Computational Geometry and Spatial Data Mining Marc van Kreveld Department of Information and...

Computational Geometry and Spatial Data Mining

Marc van KreveldDepartment of Information and

Computing Sciences

Utrecht University

Two-part presentation

• Morning:Introduction to computational geometry with examples from spatial data mining

• Afternoon:Geometric algorithms for spatial data mining (and spatio-temporal data mining)

Spatial data mining and computation

• “Geographic data mining involves the application of computational tools to revealinteresting patterns in objects and events distributed in geographic space and across time” (Miller & Han, 2001) [ data analysis ? ]

• Large data sets attempt to carefully define interesting patterns (to avoid finding non-interesting patterns) advanced algorithms needed for efficiency

Introduction to CG

• Some words on algorithms and efficiency

• Computational geometry algorithms through examples from spatial data mining– Voronoi diagrams and clustering– Arrangements and largest clusters– Approximation for the largest cluster

Algorithms and efficiency

• You may know it all already:

• Please look bored if you know all of this

• Please look bewildered if you haven’t got a clue what I’m talking about

Algorithms

• Computational problems have an input size, denoted by n– A set of n numbers– A set of n points in the plane (2n coordinates)– A simple polygon with n vertices– A planar subdivision with n vertices

• A computational problem defines desired output in terms of the input

Algorithms

• Examples of computational problems:– Given a set of n numbers, put them in sorted

order– Given a set of n points, find the two that are

closest– Given a simple polygon P with n vertices and

a point q, determine if q is inside P

P

q

Algorithms

• An algorithm is a scheme (sequence of steps) that always gives the desired output from the given input

• An algorithm solves a computational problem

• An algorithm is the basis of an implementation

Algorithms

• An algorithm can be analyzed for its running time efficiency

• Efficiency is expressed using O(..) notation, it gives the scaling behavior of the algorithm– O(n) time: the running time doubles (roughly)

if the input size doubles– O(n2) time: the running time quadruples

(roughly) if the input size doubles

Algorithms

• Why big-Oh notation?– Because it is machine-independent– Because it is programming language-

independent– Because it is compiler-independent

unlike running time in seconds

It is only algorithm/method-dependent

Algorithms

• Algorithms research is concerned with determining the most efficient algorithm for each computational problem– Until ~1978: O(n2) time– Until 1990: O(n log n) time– Now: O(n) time

polygon triangulation}

Algorithms

• For some problems, efficient algorithms are unknown to exist

• Approximation algorithms may be an option. E.g. TSP– Exact: exponential time– 2-approx: O(n log n) time– 1.5-approx: O(n3) time– (1+)-approx: O(n1/) time

Voronoi diagrams and clustering

• A Voronoi diagram stores proximity among points in a set


• Single-link clustering attempts to maximize the distance between any two points in different sets




• Algorithm (point set P; desired: k clusters):– Compute Voronoi diagram of P– Take all O(n) neighbors and sort by distance– While #clusters > k do

• Take nearest neighbor pair p and q• If they are in different clusters, then merge them

and decrement #clusters (else, do nothing)


• Analysis; n points in P:– Compute Voronoi diagram: O(n log n) time– Sort by distance: O(n log n) time– While loop that merges clusters: O(n log n)

time (using union-find structure)

• Total: O(n log n) + O(n log n) + O(n log n) = O(n log n) time


• What would an “easy” algorithm have given?– really easy: O(n3) time– slightly less easy:

O(n2 log n) time

1000 n log n

10 n2 log n

n3

100 200 300

time

Computing Voronoi diagrams

• By plane sweep

• By randomized incremental construction

• By divide-and-conquer

all give O(n log n) time


• Study the geometry, find properties– 3-point empty circle Voronoi vertex– 2-point empty circle Voronoi edge


• Some geometric properties are needed, regardless of the computational approach

• Other geometric properties are only needed for some approach


• Fortune’s sweep line algorithm (1987)– An imaginary line moves from left to right– The Voronoi diagram is computed while the

known space expands (left of the line)


• Beach line: boundary between known and unknown sequence of parabolic arcs– Geometric property: beach line is y-monotone

it can be stored in a balanced binary tree


• Events: changes to the beach line = discovery of Voronoi diagram features– Point events


• Events: changes to the beach line = discovery of Voronoi diagram features– Point events


• Events: changes to the beach line = discovery of Voronoi diagram features– Circle events


• Events: changes to the beach line = discovery of Voronoi diagram features– Circle events


• Events: changes to the beach line = discovery of Voronoi diagram features– Only point events and circle events exist


• For n points, there are– n point events– at most 2n circle events


• Handling an event takes O(log n) time due to the balanced binary tree that stores the beach line in total O(n log n) time

Intermediate summary

• Voronoi diagrams are useful for clustering (among many other things)

• Voronoi diagrams can be computed efficiently in the plane, in O(n log n) time

• The approach is plane sweep (by Fortune)

Figures from the on-line animation ofAllan Odgaard & Benny Kjær Nielsen

Arrangements and largest clusters

• Suppose we want to identify the largest subset of points that is in some small region– formalize “region” to circle– formalize “small’’ to radius r

r

Place circle to maximize point containment


• Bad idea: Try m = 1, 2, ... and test every subset of size m

• Not so bad idea: for every 3 points, compute the smallest enclosing circle, test the radius and test the other points for being inside


• Bad idea analysis: A set of n points has roughly ( ) = O(nm) subsets of size m

• Not so bad idea analysis: n points give ( ) = O(n3) triples of points. Each can be tested in O(n) time O(n4) time algorithm

nm

n3


• The placement space of circles of radius r

A circle C of radius r contains a point p

if and only if

the center of C lies inside a circle of radius r centered at p

C p


• The placement space of circles of radius r

Circles with center here contain 2 points of P

Circles with center here contain 3 points of P


• Maximum point containment is obtained for circles whose center lies in the most covered cell of the placement space

Computing the most covered cell

• Compute the circle arrangement in a topological data structure

• Fill the cells by the cover value by traversal of the arrangement

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31

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2

1

The value to be assigned to a cell is +/- 1 of its (known) neighbor

0


• Compute the circle arrangement:– by plane sweep: O(n log n + k log n) time– by randomized incremental construction in

O(n log n + k) time

where k is the complexity of the arrangement;k = O(n2)

If the maximum coverage is denoted m, then k = O(nm) and the running time is O(n log n + nm)


• Randomized incremental construction:– Put circles in random order– “Glue” them into the topological structure for

the arrangement with vertical extensions

Every cell has ≤ 4 sides (2 vertical and 2 circular)


Every cell has ≤ 4 sides (2 vertical and 2 circular)

Trace a new circle from its leftmost point to glue it into the arrangement the exit from any cell can be determined in O(1) time


• Randomized analysis can show that adding one circle C takes O(log n + k’ ) time, where k’ is the number of intersections with C

• The whole algorithm takes O(n log n + k) time, where k = k’ is the arrangement size

• The O(n + k) vertical extensions can be removed in O(n + k) time


• Traverse the arrangement (e.g., depth-first search) to fill the cover numbers in O(n + k) time

+1

-1

• into a circle

• out of a circle

Intermediate summary

• The largest cluster for a circle of radius r can be computed in O(n log n + nm) time if it has m entities

• We use arrangement construction and traversal

• The technique for arrangement construction is randomized incremental construction (Mulmuley, 1990)

Largest cluster for approximate radius

• Suppose the specified radius r for a cluster is not so strict, e.g. it may be 10% larger

r

Place circle to maximize point containment

(1+) r

If the largest cluster of radius r has m entities, we must guarantee to find a cluster of m entities and radius (1+) r

Approximate radius clustering

• The idea: snap the entity locations to grid points of a well-chosen grid

Snapping should not move points too much: less than r /4

grid spacing r /4 works



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For each grid point, collect and add the count of all grid points within distance (1+/2) r



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Collected count = 10



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• Claim: a largest approximate radius cluster is given by the highest count

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• Let Copt be a radius-r circle with the most entities inside

• Due to the grid spacing, we have a grid point within distance r /4 from the center of Copt that must have a count


• Snapping moves entities at most r /4

• C and Copt differ in radius r /2 no point in Copt can have moved outside C

• Snapped points inside C have their origins inside a circle of radius at most (1+) r no points too far from C can have entered C


• Intuition: We use the in different places– Snapping points– Trying only circle centers on grid points

... and we guarantee to test a circlethat contains all entities inthe optimal circle, but notother entities too far away


• Efficiency analysis– n entities: each gives a count to O(1/2) grid

cells– in O(n /2) time we have all collected counts

and hence the largest count

Exact or approximate?

• O(n log n + nm) versus O(n /2) time

• In practice: What is larger: m or 1 /2 ?– If the largest cluster is expected to be fairly

small, then the exact algorithm is fine– If the largest cluster may be large and we

don’t care about the precise radius, the approximate radius algorithm is better

Concluding this session

• Basic computational geometry ...

Voronoi diagrams, arrangements,-approximation techniques

... is already useful for spatial data mining

• Afternoon: spatial and spatio-temporal data mining and more geometric algorithms

Date post:	18-Dec-2015
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Computational Geometry and Spatial Data Mining Marc van Kreveld Department of Information and...

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