Fall 20051 Orthogonal Range Search deBerg et. al (Chap.5)

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Orthogonal Range Search

deBerg et. al (Chap.5)

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1D Range SearchData stored in balanced binary search tree T Input: range tree T and range [x,x’]Output: all points in the range

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Idea

Find split nodeFrom split node, find path to , the node x; report all right subtree along the pathFrom split node, find path to ’, the node x’; report all left subtree along the pathCheck and ’

x x’

split-node

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Performance

T: O(n) storage, built in O(nlogn)Query: Worst case: (n) … sounds bad Refined analysis (output-sensitive)

Output k: ReportSubTree O(k) Traverse tree down to or ’: O(logn) Total: O(logn + k)

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2D Kd-tree for 2D range search

Kd-tree: special case of BSPInput: [x,x’][y,y’]Output: all points in range

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Build kd-tree

Break at median n/2 nodesLeft (and bottom) child stores the splitting line

)log()(

2/2)()(

nnOnT

nTnOnT

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Step-by-Step (left subtree)

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1,2,3,4,5 6,7,8,9,10

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6,7,8,9,10

1,2,3 4,5

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6,7,8,9,10

4,5

1,2 3

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6,7,8,9,10

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6,7,8,9,10

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Range Search Kd-Tree

Idea: Traverse the kd-tree; visit only nodes wh

ose region intersected by query rectangle If region is fully contained, report the sub

tree If leaf is reached, query the point against

the range

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Algorithm v

lc(v) rc(v)

v

lc(v)

rc(v)

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Examplel1 l1

l2

l2 l3

l3

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Region Intersection & Containment

Each node in kd-tree implies a region in 2D (k-d in general): [xl,xh]×[yl,yh] Each region can be derived from the defining vert

ex and region of parent Note: the region can be unbounded

The query rectangle: [x, x’]×[y, y’]Containment: [xl,xh] [x, x’] [yl,yh] [y, y’]

Intersection test can be done in a similar way

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Example

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1(-4,0)2(0,3)3(1,2)4(3,-3)

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(-,0]×(-,)

(-,)×(-,)

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Example

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1(-4,0)2(0,3)3(1,2)4(3,-3)

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(-,0]×(-,)

(-,)×(-,)

(-,0]×(-,0]

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Example

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1(-4,0)2(0,3)3(1,2)4(3,-3)

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(-,)×(-,)

(0,)×(-,)

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Example

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1

1(-4,0)2(0,3)3(1,2)4(3,-3)

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(-,)×(-,)

(0,)×(-,)

(0,)×(-3,)

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Homework

Research on linear algorithm for median finding. Write a summary.

Build a kd-tree of the points on the following page. Do the range query according to the algorithm on p.13 Detail the region of each node and intersection/

containment check

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a

h

b

d

g

fc

e

Read off coordinate from the sketching layout, e.g., a=(-4,2)Query rectangle = [-2,4]×[-1,3]