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資料結構與演算法 (下 )

呂學一 (Hsueh-I Lu)

http://www.csie.ntu.edu.tw/~hil/

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Today

Segment intersection

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The problem

Input: – n line segments in the plane

Each segment is specified by the coordinates of its endpoints.

Output:– Determining whether or not there are two

input segments intersected

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Illustration

Yes No

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Naïve algorithm

O(n2) time– For each of the O(n2) pairs of input segments,

determine in O(1) time whether they are intersected or not.

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A clever algorithm

Initialization: – Sorting the 2n endpoints according to their x

coordinates– O(n log n) time

Key step:– Process each of the 2n endpoints according to

the above sorted order– Each step takes O(log n) time.

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Key idea – Sweep line

a

b

c

de

f

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Segment list L

The segments are “sorted” according to their vertical order at the sweep line

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Segment list L

a

b

c

de

f

a ab

acb

dacb

dcb

edcb

edb

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Key observation

Suppose that segments a and b have the leftmost intersection point. Then, they have to be next to each other in the segment list L at some point during the line-sweeping process.

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Segment list L

a

b

c

de

f

a ab

acb

dacb

dcb

edcb

edb

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Furthermore

Let p be the intersection point of segments a and b. Then, no segments in L change their order before the sweep-line passing point p.

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Segment list L

a

b

c

de

f

a ab

acb

dacb

dcb

edcb

edb

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Maintaining L

The segments are “sorted” according to their vertical order at the sweep line:– When reaching a starting endpoint,

we insert the segment into the list L according to the vertical order at the sweep line.

– When reaching an ending endpoint, we delete the segment from the list L according to

the vertical order at the sweep line.

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Detecting intersection

We only have to detect intersection for consecutive segments in L.– When inserting a new segment in L, we check

this new segment with its neighbors in L.– When deleting a segment from L, we check

for the segments that become neighbors in L due to the deletion.

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To be more careful

The segment list L should be “implemented” by a binary search tree (e.g., a 2-3 tree).

The binary search tree is “sorted” by their relative order of y-coordinates.

It takes O(log n) time to do insertion and deletion.

It takes O(log n) time to identify the “neighbors” of each segment.

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Segment list L

a

b

c

de

f

a ab

acb

dacb

dcb

edcb

edb

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O(n log n)-time algorithm Initialization:

– Sorting the 2n endpoints according to their x coordinates

– O(n log n) time Key step:

– Process each of the 2n endpoints from according to above sorted order

– Each step takes O(log n) time