Computing Diameter in the Streaming and
Sliding-Window Models
J. Feigenbaum, S. Kannan, J. Zhang
Introduction
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Two computational models:
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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.
More about Models
The streaming model
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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.
2.
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.
1.
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More about Models (Continued)
The sliding-window model
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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.
Dynamic Algorithm in Computational
Geometry
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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.
Computing Diameter in the Streaming Model
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A well-known diameter-approximation is streaming in
nature.
Project the points onto lines.
Requires θ ≤ 2ε such that
|π(p)π(q)| ≥ |pq| cosθ ≥ (1− θ2/2)|pq| ≥ (1−ε)|pq|
The algorithm goes through the input once. It needs
storage for O(1/ ε) points. To process each point, it
performs O(1/ ε ) projections.
Diameter Approximation in the Streaming
Model
Theorem There is a streaming ε-approximation
algorithm for diameter that needs storage for O(1/ε)
points and processes each point in O(log(1/ε)) time.
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Take the first point of the stream as the “center” and
divide the space into sectors of angle θ = ε/2(1-ε).
For each sector, keep the point furthest from the center in
that sector.
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 max{H, maxi,j Tij}
Maintaining Diameter in the Sliding-Window
Model
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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.
If we assume the minimal non-zero distance is 1, R is the
diameter in the window.
When the set of points P can be bounded in a box of size
R, we maintain the diameter for sliding windows using
polylog(R) bits of space.
Maintaining Diameter in the Sliding-Window
Model
Theorem There is an ε-approximation algorithm that
maintains the diameter for a planar point set in the
sliding-window model using
Poly(1/ε, log n, log R) bits of space.
Remove Irrelevant Points
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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”
The “Rounding” Method
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Take the newest point as the “center,” and “round” down
other points.
Divide the line into the following intervals such that |cti|
= ( 1+ε )id for some distance d (to be specified later).
Round all points in the interval [ti, ti+1) 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.
Number of Points in a Cluster
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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 ≤ log1+ε D/d = (log D/d)/log (1+ε ) ≤ (2/ε )log
D/d
When Window Starts Sliding
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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.
Multiple Clusters in a Window
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The window can be divided into clusters of level 1, 2,
…, log n.
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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” i+1.
Clusters in a Window
Merge Clusters
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Cluster c1+cluster c2 = cluster c3
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Make Ctr2 the center of cluster c3
Merge Clusters (Continued)
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Discard the points in c1 that are located between the
centers of c1 and c2.
If point p in c1 satisfies |pCtr1| ≤ (1+ε )|Ctr1Ctr2|, discard
it, too.
Merge Clusters (Continued)
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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 ε ∙ |Ctr1Ctr2|. The
number of points in a cluster is then bounded by:
(2/ε )(log R + log 1/ε )
The Algorithm in 1-d
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Update: when a new point arrives,
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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.
Space Requirement
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Let diamp be a diameter realized by point p. Each time
we do “rounding,” we introduce a displacement for p at
most ε ∙ diamp. Also p can be “rounded” at most log n
times.
Choose ε to be at most ε/(2log n) to bound the error.
There are at most 2log n clusters and in each cluster at
most O(1/ε log n (log R + log log n + log 1/ε )) points.
Keeping the age may require log n space for each point.
The total space required is:
O(1/ε log3n (log R + log log n + log 1/ε ))
Time Complexity
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Query answer time is O(1).
Worst case update time is O(1/ε log2n (log R + log log n
+ log 1/ε )) because we may have cascading merges.
The amortized update time is O(log n)
Extend the Algorithm to 2-d
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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 φ such that 1− cos φ ≤ ε/2
Use the diameter-maintenance algorithm in 1-d for each
line.
Everything will have a multiplicative overhead of
O(1/ ε ).
Lower Bound for Maintaining Exact Diameter
Theorem To maintain the exact diameter in a sliding
window model requires Ω(n) bits of space.
Consider 2n points {a1, a2, ..., a2n} with the following
properties:
– an+1, an+2, ..., a2n are located at coordinate zero.
– |a1an| ≥ |a2an+1| ≥ |a3an+2| ≥ ... ≥ |an-1a2n-2| = 1
– The coordinates of the points aj for j = 1,2,..., n-2 have the form
n∙k for some k = 1,2,..., n.
A Family of Point Sequences
We show below two sequences in the family:
an
an+1
an+2
an-1
an-2
a2
an
an+1
an+2
an-1
an-2
a2
a1
......
......
......
a1
Lower Bound for Maintaining Exact Diameter
(Countinued)
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 2n  5 


 n2 
There are at least
different sequences of 2n
points satisfying the above properties.
Need O(n) space to distinguish them.
(Note here R ≤ n2 << 2n)