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TCSS 342 Lecture NotesCourse Overview,

Review of Math Concepts,Algorithm Analysis and Big-Oh Notation

Weiss book, Chapter 5

These lecture notes are copyright (C) Marty Stepp, 2005. They may not be rehosted, sold, or modified without expressed permission from the author. All rights reserved.

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Lecture outline Introduction and course objectives

Mathematical background review exponents and logarithms arithmetic and geometric series

Algorithm analysis and Big-Oh notation the RAM model Big-Oh Big-Omega, Big-Theta, Little-Oh, Little-Omega

Comparison of algorithms to solve the Maximum Contiguous Subsequence Sum problem

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Course objectives (broad) prepare you to be a good software engineer

(specific) learn basic data structures and algorithms data structures – how data is organized algorithms – unambiguous sequence of steps to compute

something algorithm analysis – determining how long an algorithm will take to

solve a problem Who cares? Aren't computers fast enough and getting faster?

"Data Structures + Algorithms = Programs"-- Niklaus Wirth, author of Pascal language

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Given an array of 1,000,000 integers, find the maximum integer in the array.

Now suppose we are asked to find the kth largest element. (The Selection Problem)

…0 1 2 999,999

An example

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candidate solution 1 sort the entire array (from small to large), using Java's Arrays.sort()

pick out the (1,000,000 – k)th element

candidate solution 2 place the first k elements into a sorted array of size k for each of the remaining 1,000,000 – k elements,

keep the k largest in the array pick out the smallest of the k survivors and toss it

Candidate solutions

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Is either solution good? What advantages and disadvantages does each

solution have?

Is there a better solution?

What makes a solution "better" than another? Is it entirely based on runtime?

How would you go about determining which solution is better? could code them, test them could somehow make predictions and analysis of

each solution, without coding

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Why algorithm analysis? as computers get faster and problem sizes get bigger,

analysis will become more important

The difference between good and bad algorithms will get bigger

being able to analyze algorithms will help us identify good ones without having to program them and test them first

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Why data structures? when programming, you are an engineer

engineers have a bag of tools and tricks – and the knowledge of which tool is the right one for a given problem

Examples: arrays, lists, stacks, queues, trees, hash tables, heaps, graphs

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Mathematical background review

Let's review some math material that'll be useful later when computing the runtime of various algorithms.

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Math background: exponents Exponents

XY , or "X to the Yth power";X multiplied by itself Y times

Some useful identities XA XB = XA+B

XA / XB = XA-B

(XA)B = XAB

XN+XN = 2XN

2N+2N = 2N+1

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Math background:Logarithms Logarithms

definition: XA = B if and only if logX B = A intuition: logX B means:

"the power X must be raised to, to get B"

In this course, a logarithm with no base implies base 2.log B means log2 B

Examples log2 16 = 4 (because 24 = 16) log10 1000 = 3 (because 103 = 1000)

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Identities for logs with addition, multiplication, powers: log (AB) = log A + log B log (A/B) = log A – log B log (AB) = B log A

Identity for converting bases of a logarithm:

example:log432 = (log2 32) / (log2 4) = 5 / 2

1,0,,log

loglog ACBA

A

BB

C

CA

Logarithm identities

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Logarithm problem solving When presented with an expression of the form:

logaX = Y

and trying to solve for X, raise both sides to the a power.

X = aY

When presented with an expression of the form: logaX = logbY

and trying to solve for X, find a common base between the logarithms using the identity on the last slide.

logaX = logaY / logab

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Logarithm practice problems Determine the value of x in the following equation.

log7 x + log713 = 3

Determine the value of x in the following equation. log8 4 - log8 x = log8 5 + log16 6

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Series

for some expression Expr (possibly containing i ), means the sum of all values of Expr with each value of i between j and k inclusive

Example:

= (2(0) + 1) + (2(1) + 1) + (2(2) + 1) + (2(3) + 1) + (2(4) + 1)= 1 + 3 + 5 + 7 + 9= 25

Math review: Arithmetic series

k

ji

Expr

4

0

12i

i

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Series identities sum from 1 through N inclusive

is there an intuition for this identity? sum of all numbers from 1 to N

1 + 2 + 3 + ... + (N-2) + (N-1) + N

how many terms are in this sum? Can we rearrange them?

2

)1(

1

NNi

N

i

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More series identities sum from a through N inclusive

(when the series doesn't start at 1)

is there an intuition for this identity? The sum of all numbers from 4 to 10 is just the sum of all

numbers from 1 to 10, minus the sum of 1 through 3.

1

11

a

i

N

i

N

ai

iii

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Series of constants sum of constants

(when the body of the series doesn't contain the counter variable such as i)

example:

11

abkkkb

ai

b

ai

351410515510

4

10

4

ii

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Splitting series

for any constant k, splitting a sum with addition

moving out a constant multiple

b

ai

b

ai

b

ai

kiki

b

ai

b

ai

ikki

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Series of powers sum of powers of 2

1 + 2 + 4 + 8 + 16 + 32 = 64 - 1 = 63 think about binary representation of numbers...

111111 (63)+ 1 (1)1000000 (64)

when the series doesn't start at 0:

122 1

0

NN

i

i

1

00

222a

i

iN

i

iN

ai

i

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Series practice problems Give a closed form expression for the following

summation. A closed form expression is one without the or "…".

Give a closed form expression for the following summation.

2

0

2N

i

i

1

10

5N

i

i

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Algorithm analysis and Big-oh notation

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Algorithm performance How to determine how much time an algorithm A uses to solve

problem X ? Depends on input; use input size "N" as parameter Determine function f(N) representing cost

empirical analysis: code it and use timer running on many inputs

algorithm analysis: Analyze steps of algorithm, estimating amount of work each step takes

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Typically use a simple model for basic operation costs

RAM (Random Access Machine) model RAM model has all the basic operations:

+, -, *, / , =, comparisons fixed sized integers (e.g., 32-bit) infinite memory All basic operations take exactly one time unit (one CPU

instruction) to execute

RAM model

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Critique of the model Strengths:

simple easier to prove things about the model than the real machine can estimate algorithm behavior on any hardware/software

Weaknesses: not all operations take the same amount of time in a real

machine does not account for page faults, disk accesses, limited

memory, floating point math, etc

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Relative rates of growth most algorithms' runtime can be expressed as a

function of the input size N

rate of growth: measure of how quickly the graph of a function rises

goal: distinguish between fast- and slow-growing functions

we only care about very large input sizes(for small sizes, most any algorithm is fast enough)

this helps us discover which algorithms will run more quickly or slowly, for large input sizes

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Growth rate example

Consider these graphs of functions.Perhaps each one represents an algorithm:n3 + 2n2

100n2 + 1000

• Which growsfaster?

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Growth rate example

• How about now?

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Defn: T(N) = O(f(N))if there exist positive constants c , n0 such that: T(N) c · f(N) for all N n0

idea: We are concerned with how the function grows when N is large. We are not picky about constant factors: coarse distinctions among functions

Lingo: "T(N) grows no faster than f(N)."

Review: Big-Oh notation

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n = O(2n) ?

2n = O(n) ?

n = O(n2) ?

n2 = O(n) ?

n = O(1) ?

100 = O(n) ?

10 log n = O(n) ?

214n + 34 = O(2n2 + 8n) ?

Big-Oh example problems

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pick tightest bound. If f(N) = 5N, then:f(N) = O(N5)f(N) = O(N3)f(N) = O(N log N)f(N) = O(N) preferred

ignore constant factors and low order termsT(N) = O(N), not T(N) = O(5N)

T(N) = O(N3), not T(N) = O(N3 + N2 + N log N)

Wrong: f(N) O(g(N)) Wrong: f(N) O(g(N))

remove non-base-2 logarithmsf(N) = O(N log6 N)

f(N) = O(N log N) preferred

Preferred big-Oh usage

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Big-Oh of selected functions

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Ten-fold processor speedup "Who cares about choosing the fastest algorithm. If it is

too slow, we'll just upgrade the computer..."

Example data when computer is sped up by a factor of 10:

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Defn: T(N) = (g(N)) if there are positive constants c and n0 such that T(N) c g(N) for all N n0

Lingo: "T(N) grows no slower than g(N)."

Defn: T(N) = (g(N)) if and only if T(N) = O(g(N)) and T(N) = (g(N)).

Big-Oh, Omega, and Theta establish a relative ordering among all functions of N

Defn: T(N) = o(g(N)) if and only if T(N) = O(h(N)) and T(N) (h(N)).

Big omega, theta

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notation intuition

O (Big-Oh)

(Big-Omega)

(Theta) =

o (little-Oh) <

Intuition about the notations

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Fact: If f(N) = O(g(N)), then g(N) = (f(N)).

Proof: Suppose f(N) = O(g(N)).Then there exist constants c and n0 such that f(N) c g(N) for all N n0

Then g(N) (1/c) f(N) for all N n0, and so g(N) = (f(N))

More about asymptotics

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T(N) = O(f(N)) f(N) is an upper bound on T(N) T(N) grows no faster than f(N)

T(N) = (g(N)) g(N) is a lower bound on T(N) T(N) grows at least as fast as g(N)

T(N) = o(h(N)) T(N) grows strictly slower than h(N)

More terminology

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Facts about big-Oh

If T1(N) = O(f(N)) and T2(N) = O(g(N)), then T1(N) + T2(N) = O(f(N) + g(N)) T1(N) * T2(N) = O(f(N) * g(N))

If T(N) is a polynomial of degree k, then:T(N) = (Nk)

example: 17n3 + 2n2 + 4n + 1 = (n3)

logk N = O(N), for any constant k

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Hierarchy of Big-Oh Functions, ranked in increasing order of growth:

1 log log n log n n n log n n2

n2 log n n3

... 2n

n! nn

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Algebraex. f(N) = N / log N

g(N) = log Nsame as asking which grows faster, N or log2 N

Evaluate:

)(

)(lim

Ng

NfN

limit is Big-Oh relation

0 f(N) = o(g(N))

c 0 f(N) = (g(N))

g(N) = o(f(N))

no limit no relation

Techniques

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L'Hôpital's rule:

If and , then

example: f(N) = N, g(N) = log NUse L'Hôpital's rulef'(N) = 1, g'(N) = 1/N g(N) = o(f(N))

)(lim NfN

)(lim NgN

)('

)('lim

)(

)(lim

Ng

Nf

Ng

NfNN

Techniques, cont'd

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for (int i = 0; i < n; i += c) // O(n) statement(s); Adding to the loop counter means that the loop runtime grows linearly

when compared to its maximum value n. Loop executes its body exactly n / c times.

for (int i = 0; i < n; i *= c) // O(log n) statement(s); Multiplying the loop counter means that the maximum value n must grow

exponentially to linearly increase the loop runtime; therefore, it is logarithmic.

Loop executes its body exactly logc n times.

for (int i = 0; i < n * n; i += c) // O(n2) statement(s); The loop maximum is n2, so the runtime is quadratic.

Loop executes its body exactly (n2 / c) times.

Program loop runtimes

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Nesting loops multiplies their runtimes.for (int i = 0; i < n; i += c) { // O(n2) for (int j = 0; j < n; i += c) { statement;} }

Loops in sequence add together their runtimes, which means the loop set with the larger runtime dominates.for (int i = 0; i < n; i += c) { // O(n) statement;}for (int i = 0; i < n; i += c) { // O(n log n) for (int j = 0; j < n; i *= c) { statement;} }

More loop runtimes

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Loop runtime problems Compute the exact value of the variable sum after the

following code fragment, as a closed-form expression in terms of input size n.int sum = 0;

for (int i = 1; i <= n; i *= 2) {

sum++;

}

 

for (int i = 1; i <= 1000; i++) {

sum++;

sum++;

}

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Loop runtime problems Compute the exact value of the variable sum after the

following code fragment, as a closed-form expression in terms of input size n.int sum = 0;

for (int i = 1; i <= n; i++) {

for (int j = 1; j <= i / 2; j += 2) {

sum++;

}

}

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"Maximum subsequence sum" problem

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Maximum subsequence sum

The maximum subsequence sum problem:

Given a sequence of integers A0, A1, ..., An - 1,

find the maximum value of

for any integers 0 (i, j) < n.

(This sum is zero if all numbers in the sequence are negative.)

For example, the maximum subseq. sum of this list is 33:[2, 9, -4, 1, -20, 3, 15, -10, 20, 5, -100, 10, 7]

j

ikkA

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Thoughts about the problem This is a maximization problem.

There are a set of possible answers (sums of subsets of the original array), and we are being asked to find the largest one.

When solving maximization problems, one possible technique is to generate all possible solutions, compare them, and choose the appropriate one.

In the context of this problem, that would mean generating all possible subsets of the original array, adding them up to determine each of their sums, and comparing these sums to choose the best one of them all.

brute force algorithm: A method for solving a problem that proceeds in a simple and obvious way, possibly making it require more steps than necessary.

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First algorithm (brute force)generate and test all possible subsequences

// implement togetherfunction maxSubsequence(array[]): max sum = 0

for each starting index i, for each ending index j, add up the sum from Ai to Aj

if this sum is bigger than max, max sum = this sum

return max sum

Using the RAM model, how many steps does this algorithm require?

What do you believe is the runtime (Big-Oh) of this algorithm? How could we empirically test this?

What is wasteful about the algorithm? How could it be improved?

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More observations The preceding algorithm is a brute force algorithm, but

it isn't even the best brute force algorithm.

We are redundantly re-computing a large number of values many times.

For example, we compute the sum of the subsequence between indexes 2 and 5,A[2] + A[3] + A[4] + A[5]

next we compute the sum of the subsequence between indexes 2 and 6.A[2] + A[3] + A[4] + A[5] + A[6]

We already had computed the sum of 2--5, but we compute it again as part of the 2--6 computation.

Assuming from the RAM model that each addition takes 1 unit of time, we are wasting 4 add operations!

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still try all possible combinations, but don't redundantly add the sums

key observation:

in other words, we don't need to throw away partial sums

can we use this information to remove one of the loops from our algorithm?

// implement together

function maxSubsequence2(array[]):

What is the runtime (Big-Oh) of this new algorithm? Can it still be improved further?

1j

ikkj

j

ikk AAA

Second algorithm (improved)

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Observations: Claim 1 To make our code even faster, we must avoid trying all

possible combinations; to do this, we must find a way to broadly eliminate many potential combinations from consideration.

Claim #1: A subsequence with a negative sum cannot be the start of the maximum-sum subsequence.

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Claim #1, more formally: If Ai, j is a subsequence such that ,

then there is no q such that Ai,q is the maximum-sum subsequence.

Proof: (do it together in class)

Can this help us produce a better algorithm?

0

j

ikkA

Claim 1, continued

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Claim #2: when examining subsequences left-to-right, for some starting index i, if Ai,j becomes the first subsequence starting with i ,

such that

Then no part of Ai,j can be part of the maximum-sum subsequence.

(Why is this a stronger claim than Claim #1?)

Proof: (do it together in class)

0

j

ikkA

Claim 2

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Claim 2, continued

These figures show the possible contents of Ai,j

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Can we eliminate another loop from our algorithm?

// implement together

function maxSubsequence3(array[]):

What is its runtime (Big-Oh)?

Is there an even better algorithm than this third algorithm? Can you make a strong argument about why or why not?

Third (best) algorithm

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Express the running time as f(N), where N is the size of the input

worst case: your enemy gets to pick the input

average case: need to assume a probability distribution on the inputs

amortized: your enemy gets to pick the inputs/operations, but you only have to guarantee speed over a large number of operations

Types of runtime analysis