0 r0 r1 . . r1 r1 r1

u v y w

r2 r2 r2 . . r2 r2 r2 rl-1 rl-1 rl-1 r2 . . rl . . . rl rl rl

. . . Thus,

Recap for Transactional Memory

• A distributed directory protocol Spiral for

general networks that

 Has poly-logarithmic stretch  Is starvation free  Avoids race conditions  Factors in the stretch are mainly due to the parameters of the hierarchical clustering

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