Xingbo Yu ()ICS280sensors Winter 2005 Tributaries and Deltas: Efficient and Robust Aggregation in...

Xingbo Yu ( ) ICS280sensors Winter 2005

Tributaries and Deltas: Efficient and Robust Aggregation in Sensor

Networks

A. ManJhi, S. Nath P. Gibbons

CMU

Xingbo Yu ICS280sensors, Winter 2005

Introduction

• Existing approaches to in-network aggregation:Tree –based approach

Answer is generated by performing in-net aggregation along the tree

Proceed level by level from leavesExact computationSuffer from high communication failures

– “Not uncommon to loose 80% of readings”.


Introduction

Multi-path approachUse wireless broadcast mediumBroadcast partial results to multiple neighborsUse topology called rings.

– Nodes divided into levels according to hop count from BS

– Aggregation performed level by level up to the BS.

Each reading is accounted for multiple times– Robust

Suffer from: approximate answers and long message size


Approach Comparison


Tributary-Delta overview

• Combine the two approaches

• Adapting the aggregation to the current loss rate Low loss: trees are used

for low/zero approximate error and small size

High loss: multi-path For robustness


Challenges

• How do nodes decide whether to use tree or multi-path

• How do the nodes using different approaches communicate

• How do the nodes convert partial results when transitioning between approaches

• New algorithm for finding frequent items


More on multi-path

• To construct a rings topologyBS transmits and any node hearing the

transmission is in ring 1Nodes in ring I transmit and any node hearing the

transmission, but not already in a ring, is in ring I+1.

All level I nodes that hear a level i+1 partial result incorporate the result into its own result

Low communication error


More on multi-path

• Special technique to avoid double-counting: synopsis (sketches) diffusion Synopsis generation: takes a stream of local

sensor readings at a node and produces a partial result-synopsis

Synopsis fusion: takes two synopses and generate a new one

Synopsis evaluation: translates a synopsis into a query answer


More on multi-path

• Example: count distinct itemsLet n by upper bound of the counth() be a hash function from sensor ids to [1, …

lg(n)]SG function produces a bit vector of all 0’s and the

sets the h(i)’th bit to 1 when see an id of i.SF function is OR functionSE function takes a bit vector and output

2^(j-1)/0.77351, where j is the index of the lowest-order UNSET bit.


Tributary-Delta

• View aggregation as a directed graphNodes and BS are verticesDirected edge fro successful transmissionVertex labeled either M or T, for multi-path or treeEdge labeled based on source vertexThe labels may change


Tributary-Delta

• Correctness criteria of topology constructionNo two M vertices with partial results representing

an overlapping set of sensors are connected to T vertices.

• Restrict to: a node receiving from an M node uses M scheme

• Edge correctness: An M edge can never be incident on a T vertex

• Path correctness: in any directed path in G, a T edge can never appear after an M edge


Tributary-Delta• Dynamic adaptation:

An M vertex is switchable if all incoming edges are E edges, or no incoming edges (M1, M2)

A T vertex is switchable if its parent is an M vertex or it has no parent. (T3, T4, T5)

Let G’ be the connected component of G that includes the BS

“if the set of T vertices in G’ is not empty, at least one of them is switchable. If the set of M vertices in G’ is not empty, at least one of them is switchable”


Adaptation design• User specify a threshold on the minimum percentage of

nodes that should contribute to the aggregate answer• Depending on the % of nodes contributing to the current

result, the BS decides whether to shrink or expand the delta region for future result Increasing delta region increases the % contributing

• Key concern in switching nodes between tree and multi-path aggregation: transmitting and receiving synchronization

• Design choice: (to ensure switched nodes can retain current epoch)From M to T: must choose its parents from one of its

neighbors in level i-1.From T to M: transmits to all neighbors in level i-1


Adaptation strategies• TD-coarse: if the % is below the user-specified

threshold, all the current switchable T nodes is switched.

• TD: each switchable M node includes in its outgoing messages an

additional field : number of nodes in sub-tree not contributing.

Max and min of such number are maintained If % is below threshold: BS expands the delta region by

switching from T to M all children of swichable M nodes beloning to a sub-tree that has max nodes not contributing

When shrinking: switch each swichable M node whose subtree has only min nodes not contributing. ?

Trade-off: higher convergence time. (will it converge?)


Identify frequent items

• The problem:Each of m sensor nodes generates a collection of

items.Given a user-supplied error tolerancee, the toal is to

obtain from each item u, an e-deficient count c’(u) at the BS:Max {0, c(u)-e*N} <= c’(u) <= c(u)

Where N = sum(c(u))


Identify frequent items–tree algorithm

• Partial result sent by a node X to its parent is a summary:S = <N, e, {(u, c’(u))}>Each c’(u) satisfies max {0, c(u)-e*N} <= c’(u) <= c(u)

• Approach is to distribute the e among intermediate nodes in the tree. Make e(i) a function of height of a node (height of a leaf

node is 1)For correctness: e(1)<= e(2) <=… <= e(h)

As long as e(h) <= e, user guarantee is met.Called precision gradient

• At each node: summary of items with count at most e*N is dropped.


Identify frequent items–tree algorithm


Min Total-Load algorithm

• D-dominating tree: fro any d>=1, we say that a tree is d-dominating if for any i>=1,

H(i)>=(d-1)/d*(1+1/d+…+1/d^(i-1))Where H(i)=1/m*SUM(h(j)), with h(j) being the

number of nodes at height j, and m the total number of nodes.

• If a tree is d-dominating but not d+delta-dominating, refer to d as the domination factor.



• Lemma: for any d-dominating tree of m nodes, where d>1, a precision gradient setting of e(i)=e*(1-t)(1+t+…+t^(i-1)) with t=1/sqrt(d) limits total communication to (1+ 2/(sqrt(d)-1))*m/e.Follows from: step 3 of alg. 1, at most 1/(e(i)-e(i-1))

items are sent by a node at height i to its parent



• Lemma: a tree in which each internal node of height I has at least d children of height i-1 is d-dominating

• Construction of topology with large dominating factors:Each node of height i+1, if has two or more children of heigh I,

pins down any two of its children so that they can not switch parents, and flag itself.

Non-pinned nodes in each level j switch parents randomly to any other reachable non-flagged node in level j-1.

As soon as a non-flagged node has at least two flagged children of the same height, it pins both of them and the flags itself.

This makes the tree 2-dominating.


Identify frequent items–multi-path algorithm

• Replace the + operator with duplicate-insensitive addition operators

• Synopsis generation, fusion, and evaluation all depend on what duplicate-insensitive addition algorithm is used.


Results

Xingbo Yu ()ICS280sensors Winter 2005 Tributaries and Deltas: Efficient and Robust Aggregation in...

Documents

Transcript of Xingbo Yu ()ICS280sensors Winter 2005 Tributaries and Deltas: Efficient and Robust Aggregation in...