Computing Diameter in the Streaming and Sliding-Window Models

1
Computing Diameter in the Streaming and
Sliding-Window Models

• J. Feigenbaum, S. Kannan, J. Zhang

2
Introduction

• Two computational models
• Streaming model
• Sliding-window model
• The problem diameter of a point set P in R2. The
  diameter is the maximum pairwise distance between
  points in P.

3
More about Models

• The streaming model
• A data stream is a sequence of data elements a1
  a2 , ..., am .
• A streaming algorithm is an algorithm that
  computes some function over a data stream and has
  the following properties
• The input data are accessed in a sequential
  order.
• The order of the data elements in the stream is
  not controlled by the algorithm
• The length of the stream, m, is huge. Only
  space-efficient algorithms (sublinear or even
  polylog(m)) are considered.

4
Dynamic Algorithm in Computational Geometry

• Dynamic means that the set of objects under
  consideration may change. There could be
  additions and deletions to the point set P.
• Maintain the current set of geometry objects in
  certain data structures. Efficient updating and
  query answering are emphasized.
• May use linear space - different from the
  requirement of the streaming and the
  sliding-window models.

5
More about Models (Continued)

• The sliding-window model
• The input is still a stream of data elements.
• A data element arrives at each time instant it
  later expires after a number of time stamps equal
  to the window size n
• The current window at any time instant is the set
  of data elements that have not yet expired.

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Computing Diameter in the Streaming Model

• A well-known diameter-approximation is streaming
  in nature.
• Project the points onto lines.
• Requires ? such that
• p(p)p(q) pq cos? (1- ?2/2)pq
  (1-e)pq
• The algorithm goes through the input once. It
  needs storage for O(1/ ) points. To process
  each point, it performs O(1/ ) projections.

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Diameter Approximation in the Streaming Model

• Theorem 1 There is a streaming e-approximation
  algorithm for diameter that needs storage for
  O(1/e) points and processes each point in
  O(log(1/e)) time.
• Take the first point of the stream as the
  center and divide the space into sectors of
  angle ? e/2(1-e).
• For each sector, keep the point furthest from the
  center in that sector.

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Diameter Approximation in the Streaming Model

• Let H be the maximum distance between the
  center and any other point and Ti,j be the
  minimal distance between the boundary arcs of
  sector i (bb') and sector j (aa'). Approximate
  the diameter with maxH, maxi,j Tij

9
Maintaining Diameter in the Sliding-Window Model

• Our space efficient mehtod maintains the diameter
  for sliding windows when the set of points P can
  be bounded in a box that is not too large.
• Let R be the maximum, over all windows, the ratio
  of the diameter over the minimal non-zero
  distance between any two points in that window.
• That the bounding space is not too large means
  R lt 2n.

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Maintaining Diameter in the Sliding-Window Model

• Theorem 2 There is an e-approximation
  algorithm that maintains the diameter for a
  planar point set in the sliding-window model
  using
• Poly(1/e, log n, log R) bits of space.

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Remove Irrelevant Points

• Consider maintaining the diameter in 1-d.
• A point will never realize any diameter if it is
  spatially located between two newer points.
• Remove these points. The locations of the
  remaining points would look like
• (where a1 is newer than a2 which is newer than
  a3...)
• The newer points would be located inside and
  the older points would be located outside

12
The Rounding Method

• Take the newest point as the center, and
  round down other points.
• Divide the line into the following intervals such
  that cti ( 1e? )id for some distance d (to
  be specified later).
• Round all points in the interval ti, ti1) down
  to ti.
• In what follows we call the set of pints after
  rounding a cluster. If 2i original points are
  grouped into a cluster, we say the cluster is at
  level i.

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Number of Points in a Cluster

• If multiple points are rounded to the same
  location, we can discard the older ones and only
  keep the newest one.
• In each interval, we have only one point. Let D
  be the diameter, the number of points k in a
  cluster is bounded by
• k log1e? D/d (log D/d)/log (1e?) (2/e?
  )log D/d

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When Window Starts Sliding

• Need to consider addition and deletion.
• Deletion is easy, because the oldest point must
  be one of the cluster's extreme points.
• Addition is complicated, because we may need to
  update the cluster center for each point that
  arrives.
• Our solution keep multiple clusters.

15
Multiple Clusters in a Window

• We allow at most two clusters to be at each
  level.
• When the number of clusters of level i exceeds
  2, merge the oldest twe clusters to form a
  cluster at level i1.
• The window can thus be divided into clusters.

16
Clusters in a Window

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Merge Clusters

• Cluster c1cluster c2 cluster c3
• Make Ctr2 the center of cluster c3

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Merge Clusters (Continued)

• Discard the points in c1 that are located between
  the centers of c1 and c2.
• If point p in c1 satisfies pCtr1
  (1e?)Ctr1Ctr2, discard it, too.

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Merge Clusters (Continued)

• Round the points in c2 and those remaining in c1
  after the previous two steps using the center
  Ctr2.
• The value for d is lower bounded by e?
  Ctr1Ctr2. The number of points in a cluster is
  then bounded by
• (2/e? )(log R log 1/e? )

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The Algorithm in 1-d

• Update when a new point arrives,
• Check the age of the boundary points of the
  oldest cluster. If one of them has expired,
  remove it.
• Make the newly arrived point a cluster of size 1.
  Go through the clusters and merge clusters
  whenever necessary according to the rules stated
  above.
• While going throught the clusters, update the
  boundary points of any cluster changed.
• Update the window boundary points if necessary.
• Query Answer Report the distance between the
  window boundary points as the window diameter.

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Space Requirement

• Let diamp be a diameter realized by point p. Each
  time we do rounding, we introduce a
  displacement for p at most e? diamp. Also p can
  be rounded at most log n times.
• Choose e? to be at most e/(2log n) to bound the
  error.
• There are at most 2log n clusters and in each
  cluster at most O(1/e log n (log R log log n
  log 1/e )) points. Keeping the age may require
  log n space for each point. The total space
  required is
• O(1/e log3n (log R log log n log 1/e ))

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Time Complexity

• Query answer time is O(1).
• Worst case update time is O(1/e log2n (log R
  log log n log 1/e )) because we may have
  cascading merges.
• The amortized update time is O(log n)

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Extend the Algorithm to 2-d

• We will have a set of lines l0, l1, ... and
  project the points in the plane onto the lines.
• Guarantee that any paire of points will be
  projected to a line with angle f such that 1- cos
  f e/2
• Use the diameter-maintenance algorithm in 1-d for
  each line.
• Everything will have a multiplicative overhead of
  
• O(1/ ).

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Lower Bound for Maintaining Exact Diameter

• Theorem 3 To maintain the exact diameter in a
  sliding window model requires O(n) bits of space.
• Consider 2n points a1, a2, ..., a2n with the
  following properties
• an1, an2, ..., a2n are located at coordinate
  zero.
• a1an a2an1 a3an2 ... an-1a2n-2
  1
• The coordinates of the points aj for j 1,2,...,
  n-2 have the form nk for some k 1,2,..., n.

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A Family of Point Sequences
We show below two sequences in the family
an an1 an2 ......
an-1
an-2
a2 a1
......
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Lower Bound for Maintaining Exact Diameter
(Countinued)

• There are at least
  different sequences of 2n points satisfying the
  above properties.
• Need O(n) space to distinguish them.
• (Note here R n2 ltlt 2n)