j feigenbaum s kannan j zhang
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Computing Diameter in the Streaming and Sliding-Window Models

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J. Feigenbaum, S. Kannan, J. Zhang. Computing Diameter in the Streaming and Sliding-Window Models. Introduction. Two computational models: Streaming model Sliding-window model

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introduction
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.
more about models
More about Models

The streaming model

  • A data stream is a sequence of data elements a1a2 , ..., 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.
dynamic algorithm in computational geometry
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.
more about models continued
More about Models (Continued)

The sliding-window model

  • The inputis 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 sizen
  • The current window at any time instant is the set of data elements that have not yet expired.
computing diameter in the streaming model
Computing Diameter in the Streaming Model
  • A well-known diameter-approximation is streaming in nature.
  • Project the points onto lines.
  • Requires θ ≤ 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/ ) points. To process each point, it performs O(1/ ) projections.
diameter approximation in the streaming model
Diameter Approximation in the Streaming Model

Theorem 1There 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.

  • 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 model1
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
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 < 2n.
maintaining diameter in the sliding window model1
Maintaining Diameter in the Sliding-Window Model

Theorem 2There 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
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”
the rounding method
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| = ( 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
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≤ log1+εD/d = (log D/d)/log (1+ε) ≤ (2/ε )log D/d

when window starts sliding
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.
multiple clusters in a window
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” i+1.
  • The window can thus be divided into clusters.
merge clusters
Merge Clusters
  • Cluster c1+cluster c2 = cluster c3
  • Make Ctr2 the center of cluster c3
merge clusters continued
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| ≤ (1+ε)|Ctr1Ctr2|, discard it, too.
merge clusters continued1
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 ε ∙ |Ctr1Ctr2|. The number of points in a cluster is then bounded by:

(2/ε )(log R + log1/ε )

the algorithm in 1 d
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.
space requirement
Space Requirement
  • 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
Time Complexity
  • 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
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 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/ ).

lower bound for maintaining exact diameter
Lower Bound for Maintaining Exact Diameter

Theorem 3To maintain the exact diameter in a sliding window model requiresΩ(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

an

an+1

an+2

......

an-1

a2

a1

an-2

A Family of Point Sequences

We show below two sequences in the family:

an

an+1

an+2

......

an-1

a2

a1

an-2

......

lower bound for maintaining exact diameter countinued
Lower Bound for Maintaining Exact Diameter (Countinued)
  • There are at least different sequences of 2npoints satisfying the above properties.
  • Need O(n) space to distinguish them.

(Note here R ≤n2 << 2n)

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