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Presentation Transcript

Last time

- What did we talk about last time?
- Asymptotic notation

Logical warmup

- A businesswoman has two cubes on her desk
- Every day she arranges both cubes so that the front faces show the current day of the month
- What numbers do you need on the faces of the cubes to allow this?
- Note: Both cubes must be used for every day

3

Proving bounds

- Prove a bound for g(x) = (1/4)(x – 1)(x + 1) for x R
- Prove that x2 is not O(x)
- Hint: Proof by contradiction

Polynomials

- Let f(x) be a polynomial with degree n
- f(x) = anxn + an-1xn-1 + an-2xn-2 … + a1x + a0
- By extension from the previous results, if an is a positive real, then
- f(x) is O(xs) for all integers s n
- f(x) is (xr) for all integers r≤n
- f(x) is (xn)
- Furthermore, let g(x) be a polynomial with degree m
- g(x) = bmxm + bm-1xm-1 + bm-2xm-2 … + b1x + b0
- If an and bm are positive reals, then
- f(x)/g(x) is O(xc) for real numbers c> n - m
- f(x)/g(x) is not O(xc) for real numbers c < n -m
- f(x)/g(x) is (xn- m)

Extending notation to algorithms

- We can easily extend our -, O-, and - notations to analyzing the running time of algorithms
- Imagine that an algorithm A is composed of some number of elementary operations (usually arithmetic, storing variables, etc.)
- We can imagine that the running time is tied purely to the number of operations
- This is, of course, a lie
- Not all operations take the same amount of time
- Even the same operation takes different amounts of time depending on caching, disk access, etc.

Running time

- First, assume that the number of operations performed by A on input size n is dependent only on n, not the values of the data
- If f(n) is (g(n)), we say that Ais(g(n)) or that A is of order g(n)
- If the number of operations depends not only on n but also on the values of the data
- Let b(n) be the minimum number of operations where b(n) is (g(n)), then we say that in the best case, Ais(g(n)) or that A has a best case order of g(n)
- Let w(n) be the maximum number of operations where w(n) is (g(n)), then we say that in the worst case, Ais(g(n)) or that A has a worst case order of g(n)

Computing running time

- With a single for (or other) loop, we simply count the number of operations that must be performed:

int p = 0;

int x = 2;

for( inti = 2; i <= n; i++ )

p = (p + i)*x;

- Counting multiplies and adds, (n – 1) iterations times 2 operations = 2n – 2
- As a polynomial, 2n – 2 is (n)

Nested loops

- When loops do not depend on each other, we can simply multiply their iterations (and asymptotic bounds)

int p = 0;

for( inti = 2; i <= n; i++ )

for( int j = 3; j <= n; j++ )

p++;

- Clearly (n – 1)(n -2) is (n2)

Trickier nested loops

- When loops depend on each other, we have to do more analysis

int s = 0;

for( inti = 1; i <= n; i++ )

for( int j = 1; j <= i; j++ )

s = s + j*(i – j + 1);

- What\'s the running time here?
- Arithmetic sequence saves the day (for the millionth time)

Iterations with floor

- When loops depend on floor, what happens to the running time?

int a = 0;

for( inti = n/2; i <= n; i++ )

a = n - i;

- Floor is used implicitly here, because we are using integer division
- What\'s the running time? Hint: Consider n as odd or as even separately

Sequential search

- Consider a basic sequential search algorithm:

int search( int[]array, int n, int value)

{

for( inti = 0; i < n; i++ )

if( array[i] == value )

returni;

return -1;

}

- What\'s its best case running time?
- What\'s its worst case running time?
- What\'s its average case running time?

Insertion sort algorithm

- Insertion sort is a common introductory sort
- It is suboptimal, but it is one of the fastest ways to sort a small list (10 elements or fewer)
- The idea is to sort initial segments of an array, insert new elements in the right place as they are found
- So, for each new item, keep moving it up until the element above it is too small (or we hit the top)

Insertion sort in code

public static void sort( int[]array, int n)

{

for( inti = 1; i < n; i++ )

{

intnext = array[i];

int j = i - 1;

while( j != 0 && array[j] > next )

{

array[j+1] = array[j];

j--;

}

array[j] = next;

}

}

Best case analysis of insertion sort

- What is the best case analysis of insertion sort?
- Hint: Imagine the array is already sorted

Worst case analysis of insertion sort

- What is the worst case analysis of insertion sort?
- Hint: Imagine the array is sorted in reverse order

Average case analysis of insertion sort

- What is the average case analysis of insertion sort?
- Much harder than the previous two!
- Let\'s look at it recursively
- Let Ek be the average number of comparisons needed to sort k elements
- Ek can be computed as the sum of the average number of comparisons needed to sort k – 1 elements plus the average number of comparisons (x) needed to insert the kth element in the right place
- Ek = Ek-1 + x

Finding x

- We can employ the idea of expected value from probability
- There are k possible locations for the element to go
- We assume that any of these k locations is equally likely
- For each turn of the loop, there are 2 comparisons to do
- There could be 1, 2, 3, … up to k turns of the loop
- Thus, weighting each possible number of iterations evenly gives us

Finishing the analysis

- Having found x, our recurrence relation is:
- Ek = Ek-1 + k + 1
- Sorting one element takes no time, so E1 = 0
- Solve this recurrence relation!
- Well, if you really banged away at it, you might find:
- En = (1/2)(n2 + 3n – 4)
- By the polynomial rules, this is (n2) and so the average case running time is the same as the worst case

Exponential and Logarithmic Functions

Student Lecture

Exponential functions

- Well, they grow fast
- Graph 2x for -3 ≤ x ≤ 3
- When considering bx, it\'s critically important whether b > 1 (in which case bx grows very fast in the positive direction) or 0 < b < 1 (in which case bx grows very fast in the negative direction)
- Graph bx when b > 1
- Graph bxwhen 0 < b < 1
- What happens when b = 1?
- What happens when b ≤ 0?

Logarithmic function

- The logarithmic function with base b, written logb is the inverse of the exponential function
- Thus,
- by = x logbx = y for b > 0 and b 1
- Log is a "de-exponentiating" function
- Log grows very slowly
- We\'re interested in logb when b > 1, in which case logb is an increasing function
- If x1 < x2, logb(x1) < logb(x2), for b > 1 and positive x1 and x2

Applying log

- How many binary digits are needed to represent a number n?
- We can write n = 2k + ck-12k-1 + … c222 + c12 + c0 where ci is either 0 or 1
- Thus, we need no more than k + 1 digits to represent n
- We know that n < 2k + 1
- Since 2k ≤ n < 2k+1, k ≤ log2n < k+1
- The total number of digits we need k + 1 ≤ log2n + 1

Recurrence relations

- Consider the following recurrence relation
- a1 = 0
- ak = ak/2 + 1 for integers k ≥ 2
- What do you think its explicit formula is?
- It turns out that an = log n
- We can prove this with strong induction

Exponential and logarithmic orders

- For all real numbers b and r with b > 1 and r > 0
- logbx ≤ xr, for sufficiently large x
- xr ≤ bx, for sufficiently large x
- These statements are equivalent to saying for all real numbers b and r with b > 1 and r > 0
- logbx is O(xr)
- xr ≤ O(bx)

Important things

- We don\'t have time to show these things fully
- xk is O(xklogbx)
- xklogbx is O(xk+1)
- The most common case you will see of this is:
- x is O(x log x)
- x log x is O(x2)
- In other words, x log x is between linear and quadratic
- logbx is (logc x) for all b > 1 and c > 1
- In other words, logs are equivalent, no matter what base, in asymptotic notation
- 1/2 + 1/3 + … + 1/n is (log2 n)

Next time…

- Review for Exam 3
- Relations, counting, graphs, and trees

Reminders

- Study for Exam 3
- Monday in class
- Finish Assignment 9
- Due Friday by midnight
- Talk on the Shadow programming language
- Tonight in E281 at 6pm

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