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CSE 373 Optional Section Runtime AnalysisPowerPoint Presentation

CSE 373 Optional Section Runtime Analysis

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Interview Question / Runtime Analysis

- How do we find the integer square root of a number?

Integer Square Root Solution 1

- Increment through the list of numbers from 1 to x. Call this value i .
- What cases do we have?

Integer Square Root Solution 1

- Increment through the list of numbers from 1 to x. Call this value i .
- What cases do we have?
- If i× i < x
- If i× i == x
- If i× i > x

Integer Square Root Solution 1

- Increment through the list of numbers from 1 to x. Call this value i .
- What cases do we have?
- If i× i < x increment i
- If i× i == x return i
- If i× i > x return i - 1

Integer Square Root Solution 1 Implementation:

- If i× i < x increment i
- If i× i == x return i
- If i× i > x return i – 1

inti = 1

while(i * i < x)

i++

return i - 1

Integer Square Root Solution 1

- Implementation:
inti = 1

while(i * i < x)

i++

return i – 1

- Runtime
- O( ? )

Integer Square Root Solution 1

- Implementation:
inti = 1

while(i * i < x)

i++

return i – 1

- Runtime
- O(n0.5)

Integer Square Root Solution 1

- Implementation:
inti = 1

while(i * i < x)

i++

return i – 1

- Runtime
- O(n0.5)

- Implement it in Java together & add timing

Integer Square Root Solution 2

- Binary search
- Imagine all of the numbers are in an array (indexed by itself) and search for the integer root
- What number should we start with first?
- 0, 1, x / 2, x

Integer Square Root Solution 2

- Starting with i, jump = x; search(x, i, jump)
- Proposition A: i× i <= x
- Proposition B: (i + 1) × (i + 1) > x
- What do we do in each case?
- A & B
- A only
- B only
- Neither

Integer Square Root Solution 2

- Starting with i, jump = x; search(x, i, jump)
- Proposition A: i× i <= x
- Proposition B: (i + 1) × (i + 1) > x
- What do we do in each case?
- A & B return i
- A only
- B only
- Neither

Integer Square Root Solution 2

- Starting with i, jump = x; search(x, i, jump)
- Proposition A: i× i <= x
- Proposition B: (i + 1) × (i + 1) > x
- What do we do in each case?
- A & B return i
- A only jump /= 2; search(x, i + jump, jump)
- B only jump /= 2; search(x, i - jump, jump)
- Neither

Integer Square Root Solution 2

- Starting with i, jump = x; search(x, i, jump)
- Proposition A: i× i <= x
- Proposition B: (i + 1) × (i + 1) > x
- What do we do in each case?
- A & B return i
- A only jump /= 2; search(x, i + jump, jump)
- B only jump /= 2; search(x, i - jump, jump)
- Neither throw WhatJustHappenedException()

Integer Square Root Solution 2

- Implement
intSqrt(x):

return search(x, x / 2, x / 2)

Search(x, i, jump):

if(i * i <= x)

if((i+1)(i+1) > x)

return i

else

jump /= 2

return search(x, i + jump, jump)

else

jump /= 2

return search(x, i – jump, jump)

Integer Square Root Solution 2

- Runtime
- T(1) = 1
- T(N) = 1 + T(N / 2)
- O( ? )

Integer Square Root Solution 2

- Runtime
- T(1) = 1
- T(N) = 1 + T(N / 2)
- O(log(n))

Stuckon Proof by Induction?

- Khan Academy!
- https://www.khanacademy.org/math/trigonometry/seq_induction/proof_by_induction/v/proof-by-induction

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