Randomized Algorithms · Randomized Incremental Construction • Convex Hull of a set of points • Trapezoidal decomposition of a set of segments. • Convex polytope of a set of

Randomized Algorithms CS648

Lecture 18

Randomized Incremental Construction

(part II)

1

RANDOMIZED INCREMENTAL CONSTRUCTION

2


• Permute the elements of input randomly uniformly.

• Build the structure incrementally.

• Keep some data structure to perform 𝒊th iteration efficiently.

• Use Backward analysis to analyze the expected running time.

3


• Convex Hull of a set of points

• Trapezoidal decomposition of a set of segments.

• Convex polytope of a set of half-planes

• Smallest sphere enclosing a set of points.

• Linear programming in finite dimensions.

4

Trapezoidal decomposition

5

O((𝒏 + 𝒎) 𝐥𝐨𝐠 𝒏 ) time O(𝒎 + 𝒏 𝐥𝐨𝐠 𝒏 ) time

PROBLEM 3 CONVEX HULL OF POINTS

6

Convex hull of Points

Problem definition:

Given 𝒏 points in a plane, compute a convex polygon of smallest area that encloses all the points.

7

Convex hull of Points

Deterministic algorithm:

• O(𝒏 𝐥𝐨𝐠 𝒏) time algorithm.

• Many algorithms exist:

Grahams Scan, Jarvis’s march, divide and conquer,…

Randomized algorithm:

• O(𝒏 𝐥𝐨𝐠 𝒏) time algorithm.

• Based on Randomized Incremental Construction.

• Generalizable to higher dimensions.

8

Randomized algorithm for convex hull

Convex-hull-algorithm(𝑷) { Let <𝒑𝟏,𝒑𝟐,…,𝒑𝒏 > be a uniformly random permutation of 𝑷;

𝑯 triangle(𝒑𝟏,𝒑𝟐, 𝒑𝟑);

For 𝒊 = 𝟒 to 𝒏 do

insert 𝒑𝒊 and update 𝑯

return 𝑯;

}

9

A simple exercise from geometry

Exercise: Given a line L and two points p and q,

determine whether the points lie on the same/different sides of L.

10

L

p

q

q

𝒚 = 𝒎𝒙 + 𝒄

11

Conflict graph : a powerful data structure

𝒄𝟒

𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

cones 𝒏 − 𝒊 points 𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟏

12

Before entering the for loop

13

𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒏 − 𝟑 points cones

INSERTING 𝒊TH POINT

14

𝒊th iteration

15

𝒄𝟏

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒊th iteration

16

𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒏 − 𝒊 + 𝟏 points cones 𝒄𝟏

𝒊th iteration

17

𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖


𝒊th iteration

18

𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖


𝒊th iteration

19

𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒏 − 𝒊 points cones 𝒄𝟏

𝒊th iteration

20

𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖


𝒊th iteration

21

𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖


𝒊th iteration

22

𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖


𝒊th iteration

23

𝒄𝟏

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖


𝒄′

𝒄"

𝒊th iteration

24

𝒄𝟏

𝒄′

𝒄"

𝒄𝟔

𝒄𝟕

𝒄𝟖

𝒄′

𝒄"

𝒄𝟔

𝒄𝟕

𝒄𝟖


Running time of 𝒊th iteration

Running time of 𝒊th iteration is of the order of

• Number of edges of convex hull that are destroyed

• Number of new edges of convex hull that are created

• Number of points in the two adjacent cones that get created

Question: What is the max. number of new edges created in an iteration ?

Answer: 2

Number of edges created during the algorithm = O(𝒏)

Since every edge destroyed was once created, so

Total number of edges destroyed < Total number of edges created

25

Total time for 𝒏 iterations = O(𝒏)

𝑿𝒊

Backward analysis of 𝒊th iteration

𝒂 : a subset of 𝒊 points from 𝑷.

𝓔𝒂: first 𝒊 points of 𝑷 are some permutation of 𝒂

𝐄,𝑿𝒊 𝓔𝒂 = ??

26

Convex hull at the end of 𝒊th iteration





27

𝒄𝟏 𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖 𝒑𝟏

𝒑𝟐

𝒑𝟑

𝒑𝟒

𝒑𝟓

𝒑𝟔

𝒑𝟕

𝒑𝟖





28

𝒄𝟏 𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖 𝒑𝟏

𝒑𝟐

𝒑𝟑

𝒑𝟒

𝒑𝟓

𝒑𝟔

𝒑𝟕

𝒑𝟖

𝒏𝟏 + 𝒏𝟐

𝒏𝟐 + 𝒏𝟑

𝒏𝟑 + 𝒏𝟒

𝒏𝟒 + 𝒏𝟓





29

𝒄𝟏 𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖 𝒑𝟏

𝒑𝟐

𝒑𝟑

𝒑𝟒

𝒑𝟓

𝒑𝟔

𝒑𝟕

𝒑𝟖

𝒏𝟏 + 𝒏𝟐

𝒏𝟐 + 𝒏𝟑

𝒏𝟑 + 𝒏𝟒

𝒏𝟒 + 𝒏𝟓

𝒏𝟕 + 𝒏𝟖

𝒏𝟖 + 𝒏𝟏

𝟏

𝒊

𝟏

𝒊

𝟏

𝒊

𝟏

𝒊

𝟏

𝒊

𝟏

𝒊

+

+

+

+

+





Calculating 𝐄,𝑿𝒊- :

Let 𝑺𝒊 be the set of all subsets of

𝑷 of size 𝒊.

𝐄,𝑿𝒊- =

= 𝟐(𝒏−𝒊)

𝒊∙ 𝐏(𝓔𝒂) 𝒂∈𝑺𝒊

30

𝒄𝟏 𝒄𝟐

𝒄𝟑

𝒄𝟒

𝒄𝟓

𝒄𝟔

𝒄𝟕

𝒄𝟖 𝒑𝟏

𝒑𝟐

𝒑𝟑

𝒑𝟒

𝒑𝟓

𝒑𝟔

𝒑𝟕

𝒑𝟖 𝟐(𝒏 − 𝒊)

𝒊

= 𝟐(𝒏−𝒊)

𝒊

𝐄,𝑿𝒊 𝓔𝒂 ∙ 𝐏(𝓔𝒂)

𝒂∈𝑺𝒊

Running time of the algorithm

Expected running time of ith iteration

= 𝐄,𝑿𝒊- + O(1)

= O(𝒏−𝒊

𝒊)

Expected running time of the algorithm = O(𝒏 𝐥𝐨𝐠 𝒏 )

Theorem:

There is an O(𝒏 𝐥𝐨𝐠 𝒏 ) time Las Vegas algorithm for computing convex hull.

31

USING BACKWARD ANALYSIS FOR MISCELLANEOUS APPLICATIONS

32

PROBLEM 1 SMALLEST ENCLOSING CIRCLE

33

Smallest Enclosing Circle

34

Smallest Enclosing Circle

Question:

Suppose we sample 𝒌 points randomly uniformly from a set of 𝒏 points,

what is the expected no. of points that remain outside the smallest circle

enclosing the sample?

35

For 𝒌= 𝒏

𝟐, the answer is < 𝟏𝟎

PROBLEM 2 SMALLEST LENGTH INTERVAL

36

0 1

Sampling points from a unit interval

Question: Suppose we select 𝒌 points randomly uniformly from interval [0,1],

what is expected length of the smallest sub-interval ?

• for 𝒌 = 1, it is ??

• for 𝒌 = 2, it is ??

• General solution : ??

This bound can be derived using two methods.

• relationship between uniform distribution and exponential distribution.

• Backward analysis.

𝟏

𝟒

𝟏

𝟗

𝟏

𝒌 + 𝟏 𝟐

PROBLEM 3 MINIMUM SPANNING TREE

38

Minimum spanning tree

39

b

a c

d

h x

y

u

v

18

7

1

19

22

10

3 12

3

15

11

5

16

17

13

4 2

9

6


40

b

a c

d

h x

y

u

v

18

7

1

19

22

10

3 12

3

15

11

5

16

17

13

4 2

9

6

Light Edge

Definition: Let 𝒂 ⊂ 𝑬.

An edge 𝒆 ∈ 𝑬\𝒂 is said to be light with respect to 𝒂 if

Question:

If 𝒂 ⊂𝑟 𝑬 and |𝒂|= 𝒌,

how many edges from 𝑬\𝒂 are light with respect to 𝒂 on expectation ?

Answer: ??

41

b

a c

d

h x

y

u

v

18

7

1

19

22

10

3 12

3

15

11

5

16

17

13

4 2

9

6

MST(*𝒆+ ∪ 𝒂) ≠ MST(𝒂)

<𝒏

𝒌(𝒎 − 𝒌)

Smallest Enclosing Circle Let 𝑪 be smallest enclosing circle for set 𝑷 of point.

Facts from high school geometry

• 𝑪 is defined by 2 or 3 points lying on its boundary.

42




43

Not the smallest enclosing circle Since the defining points form an obtuse triangle.




44

This is the smallest enclosing Circle since the defining points form an acute angle triangle.




Lemma 1: If 𝒑 lies outside 𝑪,

𝒑 must be one of the defining point of smallest enclosing circle of 𝑷 ∪ *𝒑+ 45

Smallest Enclosing Circle Randomized Incremental Construction

Smallest-Enclosing-Circle(𝑷) { Let <𝒑𝟏,𝒑𝟐,…,𝒑𝒏 > be a uniformly random permutation of 𝑷;

𝑪 Circle(𝒑𝟏,𝒑𝟐);

For 𝒊 = 𝟑 to 𝒏 do

if ( 𝒑𝒊 lies outside 𝑪 ) update 𝑪;

return 𝑯;

}

Question: How many times will 𝑪 be updated ?

𝑿𝒊 = 𝟏 if 𝑪 is udated in ith iteration𝟎 otherwise

46

?

Smallest Enclosing Circle Question: What is P[𝑿𝒊 = 𝟏] ?

47

Smallest Enclosing Circle at the end of 𝒊th iteration

Smallest Enclosing Circle Question: What is P[𝑿𝒊 = 𝟏] ?

Use it to solve the given problem.

48

≤𝟑

𝒊 Using Lemma 1

PROBLEM 3 MINIMUM SPANNING TREE

49


50

b

a c

d

h x

y

u

v

18

7

1

19

22

10

3 12

3

15

11

5

16

17

13

4 2

9

6


Algorithms:

• Prim’s algorithm

• Kruskal’s algorithm

• Boruvka’s algorithm

51

b

a c

d

h x

y

u

v

18

7

1

19

22

10

3 12

3

15

11

5

16

17

13

4 2

9

6

Less known but it is the first algorithm for MST


𝑮 = (𝑽, 𝑬) : undirected graph with weights on edges

𝒏 = |𝑽|,

𝒎 = |𝑬|.

Deterministic algorithms:

Prim’s algorithm

1. O((𝒎 + 𝒏) log 𝒏) using Binary heap

2. O(𝒎 + 𝒏 log 𝒏) using Fibonacci heap

Best deterministic algorithm:

O(𝒎 + 𝒏 𝐥𝐨𝐠∗ 𝒏) bound

– Too complicated to design and analyze

– Fails to beat Prim’s algorithm using Binary heap

52


𝑮 = (𝑽, 𝑬) : undirected graph with weights on edges

𝒏 = |𝑽|,

𝒎 = |𝑬|.

Randomized algorithm:

Karger-Klein-Tarjan’s algorithm [1995]

1. Las Vegas algorithm

2. O(𝒎 + 𝒏) expected time

This algorithm uses

• Random sampling

• MST verification algorithm

• Boruvka’s algorithm

• Elementary data structure 53

We will discuss this milestone result in the next class.

Documents

Randomized Algorithms · Randomized Incremental Construction • Convex Hull of a set of points • Trapezoidal decomposition of a set of segments. • Convex polytope of a set of