Tributaries and Deltas: Efficient and Robust Aggregation in Sensor NetworksManJhi, S. Nath P. GibbonsCMU

IntroductionExisting approaches to in-network aggregation:Tree based approachAnswer is generated by performing in-net aggregation along the treeProceed level by level from leavesExact computationSuffer from high communication failures Not uncommon to loose 80% of readings.

IntroductionMulti-path approachUse wireless broadcast mediumBroadcast partial results to multiple neighborsUse topology called rings.Nodes divided into levels according to hop count from BSAggregation performed level by level up to the BS.Each reading is accounted for multiple timesRobust

Suffer from: approximate answers and long message size

Approach Comparison

Tributary-Delta overviewCombine the two approachesAdapting the aggregation to the current loss rateLow loss: trees are used for low/zero approximate error and small sizeHigh loss: multi-pathFor robustness

Challenges How do nodes decide whether to use tree or multi-pathHow do the nodes using different approaches communicateHow do the nodes convert partial results when transitioning between approaches

New algorithm for finding frequent items

More on multi-pathTo 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 resultLow communication error

More on multi-pathSpecial technique to avoid double-counting: synopsis (sketches) diffusionSynopsis generation: takes a stream of local sensor readings at a node and produces a partial result-synopsisSynopsis fusion: takes two synopses and generate a new oneSynopsis evaluation: translates a synopsis into a query answer

More on multi-pathExample: 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 0s 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-DeltaView 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-DeltaCorrectness 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 schemeEdge correctness: An M edge can never be incident on a T vertexPath correctness: in any directed path in G, a T edge can never appear after an M edge

Tributary-DeltaDynamic 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 BSif 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 designUser specify a threshold on the minimum percentage of nodes that should contribute to the aggregate answerDepending on the % of nodes contributing to the current result, the BS decides whether to shrink or expand the delta region for future resultIncreasing delta region increases the % contributingKey concern in switching nodes between tree and multi-path aggregation: transmitting and receiving synchronizationDesign 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 strategiesTD-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 maintainedIf % 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 contributingWhen shrinking: switch each swichable M node whose subtree has only min nodes not contributing. ?Trade-off: higher convergence time. (will it converge?)

Identify frequent itemsThe 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}

Identify frequent itemstree algorithmPartial result sent by a node X to its parent is a summary:S = Each c(u) satisfies max {0, c(u)-e*N}

Identify frequent itemstree algorithm

Min Total-Load algorithmD-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.

Min Total-Load algorithmLemma: 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

Min Total-Load algorithmLemma: a tree in which each internal node of height I has at least d children of height i-1 is d-dominatingConstruction 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 itemsmulti-path algorithmReplace the + operator with duplicate-insensitive addition operatorsSynopsis generation, fusion, and evaluation all depend on what duplicate-insensitive addition algorithm is used.

Results

Results