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## Outline
- Concept
- Data Structure
- Code Efficiency
- Big-O
## What is Data Structure?
- How data is organized in memory (or on disk)
|Data|Structure|
|----|---------|
|Atomic or composite data|A set of rules that holds the data together|
- Examples:
- Linked List
- Array
## Why do we care about Data Structure?
- Efficiency can refer to either
- Space (importance depending on architecture)
- or Time
- We will focus on **time**, So, we want our operations efficient in terms of time
## How to measure efficiency?
- Wall-clock time?
- Not so great
- Not that accurate
- can't compare across computer or programmers
- dependent on circumstances and what's running
- Count the number of operations!
### What kind of Operations?
- insert (add)
- delete (remove)
- access/retrieval
- min/max
- union/merge (less common)
## A Simple Example
- We assume each basic operation takes one step
- Example: Initialize an array (a loop setting `a[i]=0`)
- Count the steps ... how many times is ...
Using a for loop:
```
for(i = 0; i<n; i++){
a[i] = 0;
}
```
- Break down the steps:
1. `i = 0` (done once)
2. `i < n` (done 'n+1' times)
3. `a[i] = 0` (done 'n' times)
4. `i++` (done 'n' times)
- The total number of steps is `1 + (n+1) + n + n` = `3n+2`
- This is tedious and error prone, so we will teach how to count and be lazy!
## When do we care?
- Turns out we care about what happens **when n is big**, i.e., how our alogirthm behaves as n gets large
- When n is a million, 2 doesn't matter, 2000 doesn't matter, c doesn't matter for sufficiently larger n
- We only care about the one with the highest impact on efficiency
## Big-O Notation
- To classify the cost of an algorithm we use the big-O notation
- it's called that because we use a capital 'O'
- To compute the big-O runtime of a function (mathematical function):
- drop all but the fastest-growing term
- drop any coefficient on that remaining term
- So, e.g., `3n+2` is O(n)
- Drop the 2 (not the fastest growing term) and the 3 (coefficient of remaining term)
- Example: `5n^2 + 3n + 5` -> `5n^2` -> `n^2`
- Would be represented as O(n^2)
- Example: `600 + 30n` -> `30n` -> `n`
- Would be represented as O(n)
- Example: `15n + 12nlog(n) + 3` -> `12nlog(n)` -> `nlog(n)`
- Would be represented as O(nlog(n))
- Example: `14n^2 + 0.05n^3 + 18000000` -> `0.05n^3` -> `n^3`
- Would be represented as O(n^3)
## Apply Big-O on Code Analysis
- Let's analyze some loops and compute the big-O runtimes
- `for(i=0; i<n; i++) x++;`
- `3n+2` -> `n` -> O(n)
- `for(i=n; i>0; i--) x++;`
- O(n)
- `for(i=0; i<(n/2); i++) x++;`
- cut the threshold by half
- O(n/2) is **not** the finalized runtime
- O(n) is the finalized runtime
- `for(i=0; i<n; i=i+2) x++;`
- Double the step size
- O(n/2) is **not** the finalized runtime
- O(n) is the finalized runtime
## Big-O Add
- O(n) + O(n) = ?
- `for(i=0; i<n; i++) x++; O(n)`
- `for(i=0; i<n; i++) x++; O(n)`
- O(n) + O(n) = O(n + n) = O(2n) -> O(n)
- `for(i = 1; i < n; i=i*2) x++`
- Break it down:
- iteration 1: i=1
- iteration 2: i=2
- iteration 3: i=4
- iteration 4: i=8
- iteration 5: i=16
- and so on, until i<n
- Notice: iteration x: i=2^(x-1)
- The total number of iterations we need to run is x = (log2(n)+1)
- O(x) = O(log2(n)+1) -> O(log2(n)) -> O(log(n))
- log base value doesn't matter
- Example: `for(i=1; i<n; i=i*100) x++;`
- O(log100(n)) -> O(log(n))
- Example: `for(i=1; i<n; i=i/5) x++;`
- O(log(n))
Example: Nested For Loop
```
for(i=0; i<n; i++){
for(j=0; j<n; j++){
x++;
}
}
```
- How many times does this need to run?
|iteration number|outer loop|inner loop|
|----------------|----------|----------|
|1 |i=0 |n |
|2 |i=1 |n |
|3 |i=2 |n |
|n |i=n-1 |n |
- You get O(n * n) = O(n^2)
- Essentially, you multiply a big-O (outer loop) by another big-O (inner loop)
Example: Another Nested For Loop
```
for(i=1; i<=n; i++){
for(j=1; j<1; j++){
x++;
}
}
```
- These loops are no longer independent
|iteration number|outer loop|inner loop|
|----------------|----------|----------|
|1 |i=1 |0 |
|2 |i=2 |1 |
|3 |i=3 |2 |
|n |i=n |n-1 |
- Total number of iterations is x = 0+1+2+...+(n-1)
- What is x?
- x = (n-1)+(n-2)+...+2+1+0
- 2x = (n-1)\*n -> x = ((n-1)\*n)/2
- O(x) = O(((n-1)\*n)/2) -> O(n^2)
---
[04/10 ->](04-10.md)
|
