1. Analysis of algorithms and BIG-O
Tuesday, January 28, 2014
Analysis of algorithms and BIG-O
CS16: Introduction to Algorithms & Data Structures

2. Outline Running time and theoretical analysis Big-O notation
Tuesday, January 28, 2014
Outline
Running time and theoretical analysis
Big-O notation
Big-Ω and Big-Θ
Analyzing seamcarve runtime
Dynamic programming
Fibonacci sequence

3. How fast is the seamcarve algorithm?
Tuesday, January 28, 2014
How fast is the seamcarve algorithm?
What does it mean for an algorithm to be fast?
Low memory usage?
Small amount of time measured on a stopwatch?
Low power consumption?
We’ll revisit this question after developing the fundamentals of algorithm analysis

4. Tuesday, January 28, 2014
Running Time
The running time of an algorithm varies with the input and typically grows with the input size
Average case difficult to determine
In most of computer science we focus on the worst case running time
Easier to analyze
Crucial to many applications: what would happen if an autopilot algorithm ran drastically slower for some unforeseen, untested inputs?

5. How to measure running time?
Tuesday, January 28, 2014
How to measure running time?
Experimentally
Write a program implementing the algorithm
Run the program with inputs of varying size
Measure the actual running times and plot the results
Why not?
You have to implement the algorithm which isn’t always doable!
Your inputs may not entirely test the algorithm
The running time depends on the particular computer’s hardware and software speed

6. Tuesday, January 28, 2014
Theoretical Analysis
Uses a high-level description of the algorithm instead of an implementation
Takes into account all possible inputs
Allows us to evaluate speed of an algorithm independent of the hardware or software environment
By inspecting pseudocode, we can determine the number of statements executed by an algorithm as a function of the input size

7. Elementary Operations
Tuesday, January 28, 2014
Elementary Operations
Algorithmic “time” is measured in elementary operations
Math (+, -, *, /, max, min, log, sin, cos, abs, ...)
Comparisons ( ==, >, <=, ...)
Function calls and value returns
Variable assignment
Variable increment or decrement
Array allocation
Creating a new object (careful, object's constructor may have elementary ops too!)
In practice, all of these operations take different amounts of time
For the purpose of algorithm analysis, we assume each of these operations takes the same time: “1 operation”

8. Example: Constant Running Time
Tuesday, January 28, 2014
Example: Constant Running Time
function first(array): // Input: an array // Output: the first element return array[0] // index 0 and return, 2 ops
How many operations are performed in this function if the list has ten elements? If it has 100,000 elements?
Always 2 operations performed
Does not depend on the input size

9. Example: Linear Running Time
Tuesday, January 28, 2014
Example: Linear Running Time
function argmax(array): // Input: an array // Output: the index of the maximum value index = 0 // assignment, 1 op for i in [1, array.length): // 1 op per loop if array[i] > array[index]: // 3 ops per loop
index = i // 1 op per loop, sometimes return index // 1 op
How many operations if the list has ten elements? 100,000 elements?
Varies proportional to the size of the input list: 5n + 2
We’ll be in the for loop longer and longer as the input list grows
If we were to plot, the runtime would increase linearly

10. Example: Quadratic Running Time
Tuesday, January 28, 2014
Example: Quadratic Running Time
function possible_products(array): // Input: an array // Output: a list of all possible products // between any two elements in the list products = [] // make an empty list, 1 op for i in [0, array.length): // 1 op per loop for j in [0, array.length): // 1 op per loop per loop products.append(array[i] * array[j]) // 4 ops per loop per loop return products // 1 op
Requires about 5n2 + n + 2 operations (okay to approximate!)
If we were to plot this, the number of operations executed grows quadratically!
Consider adding one element to the list: the added element must be multiplied with every other element in the list
Notice that the linear algorithm on the previous slide had only one for loop, while this quadratic one has two for loops, nested. What would be the highest-degree term (in number of operations) if there were three nested loops?

11. Summarizing Function Growth
Tuesday, January 28, 2014
Summarizing Function Growth
For very large inputs, the growth rate of a function becomes less affected by:
constant factors or
lower-order terms
Examples
105n n and n2 both grow with same slope despite differing constants and lower-order terms
10n and n both grow with same slope as well
105n n n2
10n + 105 n
T(n)
When studying growth rates, we only care about what happens for very large inputs (as n approaches infinity…) n
In this graph (log scale on both axes), the slope of a line corresponds to the growth rate of its respective function

12. Big-O Notation if there exist positive constants c and n0 such that
Tuesday, January 28, 2014
Big-O Notation
Given functions f(n) and g(n), we say that
f(n) is O(g(n))
if there exist positive constants c and n0 such that
f(n) ≤ cg(n) for all n ≥ n0
Example: 2n + 10 is O(n)
Pick c = 3 and n0 = 10
2n + 10 ≤ 3n
2(10) + 10 ≤ 3(10)
30 ≤ 30

13. Big-O Notation (continued)
Tuesday, January 28, 2014
Big-O Notation (continued)
Example: n2 is not O(n)
n2 ≤ cn
n ≤ c
The above inequality cannot be satisfied because c must be a constant, therefore for any n > c the inequality is false

14. Tuesday, January 28, 2014
Big-O and Growth Rate
Big-O notation gives an upper bound on the growth rate of a function
We say “an algorithm is O(g(n))” if the growth rate of the algorithm is no more than the growth rate of g(n)
We saw on the previous slide that n2 is not O(n)
But n is O(n2)
And n2 is O(n3)
Why? Because Big-O is an upper bound!

15. Tuesday, January 28, 2014
Summary of Big-O Rules
If f(n) is a polynomial of degree d, then f(n) is O(nd). In other words:
forget about lower-order terms
forget about constant factors
Use the smallest possible degree
It’s true that 2n is O(n50), but that’s not a helpful upper bound
Instead, say it’s O(n), discarding the constant factor and using the smallest possible degree

16. Constants in Algorithm Analysis
Tuesday, January 28, 2014
Constants in Algorithm Analysis
Find the number of primitive operations executed as a function (T) of the input size
first: T(n) = 2
argmax: T(n) = 5n + 2
possible_products: T(n) = 5n2 + n + 3
In the future we can skip counting operations and replace any constants with c since they become irrelevant as n grows
first: T(n) = c
argmax: T(n) = c0n + c1
possible_products: T(n) = c0n2 + n + c1

17. Big-O in Algorithm Analysis
Tuesday, January 28, 2014
Big-O in Algorithm Analysis
Easy to express T in big-O by dropping constants and lower-order terms
In big-O notation
first is O(1)
argmax is O(n)
possible_products is O(n2)
The convention for representing T(n) = c in big-O is O(1).

18. Tuesday, January 28, 2014
Big-Omega (Ω)
Recall that f(n) is O(g(n)) if f(n) ≤ cg(n) for some constant as n grows
Big-O expresses the idea that f(n) grows no faster than g(n)
g(n) acts as an upper bound to f(n)’s growth rate
What if we want to express a lower bound?
We say f(n) is Ω(g(n)) if f(n) ≥ cg(n)
f(n) grows no slower than g(n)
Big-Omega

19. f(n) is O(g(n)) and Ω(g(n))
Tuesday, January 28, 2014
Big-Theta (Θ)
What about an upper and lower bound?
We say f(n) is Θ(g(n)) if
f(n) is O(g(n)) and Ω(g(n))
f(n) grows the same as g(n) (tight-bound)
Big-Theta

20. Some More Examples an + b Θ(n) an2 + bn + c Θ(n2) a Θ(1) 3n + an40
Tuesday, January 28, 2014
Some More Examples
Function, f(n)
Big-O
Big-Ω
Big-Θ
an + b ?
Θ(n)
an2 + bn + c
Θ(n2) a
Θ(1)
3n + an40
Θ(3n)
an + b log n

21. Tuesday, January 28, 2014
Back to Seamcarve
How many distinct seams are there for an w × h image?
At each row, a particular seam can go down to the left, straight down, or down to the right: three options
Since a given seam chooses one of these three options at each row (and there are h rows), from the same starting pixel there are 3h possible seams!
Since there are w possible starting pixels, the total number of seams is: w × 3h
For a square image with n total pixels, that means there are possible seams

22. Tuesday, January 28, 2014
Seamcarve
An algorithm that considers every possible solution is known as an exhaustive algorithm
One solution to the seamcarve problem would be to consider all possible seams and choose the minimum
What would be the big-O running time of that algorithm in terms of n input pixels?
: exponential and not good

23. Seamcarve What’s the runtime of the solution we went over last class?
Tuesday, January 28, 2014
Seamcarve
What’s the runtime of the solution we went over last class?
Remember: constants don’t affect big-O runtime
The algorithm:
Iterate over every pixel from bottom to top to populate the costs and dirs arrays
Create a seam by choosing the minimum value in the top row and tracing downward
How many times do we evaluate each pixel?
A constant number of times
Therefore the algorithm is linear, or O(n), where n is the number of pixels
Hint: we also could have looked back at the pseudocode and counted the number of nested loops!

24. Seamcarve: Dynamic Programming
Tuesday, January 28, 2014
Seamcarve: Dynamic Programming
How did we go from an exponential algorithm to a linear algorithm!?
By avoiding recomputing information we already calculated!
Many seams cross paths, and we don’t need to recompute the sum of importances for a pixel if we’ve already calculated it before
That’s the purpose of the additional costs array
This strategy, storing computed information to avoid recomputing later, is what makes the seamcarve algorithm an example of dynamic programming

25. Fibonacci: Recursive 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …
Tuesday, January 28, 2014
Fibonacci: Recursive
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …
The Fibonacci sequence is usually defined by the following recurrence relation:
F0 = 0, F1 = 1
Fn = Fn-1 + Fn-2
This lends itself very well to a recursive function for finding the nth Fibonacci number
function fib(n):
if n = 0:
return 0
if n = 1:
return 1
return fib(n-1) + fib(n-2)

26. Tuesday, January 28, 2014
Fibonacci: Recursive
In order to calculate fib(4), how many times does fib() get called?
fib(3)
fib(2)
fib(1)
fib(0)
fib(4)
fib(1) alone gets recomputed 3 times!
At each level of recursion, the algorithm makes twice as many recursive calls as the last. So for fib(n), the number of recursive calls is approximately 2n, making the algorithm O(2n)!

27. Fibonacci: Dynamic Programming
Tuesday, January 28, 2014
Fibonacci: Dynamic Programming
Instead of recomputing the same Fibonacci numbers over and over, we’ll compute each one only once, and store it for future reference.
Like most dynamic programming algorithms, we’ll need a table of some sort to keep track of intermediary values.
function dynamicFib(n):
fibs = [] // make an array of size n
fibs[0] = 0
fibs[1] = 1
for i from 2 to n:
fibs[i] = fibs[i-1] + fibs[i-2]
return fibs[n]

28. Fibonacci: Dynamic Programming (2)
Tuesday, January 28, 2014
Fibonacci: Dynamic Programming (2)
What’s the runtime of dynamicFib()?
Since it only performs a constant number of operations to calculate each fibonacci number from 0 to n, the runtime is clearly O(n).
Once again, we have reduced the runtime of an algorithm from exponential to linear using dynamic programming!

29. Readings Dasgupta Section 0.2, pp 12-15 Dasgupta Section 0.3, pp 15-17
Tuesday, January 28, 2014
Readings
Dasgupta Section 0.2, pp 12-15
Goes through this Fibonacci example (although without mentioning dynamic programming)
This section is easily readable now
Dasgupta Section 0.3, pp 15-17
Describes big-O notation far better than I can
If you read only one thing in Dasgupta, read these 3 pages!
Dasgupta Chapter 6, pp
Goes into detail about Dynamic Programming, which it calls one of the “sledgehammers of the trade” – i.e., powerful and generalizable.
This chapter builds significantly on earlier ones and will be challenging to read now, but we’ll see much of it this semester.