1. Xingbo Yu ()ICS280sensors Winter 2005 Tributaries and Deltas: Efficient and Robust Aggregation in Sensor Networks A.ManJhi, S. Nath P. Gibbons CMU

2. Xingbo YuICS280sensors, 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 leaves  Exact computation  Suffer from high communication failures –“Not uncommon to loose 80% of readings”.

3. Xingbo YuICS280sensors, Winter 2005 Introduction 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

4. Xingbo YuICS280sensors, Winter 2005 Approach Comparison

5. Xingbo YuICS280sensors, Winter 2005 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

6. Xingbo YuICS280sensors, Winter 2005 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

7. Xingbo YuICS280sensors, Winter 2005 More on multi-path 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

8. Xingbo YuICS280sensors, Winter 2005 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

9. Xingbo YuICS280sensors, Winter 2005 More on multi-path 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.

10. Xingbo YuICS280sensors, Winter 2005 Tributary-Delta 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

11. Xingbo YuICS280sensors, Winter 2005 Tributary-Delta 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

12. Xingbo YuICS280sensors, Winter 2005 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”

13. Xingbo YuICS280sensors, Winter 2005 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

14. Xingbo YuICS280sensors, Winter 2005 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?)

15. Xingbo YuICS280sensors, Winter 2005 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))

16. Xingbo YuICS280sensors, Winter 2005 Identify frequent items–tree algorithm Partial result sent by a node X to its parent is a summary: S = 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.

17. Xingbo YuICS280sensors, Winter 2005 Identify frequent items–tree algorithm

18. Xingbo YuICS280sensors, Winter 2005 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.

19. Xingbo YuICS280sensors, Winter 2005 Min Total-Load algorithm 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

20. Xingbo YuICS280sensors, Winter 2005 Min Total-Load algorithm 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.

21. Xingbo YuICS280sensors, Winter 2005 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.

22. Xingbo YuICS280sensors, Winter 2005 Results

23. Xingbo YuICS280sensors, Winter 2005 Results