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Optimal Nearest Neighbor Queries in Sensor Networks Gokarna Sharma and Costas Busch
Division of Computer Science and Eng. Louisiana State University
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Problem Description
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Optimal Nearest Neighbor Queries in Sensor Networks Gokarna Sharma - - PowerPoint PPT Presentation
Optimal Nearest Neighbor Queries in Sensor Networks Gokarna Sharma - - PowerPoint PPT Presentation
Optimal Nearest Neighbor Queries in Sensor Networks Gokarna Sharma and Costas Busch Division of Computer Science and Eng. Louisiana State University 1 Problem Description Distributed sensor network 2 Problem Description Objects are tracked
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Problem Description
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Objects are tracked by the sensors Red nodes are proxy nodes that detect the objects (one proxy per object)
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Problem Description
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Query from a node: find the closest object Where is the closest object?
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Problem Description
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Response to query: proxy to closest object Closeness is measured with graph hops (or graph distance)
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Problem Challenges
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- Queries may initiate from arbitrary nodes
- Objects can move
- Unknown number of objects
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Performance Metrics
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Use a distributed data structure to maintain position information for the objects Two performance cost metrics: Query cost: distance to returned proxy node Update cost: total communication cost to maintain data structure as the objects move
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Update Cost
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Update cost lower bound: the total distance that the objects (proxies) have moved in the graph
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d
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d
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d
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d
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d
i
d
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Contributions
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A distributed directory approach based on a hierarchical clustering of the graph Low doubling dimension graphs: Query cost approximation: Update cost approximation:
1 O
n O log n is the number of nodes
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Contributions
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General weighted graphs: Query cost approximation: Update cost approximation:
n O
4
log
n O
3
log n is the number of nodes
The results can be generalized to general graphs
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Closeness
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Due to the directory structure, the absolute nearest object may be missed We guarantee closeness in low doubling dimension graphs
d ) (d O
Query node closest proxy node returned proxy
) 1 ( O
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Distributed Directory Approaches (for single object)
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Protocol Move Cost 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 Spriral [IPDPS/12] Arbitrary Graphs Hierarchical directory with sparse covers
➢ D is the diameter of the network kind ➢ S is the stretch of the tree used
D n O log log2
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Problem of Spanning Tree Approaches These two nodes are far in tree
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Network graph
Hierarchical Clustering
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Alternative representation as a hierarchy tree with leader nodes
Hierarchical Clustering
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At the lowest level (level 0) every node is a cluster
Hierarchical Clustering
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p-Doubling Dimension Graphs
Every ball of radius R can be completely covered by at most balls of radius R/2.
2-doubling dimension
2
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Overlay Tree
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Layers of independent sets of leaders Graph nodes root
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2
k 2
2
k 3
2
k
D k log
k
1 k 2 k
… ……
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Parent Node
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Level i+1 Level i
1
2
i i
2
Refinement of independent sets
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Parent Set
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Level i+1 Level i
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2
i i
2
Useful for query lookups Constant number of potential parent nodes
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General Graphs
The clusters are built from sparse covers Each cluster has diameter The parent set size is at most
) log 2 (
1
n O
i
) (logn O
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Multiple Objects
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proxy node
root
➢ Assume that is the creator of which invokes the Publish operation ➢ Nodes know their parent in the hierarchy
ξ ξ
Publish Object
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root
Send request to the parent leader
Publish Object
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root
Continue up phase
Sets downward pointer while going up
Publish Object
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root
Continue up phase
Sets downward pointer while going up
Publish Object
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root
Root node found, stop up phase
Publish Object
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root
A successful Publish operation ξ
Publish Object
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root
The publish operation ends at any node with downward path
Publish Object – multiple objects
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destination
- rigin
root
➢ Initially, nodes point downward to object owner (predecessor node) due to Publish operation ➢ Nodes know their parent in the hierarchy
ξ
Move Object
Classic directory approach
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Send request to parent leader node
root
Move Object
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Continue up phase until downward pointer found
root
Sets downward path while going up
Move Object
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Continue up phase
root
Sets downward path while going up
Move Object
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Continue up phase
root
Sets downward path while going up
Move Object
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Downward pointer found, start down phase
root
Discards path while going down
Move Object
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Continue down phase
root
Discards path while going down
Move Object
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Continue down phase
root
Discards path while going down
Move Object
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Predecessor reached, object move completed
root
Query Lookup is similar without changing the directory structure
Move Object
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Move Object - Limitation
For multiple objects the origin path and destination path may not be matched with the same object. move delete wrong path
x
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Concurrent -
Insert operations from destination
& Synchronous
Delete operations from source
Common ancestor
Move Object – new scenario
insert delete Inserts and delete eventually meet at lowest common ancestor
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Move Object – 2 approaches
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Conclusions
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O(1) query lookup, and polylog update cost In low doubling dimension graphs Polylog approximation for general graphs Open problem:
- Prove matching lower bounds for move costs
- r improve upper bounds
- Analyze concurrent multiple moves