Neighborhood-Based Neighborhood-Based Topology Recognition in Topology Recognition in

Sensor NetworksSensor Networks

S.P. Fekete, A. Kröller, D. S.P. Fekete, A. Kröller, D. Pfisterer, S. Fischer, andPfisterer, S. Fischer, and

C. BuschmannC. Buschmann

Corby Ziesman

Finding BoundariesFinding Boundaries

Useful for:Useful for:Keeping tracks of events entering or leaving Keeping tracks of events entering or leaving

the regionthe regionCommunication purposes to the outsideCommunication purposes to the outside

Routing along shortest paths puts increased Routing along shortest paths puts increased loads on boundary nodesloads on boundary nodesExhausting energy supplyExhausting energy supplyModerately sized holes caused by failed nodes or Moderately sized holes caused by failed nodes or

obstacles may tend to grow larger and largerobstacles may tend to grow larger and larger

Exact Coordinates?Exact Coordinates? Computing exact coordinates requires use of Computing exact coordinates requires use of

special hardware like GPS or scanning devicesspecial hardware like GPS or scanning devices Limits size and structure of networkLimits size and structure of network

Continuous range modulationContinuous range modulation

Inaccuracies from local measurements Inaccuracies from local measurements accumulate to become significant errorsaccumulate to become significant errors

Can be cumbersome if desiring high accuracyCan be cumbersome if desiring high accuracy

But we can calculate the boundaries without But we can calculate the boundaries without needing the exact coordinates….needing the exact coordinates….

How to detect boundaries?How to detect boundaries?

Assumptions for this approach:Assumptions for this approach:Positioning of nodes is a result of a random Positioning of nodes is a result of a random

distributiondistributionReasonable node densityReasonable node densityEach node can communicate with at least 100 Each node can communicate with at least 100

other nodesother nodesNetwork is overall connectedNetwork is overall connected

How to detect boundaries?How to detect boundaries? Communication range of boundary nodes intersects a Communication range of boundary nodes intersects a

smaller than average portion of the regionsmaller than average portion of the region May be natural fluctuations in density, so probabilistic May be natural fluctuations in density, so probabilistic

tools are employedtools are employed Using a simple local rule to let nodes decide whether Using a simple local rule to let nodes decide whether

they are close to a boundarythey are close to a boundary Node density μNode density μ Threshold αThreshold α Check if number of neighbors falls below αμCheck if number of neighbors falls below αμ

It’s important to get good estimates for the average It’s important to get good estimates for the average density μ of fully connected nodes, and determining a density μ of fully connected nodes, and determining a good threshold αgood threshold α

Determining μDetermining μ

Compute node degree histogramCompute node degree histogram

Δ = max neighborhood sizeΔ = max neighborhood size

μ = average densityμ = average density

Determining αDetermining α If α is too small, no node will be part of the boundaryIf α is too small, no node will be part of the boundary As α increases, connected boundary pieces grow until As α increases, connected boundary pieces grow until

different pieces of the same boundary merge together different pieces of the same boundary merge together correctlycorrectly

If α is too large, false boundaries in low density areas If α is too large, false boundaries in low density areas emerge, until eventual the entire network is a single emerge, until eventual the entire network is a single boundaryboundary

Plateau indicates correct boundaries

α can be calculated α can be calculated then through sampling then through sampling values of α and values of α and keeping track of # keeping track of # boundariesboundaries

CommunicatingCommunicating Using a min spanning tree in a graph with n nodes in a distributed fashion, Using a min spanning tree in a graph with n nodes in a distributed fashion,

using only local communication*using only local communication* Root first queries tree for Δ, and then for historgram… then determines Root first queries tree for Δ, and then for historgram… then determines

μ_estμ_est Network flood passing on value of αμ_estNetwork flood passing on value of αμ_est Nodes then decide if they belong to a boundary before passing on the floodNodes then decide if they belong to a boundary before passing on the flood Forms connected boundaries by constructing tree (two nodes connected if Forms connected boundaries by constructing tree (two nodes connected if

hop distance is at most 2)hop distance is at most 2) Root assigns resulting tree a unique ID (it’s node ID), which is broadcastRoot assigns resulting tree a unique ID (it’s node ID), which is broadcast Nodes then determine hop count to closest boundary (nodes equidistant to Nodes then determine hop count to closest boundary (nodes equidistant to

different boundaries are called Voronoi nodes)different boundaries are called Voronoi nodes) Root sends out message token, recipients decide who to send token to Root sends out message token, recipients decide who to send token to

based on smallest common neigborhoodbased on smallest common neigborhood Neighbors not selected to pass token along to exclude themselves from Neighbors not selected to pass token along to exclude themselves from

further token passingfurther token passing After awhile, the root ID is prioritized when searching for the token’s next After awhile, the root ID is prioritized when searching for the token’s next

hop, closing the loop, forming the boundaryhop, closing the loop, forming the boundary

* R. G. Gallager, P. A. Humblet, and P. M. Spira. A distributed algorithm for minimum-weight spanning trees. ACM Transactions on Programming Languages and Systems, 1983.

Inner and Outer BoundariesInner and Outer Boundaries

If the geometry of the boundaries are not If the geometry of the boundaries are not too convoluted, it can be assumed that the too convoluted, it can be assumed that the outer boundary will be the longest, and outer boundary will be the longest, and consist of the largest number of nodesconsist of the largest number of nodes

Future work may involve taking into Future work may involve taking into account possibility of convoluted inner account possibility of convoluted inner boundaries by keeping track of curvature boundaries by keeping track of curvature along the boundaryalong the boundary

Example Example network with network with inner and inner and outer outer boundariesboundaries

Boundaries Boundaries are detected are detected by choosing by choosing a correct αa correct α

ConclusionsConclusions

Topology of a large, dense sensor network Topology of a large, dense sensor network is possible without location hardwareis possible without location hardware

Future work may involve taking into Future work may involve taking into account higher-order information of the account higher-order information of the neighborhood structure to overcome the neighborhood structure to overcome the limitation of requiring high node densitylimitation of requiring high node density (May lead to routing and energy management)(May lead to routing and energy management)