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Tributaries and Deltas: Efficient and Robust Aggregation in Sensor Networks

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Tributaries and Deltas: Efficient and Robust Aggregation in Sensor Networks

ManJhi, S. Nath P. Gibbons

CMU

ICS280sensors Winter 2005

- 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 leaves
- Exact computation
- Suffer from high communication failures
- “Not uncommon to loose 80% of readings”.

- Tree –based approach

ICS280sensors, Winter 2005

- Multi-path approach
- Use wireless broadcast medium
- Broadcast partial results to multiple neighbors
- Use 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

ICS280sensors, Winter 2005

ICS280sensors, Winter 2005

- 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

- Low loss: trees are used

ICS280sensors, Winter 2005

- 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

ICS280sensors, Winter 2005

- To construct a rings topology
- BS transmits and any node hearing the transmission is in ring 1
- Nodes 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

ICS280sensors, Winter 2005

- 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

ICS280sensors, Winter 2005

- Example: count distinct items
- Let n by upper bound of the count
- h() 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 function
- SE function takes a bit vector and output 2^(j-1)/0.77351, where j is the index of the lowest-order UNSET bit.

ICS280sensors, Winter 2005

- View aggregation as a directed graph
- Nodes and BS are vertices
- Directed edge fro successful transmission
- Vertex labeled either M or T, for multi-path or tree
- Edge labeled based on source vertex
- The labels may change

ICS280sensors, Winter 2005

- Correctness criteria of topology construction
- No 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

ICS280sensors, Winter 2005

- 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”

ICS280sensors, Winter 2005

- 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

ICS280sensors, Winter 2005

- 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?)

ICS280sensors, Winter 2005

- 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))

ICS280sensors, Winter 2005

- 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.

ICS280sensors, Winter 2005

ICS280sensors, Winter 2005

- 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.

ICS280sensors, Winter 2005

- 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

ICS280sensors, Winter 2005

- 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.

ICS280sensors, Winter 2005

- Replace the + operator with duplicate-insensitive addition operators
- Synopsis generation, fusion, and evaluation all depend on what duplicate-insensitive addition algorithm is used.

ICS280sensors, Winter 2005

ICS280sensors, Winter 2005

ICS280sensors, Winter 2005