CS648 Lecture 17 Miscellaneous applications of Backward analysis 1 Minimum spanning tree 2 Minimum spanning tree 3 b a c d h x y u v 18 7 1 19 22 10 3 12 3 15 11 5 ID: 274407
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Slide1
Randomized AlgorithmsCS648
Lecture 17Miscellaneous applications of Backward analysis
1Slide2
Minimum spanning tree2Slide3
Minimum spanning tree 3
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Minimum spanning tree Algorithms: Prim’s
algorithmKruskal’s algorithmBoruvka’s algorithm4
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Less known but it is the first algorithm for MSTSlide5
Minimum spanning tree
: undirected graph with weights on edges,
.
Deterministic algorithms
:
Prim’s algorithm
O
((
+
)
log
) using
Binary
heap
O
( +
log ) using
Fibonacci heapBest deterministic algorithm: O(
+
) bound
Too complicated to design and analyzeFails to beat Prim’s algorithm using Binary heap
5Slide6
Minimum spanning treeWhen finding an efficient solution of a problem appears hard, one should strive to design an efficient verification algorithm.
MST verification algorithm: [King, 1990]Given a graph and a spanning tree
, it takes
O
(
+
) time to
d
etermine if
is MST of
.
Interestingly, no deterministic algorithm for MST could use this algorithm to achieve
O
(
+
)
time.
6Slide7
Minimum spanning tree
: undirected graph with weights on edges,
.
Randomized algorithm
:
Karger
-Klein-
Tarjan
’s
algorithm [
1995
]
Las Vegas algorithm
O
(
+
) expected timeThis algorithm uses Random samplingMST verification algorithm
Boruvka’s algorithmElementary data structure
7Slide8
Minimum spanning tree
: undirected graph with weights on edges,
.
Randomized algorithm
:
Karger
-Klein-
Tarjan
’s
algorithm [
1994
]
Las Vegas algorithm
O
(
+
) expected timeRandom sampling :
How close is MST of a random sample of edges to MST of original graph ?
The notion of closeness is formalized in the following slide.
8Slide9
Light Edge
Definition: Let . An edge
is said to be
light
with respect to
if
Question
:
If
and |
|=
,
how many edges from
are light with respect to
on expectation ?
Answer
: ??
9
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MST
(
)
MST
(
)
Slide10
USING Backward analysis forMiscellaneous Applications10Slide11
problem 1SMALLEST Enclosing circle11Slide12
Smallest Enclosing Circle12Slide13
Smallest Enclosing CircleQuestion: Suppose we sample
points randomly uniformly from a set of points, what is the expected number of points that remain outside the smallest circle enclosing the sample?
13
For
=
, the answer is
Slide14
problem 2smallest length interval14Slide15
0
1
Sampling points from a
unit interval
Question:
Suppose we select
points from interval [
0
,
1
], what is expected length of the smallest sub-interval
?
for
, it is ??
for
, it is ?? General solution : ??
This bound can be derived using two methods.One method is based on establishing a relationship between uniform distribution and exponential distribution.
Second method (for nearly same asymptotic bound) using Backward analysis.
Slide16
problem 3Minimum spanning tree16Slide17
Light Edge
Definition: Let . An edge
is said to be
light
with respect to
if
Question
:
If
and |
|=
,
how many edges from
are light with respect to
on expectation ?
17
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MST
(
)
MST
(
)
Slide18
using Backward analysis forThe 3 problems :A General framework
18Slide19
A General frameworkLet be the desired random variable in any of these problems/random experiment.
Step 1: Define an event related to the random variable . Step 2: Calculate probability of event
using
standard
method
based on
definition
. (This establishes a relationship between )
Step
3:
Express the underlying
random experiment
as a Randomized incremental construction and calculate the probability of the event
using Backward analysis.
Step 4
: Equate the expressions from Steps 1
and 2 to calculate E[
].
19Slide20
problem 3Minimum spanning tree20Slide21
A Better understanding of light edges21Slide22
Minimum spanning tree 22
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Random sampling
Slide23
Minimum spanning tree 23
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MST(
)
Slide24
Minimum spanning tree 24
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MST(
)
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Minimum spanning tree 25
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MST(
)
LightSlide26
First useful insight
Lemma1: An edge is light with respect to if and only if
belongs to
MST
(
).
26Slide27
Minimum spanning tree 27
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MST(
)
Light
heavySlide28
Minimum spanning tree 28
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MST(
)
Light
heavy
MST(
)
Is there any relationship among MST
(
),
MST(
)
and Light edges from
?
Slide29
Second useful insightLemma2: Let
and be the set of all edges from
that are
light
with respect to
.
Then
MST
(
)
=
MST(
)
This lemma is used in the design of randomized algorithm for MST as follows (just a sketch):
Compute MST of a sample of
edges (recursively). Let it be T’.There will be expected
edges light edges among all unsampled
edges.Recursively compute MST of
T’
edges which are less than
on expectation.
29Slide30
Light Edge
Definition: Let . An edge
is said to be
light
with respect to
if
Question
:
If
and |
|=
,
how many edges from
are
light
with respect to
on expectation ?
30
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a
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MST
(
)
MST
(
)
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.Slide31
31
MST(
)
Slide32
32
MST(
)
LightSlide33
33
Light
heavy
MST(
)
Slide34
: random variable for the number of light edges in
when is a random sample of edges.
: set of all subsets of
of size
.
: number of light edges in
when
.
= ??
34
Can you express
in terms of
and
only ?
Slide35
Step 1Question: Let
be a uniformly random sample of edges from .What is the prob. that an edge selected randomly from
is a light edge ?
35
Two methods to find
Slide36
Step 2Calculating using definition
36Slide37
Step 2 Calculating using definition
37
MST(
)
Light
heavy
Light edges
=
Slide38
Step 2Calculating using definition
: set of all subsets of of size .The probability
is equal to
38
Slide39
Step 3Expressing the entire experiment as Randomized Incremental ConstructionA 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.
39Slide40
random sample and random permutation Observation:
is indeed a uniformly random sample of
40
Random permutation of
Slide41
Step 3The underlying random experiment as Randomized Incremental Construction:
Permute the edges randomly uniformly.Find the probability that th edge is light relative to the first edges.
Question:
Can you now calculate probability
?
Spend some time on this question before proceeding further.
41Slide42
Step 3
42
Random permutation of
…
Slide43
Step 3
: a random variable taking value 1 if is a light
edge with respect to
MST
(
).
43
Random permutation of
…
Slide44
Step 3
: a random variable taking value 1 if is a
light
edge with respect to
MST
(
).
Question:
What is relation between
and
’s?
Answer:
??
44
Random permutation of
…
Slide45
Calculating ).
: set of all subsets of
of size
.
) =
depends upon
45
Forward analysis
…
MST(
)
Random permutation of
Slide46
Calculating ).
: set of all subsets of
of size
.
)=
46
Backward analysis
…
Random permutation of
Slide47
= ??
??
47
Backward analysis
…
MST(
)
Random permutation of
Use Lemma 2.
Slide48
Calculating )
: set of all subsets of
of size
.
)=
48
Backward analysis
…
Random permutation of
Slide49
Combining the two methods for calculating
Using method 1:
Using
method
2
:
)
Hence:
49Slide50
Theorem: If we sample
edges uniformly randomly from an undirected graph on vertices and edges, the number of light edges among the unsampled edges will be less than
on expectation.
50