1. Randomized Algorithms
CS648
Lecture 17
Miscellaneous applications of Backward analysis
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2. MINIMUM SPANNING TREE
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3. Minimum spanning tree
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4. Minimum spanning tree
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Algorithms:
• Prim’s algorithm
• Kruskal’s algorithm
• Boruvka’s algorithm
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Less known but it is the first algorithm for MST
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5. Minimum spanning tree
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6. Minimum spanning tree
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7. Minimum spanning tree
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8. Minimum spanning tree
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9. Light Edge
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10. USING BACKWARD ANALYSIS FOR
MISCELLANEOUS APPLICATIONS
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11. PROBLEM 1
SMALLEST ENCLOSING CIRCLE
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12. Smallest Enclosing Circle
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13. Smallest Enclosing Circle
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14. PROBLEM 2
SMALLEST LENGTH INTERVAL
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15. Sampling points from a unit interval
0
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16. PROBLEM 3
MINIMUM SPANNING TREE
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17. Light Edge
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18. USING BACKWARD ANALYSIS FOR
THE 3 PROBLEMS :
A GENERAL FRAMEWORK
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19. A General framework
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20. PROBLEM 3
MINIMUM SPANNING TREE
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21. A BETTER UNDERSTANDING OF
LIGHT EDGES
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22. Minimum spanning tree
d
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h
v
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3
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1
a
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b
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42
u
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3
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5
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c
x
y
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7
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Random sampling
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d
h
v
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4
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b
a
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c
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u
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x
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23. Minimum spanning tree
d
17
h
v
19
6
3
22
1
a
10
b
16
42
u
4
2
31
3
19
5
11
c
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13
16
d
v
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4
b
a
5
31
22
1
y
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7
h
x
c
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u
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11
x
y
18
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24. Minimum spanning tree
d
17
h
v
19
6
3
22
a
1
10
b
42
u
4
31
3
1
v
10
7
y
18
15
16
d
3
b
x
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a
5
11
c
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6
19
2
7
h
16
42
u
4
3
19
2
c
11
5
x
y
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23
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25. Minimum spanning tree
d
17
h
v
19
6
3
22
a
1
10
b
42
u
4
31
3
1
v
10
7
y
18
15
16
d
3
b
x
23
19
a
5
11
c
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6
19
2
7
h
16
42
u
4
3
19
2
c
11
5
x
y
Light
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26. First useful insight
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27. Minimum spanning tree
d
17
h
v
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6
3
22
a
1
10
b
42
u
4
31
3
1
v
10
7
y
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d
3
b
x
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a
5
11
c
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6
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2
7
h
16
42
u
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3
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2
c
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5
x
y
Light
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23
heavy
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28. Minimum spanning tree
d
v
h
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3
a
1
4
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x
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2
x
y
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b
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d
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h
1
v
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3
a
10
b
7
16
42
u
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2
c
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y
Light
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heavy
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29. Second useful insight
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30. Light Edge
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We shall answer the above question using the Generic framework.
But before that, we need to get a better understanding of the
corresponding random variable.
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31. 31

32. Light
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33. Light
heavy33

34. 34

35. Step 1
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36. Step 2
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37. Step 2
Light
heavy37

38. Step 2
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39. Step 3
Expressing the entire experiment as Randomized Incremental Construction
A slight difficulty in this process is the following:
• The underlying experiment talks about random sample from a set.
• But RIC involves analyzing a random permutation of a set of elements.

Question: What is relation between random sample from a set and a random
permutation of the set ?
Spend some time on this question before proceeding further.
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40. random sample and random permutation
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41. Step 3
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42. Step 3
…
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43. Step 3
…
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44. Step 3
…
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45. …
Forward analysis
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46. …
Backward analysis
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47. …
Backward analysis
Use Lemma 2.
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48. …
Backward analysis
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49. 49

50. 50