# CSE 202 - Algorithms - PowerPoint PPT Presentation

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Recurrences Master Method. CSE 202 - Algorithms. Your turn. For one of the following recursion trees ... T(n) = 3T(n/4) + 5n for n = 256, T(1)=5 T(n) = 2T(n/2) + 5n for n = 32, T(1)=5 T(n) = 3T(n/2) + 5n for n = 32, T(1)=5 T(n) = 3T(n/2) + n 2 for n = 32, T(1)=1

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

CSE 202 - Algorithms

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Recurrences

Master Method

## CSE 202 - Algorithms

CSE 202 - Recurrences

For one of the following recursion trees ...

• T(n) = 3T(n/4) + 5n for n = 256, T(1)=5

• T(n) = 2T(n/2) + 5n for n = 32, T(1)=5

• T(n) = 3T(n/2) + 5n for n = 32, T(1)=5

• T(n) = 3T(n/2) + n2 for n = 32, T(1)=1

• T(n) = 4T(n/2) + n2 for n = 32, T(1)=1

...figure out the total “conquer time” (the last term) needed at each level.

Answer should be a sequence of 5 or 6 numbers.

CSE 202 - Recurrences

c

c

c

c

c

c

c

c

c

### Recursion Tree for T(n) = aT(n/b) + cn

c n

1 node at depth 0

a nodes at depth 1

c n/b

c n/b

c n/b

...

a2 nodes at depth 2

c n/b2

c n/b2

c n/b2

c n/b2

...

...

...

...

...

alogbn nodes at

depth logb n

T(n) = cn+ ac(n/b) + a2c(n/b2) + ... + alogbnc(n/ blogbn)

= cn ( 1 + a/b + (a/b)2 + ... + (a/b)logbn ).

CSE 202 - Recurrences

### How does your tree grow?

What’s T(n) = cn ( 1 + a/b + (a/b)2 + ... + (a/b)logbn ) ?

The largest term of a geometric series “dominates”.

If a/b < 1, the first term dominates

Thus, T(n)  (n).

If a/b > 1, the last term dominates

So T(n)  (n(a/b)logbn) = (n(a logbn/blogbn) ) = (n(a logbn/n) ) = (a logbn) = (nlogba).

If a/b = 1, all terms are equal. There are logbn terms. So T(n)  (n logbn ) = (n lg n).

CSE 202 - Recurrences

### Where are we ??

In Divide & Conquer ... if T(n) = aT(n/b) + cn,

(i.e. if you can combine pieces with linear work)

then there are three cases:

if a>b, then T(n) is (nlogba) (there are so many tiny subproblems, they dominate the time)

if a=b, then T(n) is (n lg n) (just like merge sort)

if a<b, then T(n) is (n) (big step is most expensive)

CSE 202 - Recurrences

### What if combining takes f(n) work??

In Divide & Conquer ... if T(n) = aT(n/b) + f(n), then three corresponding cases are:

• The tiny subproblems dominate the run time

• Happens when f(n) < c a f(n/b) for some c<1 and all n

• If so, T(n)  (alogbn) = (nlogba).

• All levels take about the same time

• Happens when f(n) is (alogbn).

• If so, T(n) = (f(n) lgn).

• Big step is most expensive

• Happens when f(n) > c a f(n/b) for some c>1 and all n.

• If so, T(n) = (f(n)).

CSE 202 - Recurrences

### Previous slide is “Master Method”

Slight differences:

Book’s condition on case 1, “f(n) is O(nlogba-)”, is slightly more general.

It allows f(n) to be less uniform.

Case 2 – remember alogbn = nlogba.

Book has (unnecessary) extra condition in case 3

“f(n) is (nlogba+)” is implied by “f(n) > c a f(n/b), c>1”

Master method can “interpret n/b to mean either n/b or n/b”

CSE 202 - Recurrences

### Other recurrences

Master Method doesn’t always apply:

What if, in MergeSort, we divided 75% - 25% ?

T(n) = T(3n/4) + T(n/4) + c n.

Or we divided into 1000 and n-1000 sized pieces?

T(n) = T(1000) + T(n-1000) + cn (for n>1000).

Or consider:

T(n) = 2T(n/2) + n lg n.

CSE 202 - Recurrences
The text passage is a PowerPoint presentation on algorithms, specifically discussing recurrences and the Master Method. It includes several mathematical questions and provides answers to some of them.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# CSE 202 - Algorithms - PowerPoint PPT Presentation

1 / 8

Recurrences Master Method. CSE 202 - Algorithms. Your turn. For one of the following recursion trees ... T(n) = 3T(n/4) + 5n for n = 256, T(1)=5 T(n) = 2T(n/2) + 5n for n = 32, T(1)=5 T(n) = 3T(n/2) + 5n for n = 32, T(1)=5 T(n) = 3T(n/2) + n 2 for n = 32, T(1)=1

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

CSE 202 - Algorithms

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Recurrences

Master Method

## CSE 202 - Algorithms

CSE 202 - Recurrences

For one of the following recursion trees ...

• T(n) = 3T(n/4) + 5n for n = 256, T(1)=5

• T(n) = 2T(n/2) + 5n for n = 32, T(1)=5

• T(n) = 3T(n/2) + 5n for n = 32, T(1)=5

• T(n) = 3T(n/2) + n2 for n = 32, T(1)=1

• T(n) = 4T(n/2) + n2 for n = 32, T(1)=1

...figure out the total “conquer time” (the last term) needed at each level.

Answer should be a sequence of 5 or 6 numbers.

CSE 202 - Recurrences

c

c

c

c

c

c

c

c

c

### Recursion Tree for T(n) = aT(n/b) + cn

c n

1 node at depth 0

a nodes at depth 1

c n/b

c n/b

c n/b

...

a2 nodes at depth 2

c n/b2

c n/b2

c n/b2

c n/b2

...

...

...

...

...

alogbn nodes at

depth logb n

T(n) = cn+ ac(n/b) + a2c(n/b2) + ... + alogbnc(n/ blogbn)

= cn ( 1 + a/b + (a/b)2 + ... + (a/b)logbn ).

CSE 202 - Recurrences

### How does your tree grow?

What’s T(n) = cn ( 1 + a/b + (a/b)2 + ... + (a/b)logbn ) ?

The largest term of a geometric series “dominates”.

If a/b < 1, the first term dominates

Thus, T(n)  (n).

If a/b > 1, the last term dominates

So T(n)  (n(a/b)logbn) = (n(a logbn/blogbn) ) = (n(a logbn/n) ) = (a logbn) = (nlogba).

If a/b = 1, all terms are equal. There are logbn terms. So T(n)  (n logbn ) = (n lg n).

CSE 202 - Recurrences

### Where are we ??

In Divide & Conquer ... if T(n) = aT(n/b) + cn,

(i.e. if you can combine pieces with linear work)

then there are three cases:

if a>b, then T(n) is (nlogba) (there are so many tiny subproblems, they dominate the time)

if a=b, then T(n) is (n lg n) (just like merge sort)

if a<b, then T(n) is (n) (big step is most expensive)

CSE 202 - Recurrences

### What if combining takes f(n) work??

In Divide & Conquer ... if T(n) = aT(n/b) + f(n), then three corresponding cases are:

• The tiny subproblems dominate the run time

• Happens when f(n) < c a f(n/b) for some c<1 and all n

• If so, T(n)  (alogbn) = (nlogba).

• All levels take about the same time

• Happens when f(n) is (alogbn).

• If so, T(n) = (f(n) lgn).

• Big step is most expensive

• Happens when f(n) > c a f(n/b) for some c>1 and all n.

• If so, T(n) = (f(n)).

CSE 202 - Recurrences

### Previous slide is “Master Method”

Slight differences:

Book’s condition on case 1, “f(n) is O(nlogba-)”, is slightly more general.

It allows f(n) to be less uniform.

Case 2 – remember alogbn = nlogba.

Book has (unnecessary) extra condition in case 3

“f(n) is (nlogba+)” is implied by “f(n) > c a f(n/b), c>1”

Master method can “interpret n/b to mean either n/b or n/b”

CSE 202 - Recurrences

### Other recurrences

Master Method doesn’t always apply:

What if, in MergeSort, we divided 75% - 25% ?

T(n) = T(3n/4) + T(n/4) + c n.

Or we divided into 1000 and n-1000 sized pieces?

T(n) = T(1000) + T(n-1000) + cn (for n>1000).

Or consider:

T(n) = 2T(n/2) + n lg n.

CSE 202 - Recurrences
user
The text passage is a PowerPoint presentation on algorithms, specifically discussing recurrences and the Master Method. It includes several mathematical questions and provides answers to some of them.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS