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

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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.

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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

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Approach Comparison
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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

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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

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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 I1.
• All level I nodes that hear a level i1 partial
  result incorporate the result into its own result
• Low communication error

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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

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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 0s 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.

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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

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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

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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

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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

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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?)

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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)-eN lt c(u) lt c(u)
• Where N sum(c(u))

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Identify frequent itemstree algorithm

• Partial result sent by a node X to its parent is
  a summary
• S ltN, e, (u, c(u))gt
• Each c(u) satisfies max 0, c(u)-eN lt c(u)
  lt 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)lt e(2) lt lt e(h)
• As long as e(h) lt e, user guarantee is met.
• Called precision gradient
• At each node summary of items with count at most
  eN is dropped.

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Identify frequent itemstree algorithm
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Min Total-Load algorithm

• D-dominating tree fro any dgt1, we say that a
  tree is d-dominating if for any igt1,
• H(i)gt(d-1)/d(11/d1/d(i-1))
• Where H(i)1/mSUM(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
  ddelta-dominating, refer to d as the domination
  factor.

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Min Total-Load algorithm

• Lemma for any d-dominating tree of m nodes,
  where dgt1, a precision gradient setting of
  e(i)e(1-t)(1tt(i-1)) with t1/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

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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 i1, 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.

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Identify frequent itemsmulti-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.

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Results
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Results