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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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J feigenbaum s kannan j zhang

J. Feigenbaum, S. Kannan, J. Zhang

Computing Diameter in the Streaming and Sliding-Window Models


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)