Energy Efficient Exact kNN Search in Wireless Broadcast Environments

1
Energy Efficient Exact kNN Search in Wireless
Broadcast Environments

• Bugra Gedik, Aameek Singh, Ling Liu
• College of Computing
• Georgia Institute of Technology

2
Motivation

• There is strong interest in location based
  services (LBSs)
• Popularity of mobile wireless communications
• Emergence of positioning technologies like GPS
• LBSs provide mobile users with information that
  is specialized to their location, ex
• the nearest gas station or the five closest
  restaurants
• LBSs are accessed through a common wireless
  channel, which connects the users to the service
  provider
• Limited network bandwidth -gt Wireless
  broadcast
• Power constrained mobile devices -gt Energy
  efficient search
• An interesting problem in this domain is,
• Investigation of indexing and searching
  mechanisms for energy efficient querying of
  location dependent data in wireless broadcast
  environments

3
Problem Definition

• kNN Search on the Air problem
• Broadcasting location dependent data together
  with a spatial index on the wireless medium and
  searching this broadcast on mobile client devices
  in order to answer k nearest neighbor (kNN)
  queries in an energy efficient way
• Traditional mechanisms do not work well
• Access to the medium is sequential
• Methods based on approximate results 30
• How about exact search?

4
Some Example Applications

• Commercial Setting
• The server can broadcast locations of gas
  stations
• A car driver can pose a query like Give me the
  positions of the 3 nearest gas stations
• Military Setting
• The server can build a spatial index on positions
  of military units and broadcast it to the
  battlefield (possibly encrypted)
• Then the mobile devices can tune in and process
  the broadcast to answer spatial queries
• One such query can be posed by a tank as Give
  me the positions and names of the 10 nearest
  friendly units
• kNN search on low dimensional spaces for
  sequential access mediums

5
Background Air Indexing

• Air indexes
• In wireless broadcast, it is crucial that energy
  is conserved on the mobile unit side when
  answering queries on the broadcasted data.
• To alleviate the vast energy consumption problem
  of searching un-indexed data, air indexes were
  introduced.
• Access Time latency measure
• the time difference between the point at which a
  query is posed and the point at which result of
  the query is fully computed
• Tune-in Time energy consumption measure
• the total time during which the mobile unit was
  listening to data from the wireless medium ( of
  packets read)
• Air indexing strives to decrease tune-in time
  while keeping the increase in access latency due
  to the broadcast of extra index information
  minimal.

6
Background R-trees

• R-trees are spatial index structures widely used
  to index n-dimensional points or rectangles
• Practical for secondary storage, nodes correspond
  to disk blocks (packets in wireless broadcast)
• R-trees can be thought of as the multidimensional
  version of B-trees

7
Background R-trees (kNN search)

• A branch and bound algorithm
• Uses a heuristic to select branches to follow
• The need for backtracking makes the algorithm ill
  suited for wireless medium lt- sequential access
  nature
• Metrics used in the algorithm
• Given a point P and the mbr of a node N,
• MINDIST(P,N) is the minimum distance from P to
  Ns mbr
• MAXDIST(P,N) is the maximum distance from P to
  Ns mbr
• MINMAXDIST(P,N) is the maximum possible minimum
  distance between P and the mbr of closest object
  residing in Ns mbr

8
Background R-trees (kNN search)

• ItemQueue is a priority queue that stores nodes
  to explore. It is sorted on the MINDIST measure
  and does not have a predefined size.
• ResultQueue is a list of size maximum k, that
  stores the best k nodes seen so far. It is sorted
  on the MINMAXDIST measure.
• kthdist MINMAXDIST value of the kth item in
  ResultQueue, infinity if less than k items are
  present. It is an upper bound on the distance of
  the kth nearest neighbor.

kthdist
ItemQueue
ResultQueue
?
(R0,0)
(R1,0) (R2,a)
(R2,b) (R1,c)
c
k2
9
Background R-trees (kNN search)

• ItemQueue is a priority queue that stores nodes
  to explore. It is sorted on the MINDIST measure
  and does not have a predefined size.
• ResultQueue is a list of size maximum k, that
  stores the best k nodes seen so far. It is sorted
  on the MINMAXDIST measure.
• kthdist MINMAXDIST value of the kth item in
  ResultQueue, infinity if less than k items are
  present. It is an upper bound on the distance of
  the kth nearest neighbor.

kthdist
ItemQueue
ResultQueue
?
(R0,0)
(R1,0) (R2,a)
(R2,b) (R1,c)
c
(R4,d) (R3,f) (R5,h) (R2,a)
(R3,g) (R2,b)
b
k2
10
Background R-trees (kNN search)

• ItemQueue is a priority queue that stores nodes
  to explore. It is sorted on the MINDIST measure
  and does not have a predefined size.
• ResultQueue is a list of size maximum k, that
  stores the best k nodes seen so far. It is sorted
  on the MINMAXDIST measure.
• kthdist MINMAXDIST value of the kth item in
  ResultQueue, infinity if less than k items are
  present. It is an upper bound on the distance of
  the kth nearest neighbor.

kthdist
ItemQueue
ResultQueue
?
(R0,0)
(R1,0) (R2,a)
(R2,b) (R1,c)
c
(R4,d) (R3,f) (R5,h) (R2,a)
(R3,g) (R2,b)
b
(R3,f) (R5,h) (R2,a)
(p13,d13) (R3,g)
g
p11, p12 are pruned since d11, d12 are both gt b
k2
11
Background R-trees (kNN search)

• ItemQueue is a priority queue that stores nodes
  to explore. It is sorted on the MINDIST measure
  and does not have a predefined size.
• ResultQueue is a list of size maximum k, that
  stores the best k nodes seen so far. It is sorted
  on the MINMAXDIST measure.
• kthdist MINMAXDIST value of the kth item in
  ResultQueue, infinity if less than k items are
  present. It is an upper bound on the distance of
  the kth nearest neighbor.

kthdist
ItemQueue
ResultQueue
?
(R0,0)
(R1,0) (R2,a)
(R2,b) (R1,c)
c
(R4,d) (R3,f) (R5,h) (R2,a)
(R3,g) (R2,b)
b
(R3,f) (R5,h) (R2,a)
(p13,d13) (R3,g)
g
(R5,h) (R2,a)
(p13,d13) (p1,d1)
d1
Result p13, p1
k2
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Adapting the Algorithm for Wireless Medium
(w-conv alg)

• It is not possible to sort ItemQueue on the
  MINDIST measure
• MINDIST ordering of tree nodes is not consistent
  with their order of appearance in the broadcast
• Reading a node from the medium based on the
  topmost item in the MINDIST sorted ItemQueue may
  result in leaving behind other tree nodes that
  have entries in ItemQueue
• As a result, the items in ItemQueue have to be
  sorted based on their appearance order on the
  medium
• We cannot immediately stop the search when the
  ResultQueue consists of objects only
• It is not guaranteed that the rest of the items
  in ItemQueue cannot generate an object closer
  than the current k
• Because the queue is no more sorted on MINDIST
  measure
• As a result, the search halts when the ItemQueue
  becomes empty

13
Optimizing the Algorithm for Wireless Medium
(w-opt alg)

• After the root node is processed
• ItemQueue u, v, w
• ResultQueue (v,5), (w,14)
• kthdist 14
• Node v cannot be pruned
• In fact, in this example it is possible to prune
  it

• Knowledge
• If the minimum fanout of the tree is f, then we
  know that there are at least f l-1 objects under
  node ws mbr

14
Optimizing the Algorithm for Wireless Medium
(w-opt alg)

• Given this knowledge, at the time when we add w
  to ItemQueue
• There is one object at most 5 away from the query
  point
• There are at least f l-11 ? 1 objects at most 10
  away from the query point
• Then kthdist 10 after the root node is
  processed
• Now v can be pruned

• New Rule (w-opt algorithm)
• While adding a node, say node N at level i, with
  its MINMAXDIST measure to ResultQueue, we also
  insert f i -1 additional entries with the MAXDIST
  measure of node N. ResultQueue is sorted on the
  associated measures of the entries

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Does Serialization Order Matter?

• The way a spatial index is organized on the
  broadcast medium
• of index nodes read by the kNN search, tune in
  time
• Previous work on range and kNN search in
  broadcast environments 28, 10, 30, 15 used a
  depth first search (dfs) order serialization of
  the tree,
• conventional algorithm is based on dfs guided by
  a heuristic

Proof In the paper.
16
Does Serialization Order Matter?

• Result circle The circle formed around the query
  point using its distance from the kth nearest
  neighbor
• A wrong decision causes high cost in terms of
  tune-in time, as we are trapped in a branch
• BFS serialization does not share the same
  problem, but it may result in having a large
  ItemQueue size

result circle
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Histograms for Better Pruning? (w-hist alg)

• A simple grid-like histogram can be used to
  obtain an upper bound on size of the result
  circle of a kNN query
• Given a query point P and any set of k objects,
  the distance between P and the furthest object in
  the given set of objects from P, is larger than
  the radius of the result circle
• As a result, any circle centered at point P that
  covers some set histogram cells such that the sum
  of number of objects located under these
  histogram cells is larger than k, covers the
  result circle.

• Pick the closest non-empty cells to the query
  point, such that the total number of objects
  contained in these cells is at least k
• Set r as the maximum of MAXDISTs of all cells
  that are picked
• Circle centered at P with radius r is called
  the pruning circle (PC)

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Histograms for Proving Bounds? (w-alg)

• New Rule (w-hist algorithm)
• When a new node is to be added to ItemQueue, it
  can be discarded if its mbr does not intersect
  with the pruning circle
• The histogram should not be too large, otherwise
  the gain from pruning cannot compensate the cost
  of reading the histogram
• In fact we have a formula for an upper bound on
  the tune-in time (for uniformly distributed
  data)
• Histogram cell size cannot be taken smaller than
  the value that maximizes the above equation

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Histograms for Proving Bounds?

• Histograms help decrease the max. length of the
  ItemQueue through pruning
• They may not always decrease the tune-in time,
  especially when the distribution is skewed
• As a result, the tradeoffs require further study

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Non-spatial Predicates

• Non-spatial attributes that may need to be taken
  into account when answering queries, ex type of
  an object
• We consider two techniques to answer kNN queries
  that may specify an optional type constraint
• The t-index Method
• Use t separate indexes each indexing only its
  associated type
• Keep a lookup structure that has pointers for
  each type
• Really efficient for queries with type constrains
• Costly for other queries, needs to lookup all
  indexes
• Improve the performance of queries without type
  constraints, the order of the indexes can be
  selected such that an index corresponding to type
  i comes before the one corresponding to j on the
  broadcast medium if ni gt nj .

21
Non-spatial Predicates

• The t-hist/1-index Method
• Use a single spatial index which indexes all
  objects
• Include t histograms, each one built for a
  particular type
• Keep a lookup structure that has pointers for
  each type
• Let the leaf nodes of the tree also mark the type
  of each object
• Queries with type constraints can be processed by
  only using the pruning circle derived from the
  associated histogram of the given type
• Really efficient for queries without type
  constraints
• To improve the performance of queries with type
  constraints, we can prune a node whose mbr does
  not intersect with a non-empty cell of the
  histogram of the associated type
• We name this latter optimization as
  t-hist/1-index/hp method
• Note that it is only fair to compare them when
  the total index size (together with histograms)
  occupied by the two methods is same

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Tune-in Time and Queue Size

• tune-in time
• with dfs organization
• w-opt 33 better than w-conv
• with bfs organization
• w-opt 30 better than w-conv
• bfs organization provides up to 55 improvement
  over dfs organization
• queue size
• bfs organization has larger memory requirement
  than dfs
• w-opt decreases the memory requirement for both
  organizations
• tune-in time
• after k 4 w-opt starts providing significant
  improvement over w-conv
• after k 8 w-opt performs better than w-conv
  independent of the index serialization order
• queue size
• memory requirement of bfs organization grows fast
  with increasing k when w-conv is used and
  significantly drops when w-opt is used.

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Scalability w.r.t. Number of Objects

• Similar trends are observed
• w-opt search and bfs organization prevailing over
  other configurations.
• However, comparing Figure 9 and Figure 7 reveals
  that the improvement in tune-in time increases
  with increasing k
• In fact one can prove that for NN search (k 1)
  w-opt reduces to w-conv.

24
Merits and Demerits of Histograms

• Figure 10
• a single packet histogram improves the tune-in
  time
• for this setup, histograms with larger sizes do
  not provide better tune-in times
• the effect of using a histogram is more prominent
  with the dfs organization
• with histograms the bfs and dfs organizations are
  effectively the same
• Figure 11
• with skewed data the latter observation no more
  holds bfs organization with w-opt search
  outperforms all alternatives
• this does not indicate that histograms are
  useless for skewed data sets
• a small histogram significantly (more than 50)
  decreases the queue size increases the tune-in
  time marginally (around 5)

25
More on Histograms andSearch with Non-spatial
Predicates

• Figure 12
• for larger number of objects it is better to use
  larger histograms
• dfs organization is more sensitive to the
  increase in the number of objects
• Figure 15
• t-index performs much better for queries with
  type constraints
• t-hist/1-index/hp method (powered by the
  additional histogram pruning), achieves
  significant improvement in tune-in time,
  especially for queries having constraints on
  infrequent types
• Figure 16
• t-index performs poorly for queries without type
  constraints
• Although not outperforming t-index for type
  constrained queries, t-hist/1-index/hp strikes a
  good balance between two types of queries

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Conclusions

• The introduced w-opt search technique
  significantly decreases tune-in time,
  irrespective of how the index is organized (bfs
  or dfs)

• Organizing the index in bfs manner provides
  considerably better tune-in time but a higher
  memory requirement due to queue size

• Using histograms can improve tune-in time,
  although the improvement is marginal for bfs
  organization as opposed dfs organization

• The use of histograms can be extended to support
  answering kNN queries with type constraints.

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The End !

• Thanks
• ?? Questions ??