CS 150: Analysis of Algorithms

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Goals for this Unit

• Begin a focus on data structures and algorithms

• Understand the nature of the performance of algorithms

• Understand how we measure performance– Used to describe performance of various data

structures• Begin to see the role of algorithms in the

study of Computer Science

Algorithm• An algorithm is

– a detailed step-by-step method for– solving a problem

• Computer algorithms• Properties of algorithms

– Steps are precisely stated– Determinism: based on inputs, previous

steps– Algorithm terminates– Also: correctness, generality, efficiency

C_M_L_X_T_

1. R O U F O2. U B I R A3. O U M A J4. I P E Y O

This is an important CS term for this unit. Can you guess? Choose a the set of letters below (their order is jumbled).

What makes an algorithm “better”?

• Let’s look again at the Fibonacci example

• Why does one version run much much slower than the other?

• If recursion is this slow, why ever use it?

Why Not Just Time Algorithms?• We want a measure of work that

gives us a direct measure of the efficiency of the algorithm – independent of computer, programming

language, programmer, and other implementation details.

– Usually depending on the size of the input

– Also often dependent on the nature of the input• Best-case, worst-case, average

Analysis of Algorithms

• Use mathematics as our tool for analyzing algorithm performance– Measure the algorithm itself, its nature– Not its implementation or its execution

• Need to count something– Cost or number of steps is a function of

input size n: e.g. for input size n, cost is f(n)

– Count all steps in an algorithm? (Hopefully avoid this!)

Counting Operations• Strategy: choose one operation or one

section of code such that– The total work is always roughly proportional to

how often that’s done• So we’ll just count:

– An algorithm’s “basic operation”– Or, an algorithms’ “critical section”

• Sometimes the basic operation is some action that’s fundamentally central to how the algorithm works– Example: Search a List for a target involves

comparing each list-item to the target.– The comparison operation is “fundamental.”

Asymptotic Analysis• Given some formula f(n) for the count/cost of

some thing based on the input size– We’re going to focus on its “order”– f(n) = 2n ---> Exponential function– f(n) = 100n2 + 50n + 7 ---> Quadratic function– f(n) = 30 n lg n – 10 ---> Log-linear function– f(n) = 1000n ---> Linear function

• These functions grow at different rates– As inputs get larger, the amount they increase

differs• “Asymptotic” – how do things change as the

input size n gets larger?

Comparison of Growth Rates

Comparison of Growth Rates (2)

0

250

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

f(n) = n

f(n) = log(n)

f(n) = n log(n)f(n) = n 2̂

f(n) = n 3̂

f(n) = 2 n̂

Comparison of Growth Rates (3)

0

1000

1 3 5 7 9 11 13 15 17 19

f(n) = n

f(n) = log(n)

f(n) = n log(n)f(n) = n 2̂

f(n) = n 3̂

f(n) = 2 n̂

Order Classes• For a given algorithm, we count something:

– f(n) = 100n2 + 50n + 7 ---> Quadratic function– How different is this than this?

f(n) = 20n2 + 7n + 2– For large inputs?

• Order class: a “label” for all functions with the same highest-order term– Label form: O(n2) or Θ(n2) or a few others– “Big-Oh” used most often

Growth Notations

• g O(f) (“Big-Oh”)g grows no faster than f (upper bound)

• g (f) (“Theta”)g grows as fast as f (tight bound)

• g (f) (“Omega”)g grows no slower than f (lower bound)Which one would we most like to know?

Meaning of O (“big Oh”)

g is in O (f) iff:There are positive constantsc and n0 such that g(n) ≤ cf(n) for all n ≥ n0.

O Examples

Is n in O (n2)? Yes, c = 1 and n0=1 works.Is 10n in O (n)? Yes, c = .09 and n0=1 works.

Is n2 in O (n)? No, no matter what c we pick,cn2 > n for big enough n (n > c)

g is in O (f) iff there are positive constantsc and n0 such that g(n) ≤ cf(n) for all n ≥ n0.

Back to Order Classes• Order classes group “equivalently” efficient

algorithms– O(1) – constant time! Input size doesn’t matter– O(lg n) – logarithmic time. Very efficient. E.g.

binary search (after sorting)– O(n) – linear time– O(n lg n) – log-linear time. E.g. best sorting

algorithms– O(n2) – quadratic time. E.g. poorer sorting

algorithms– O(n3) – cubic time– ….– O(2n) – exponential time. Many important

problems, often about optimization.

Ω (“Omega”): Lower Bound

g is in Ω (f) iff there are positive constants c and n0 such that

for all n ≥ n0.

g is in O (f) iff there are positive constantsc and n0 such that g(n) ≤ cf(n) for all n ≥ n0.

g(n) ≥ cf(n)

Example: Watch Code Run

• Two implementations to calculateFibonacci numbers: F(0) = F(1) = 1

F(n) = F(n-1) + F(n-2) for n > 1• Both correct!• Let’s run both for n=8, 16, 32, 64

When Does this Matter?

• Size of input matters a lot!– For small inputs, we care a lot less– But what’s a big input?

• Hard to know. For some algorithms, smaller than you think!

Two Important Problems• Search

– Given a list of items and a target value– Find if/where it is in the list

• Return special value if not there (“sentinel” value)– Note we’ve specified this at an abstract level

• Haven’t said how to implement this• Is this specification complete? Why not?

• Sorting– Given a list of items– Re-arrange them in some non-decreasing order

• With solutions to these two, we can do many useful things!

• What to count for complexity analysis? For both, the basic operation is: comparing two list-elements

Sequential Search Algorithm

• Sequential Search– AKA linear search– Look through the list until we reach the

end or find the target– Best-case? Worst-case?– Order class: O(n)– Advantages: simple to code, no

assumptions about list

Binary Search Algorithm• Binary Search

– Input list must be sorted– Strategy: Eliminate about half items left with

one comparison• Look in the middle• If target larger, must be in the 2nd half• If target smaller, must be in the 1st half

• Complexity: O(lg n)• Must sort list first, but…• Much more efficient than sequential search

– Especially if search is done many times (sort once, search many times)

• Note: Java provides static binarySearch()

Class Activity

• We’ll give you some binary search inputs– Array of int values, plus a target to find– You tell us

• what index is returned, and• the sequence of index values that were

compared to the target to get that answer• Work in two’s or three’s, and we’ll

call on groups to explain

Binary Search Example #1

Input: -1 4 5 11 13 and target 4Index returned is?Vote on indices (positions) compared to target-value:

1. -1 42. 0 13. 5 -1 44. 2 0 15. other

Binary Search Example #2

Input: -5 -2 -1 4 5 11 13 14 17 18 and target 3Index returned is?Vote on indices compared below:

1. 5 2 3 42. 4 1 2 33. 4 2 3 44. other

Binary Search Example #3

Input: -1 4 5 11 13 and target 13Index returned is?Vote on indices compared below:

1. 0 1 2 3 42. 2 43. 2 3 44. other

Note on last examples

• Input size doubled from 5 to 10• How did the worst-case number of

comparisons change?– Also double?– Something else

• Note binary search is O(lg n)– What does this imply here?

How to Sort?• Many sorting algorithms have been found!

– Problem is a case-study in algorithm design• Some “straightforward” sorts

– Insertion Sort, Selection Sort, Bubble Sort– O(n2)

• More efficient sorts– Quicksort, mergesort, heapsort– O(n lg n)

• Note: these are for sorting in memory, not on disk

Reminder Slide: Order Classes• For a given algorithm, we count something:

– f(n) = 100n2 + 50n + 7 ---> Quadratic function– How different is this than this?

f(n) = 20n2 + 7n + 2– For large inputs?

• Order class: a “label” for all functions with the same highest-order term– Label form: O(n2) or Θ(n2) or a few others– “Big-Oh” used most often

Order Classes Details• What does the label mean? O(n2)

– Set of all functions that grow at the same rate as n2

or more slowly– I.e. as efficient as any “n2” or more efficient,

but no worse– So this is an upper-bound on how inefficient an

algorithm can be• Usage: We might say: Algorithm A is O(n2)

– Means Algorithm A’s efficiency grows like a quadratic algorithm or grows more slowly. (As good or better)

• What about that other label, Θ(n2)?– Set of all functions that grow at exactly the same

rate– A more precise bound

Reminders: Input and Performance

• Input matters:– First, we said for small size of inputs,

often no difference• Also, often focus on the worst-case

input– Why?

• Sometimes interested in the average-case input– Why?

• Also the best-case, but do we care?

Reminder: What to Count

• Often count some “basic operation”• Or, we count a “critical section”• Examples:

– The block of code most deeply nested in a nested set of loops

– An operation like comparison in sorting– An expensive operation like

multiplication or database query

Back to Order Classes• Order classes group “equivalently” efficient

algorithms– O(1) – constant time! Input size doesn’t matter– O(lg n) – logarithmic time. Very efficient. E.g.

binary search (after sorting)– O(n) – linear time– O(n lg n) – log-linear time. E.g. best sorting

algorithms– O(n2) – quadratic time. E.g. poorer sorting

algorithms– O(n3) – cubic time– ….– O(2n) – exponential time. Many important

problems, often about optimization.

Discussion Question

• Binary search is faster than sequential search– But extra cost! Must sort the list first!

• When do you think it’s worth it?

Bigger Issues in Complexity• Many practical problems seem to require

exponential time– Θ(2n)– This grows much faster more quickly than any

polynomial– Such problems are called intractable problems– A famous subset of these are called

NP-complete problems• Most famous theoretical question in CS:

– Is it really impossible to find a polynomial algorithm for the NP-complete problems? (Does P=NP?)

Example• Weighted-graph problems• Find the cheapest way to connect things?

– O(n2)• Find the shortest path between two

points? All pairs of points?– O(n2) or better

• But, find the shortest path that visits all points in the graph?– Best we can do is exponential, Θ(2n) or worse– Is it impossible to do better? No one knows!

Summary and Major Points• When we measure algorithm complexity:

– Base this on size of input– Count some basic operation or how often a

critical section is executed– Get a formula f(n) for this– Then we think about it in terms of its “label”,

the order class O( f(n) )• “Big-Oh” means as efficient as “class f(n) or better

– We usually use order-class to compare algorithms

– We can measure worst-case, average-case