# CS2110: SW Development Methods Analysis of Algorithms ```CS2110: SW Development Methods
Analysis of Algorithms
• Reading: Chapter 5 of MSD text
– Except Section 5.5 on recursion (next unit)
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
• Note: CS2110 starts the study of this
– Continued in CS2150 and CS4102
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_
This is an important CS term for
this unit. Can you guess?
Choose a the set of letters below
(their order is jumbled).
1.
2.
3.
4.
ROUFO
UBIRA
OUMAJ
IPEYO
Data Structures
•
Some definitions of data structure:
A. a scheme for organizing related pieces of
information
B. A logical relationship among data elements that is
designed to support specific data manipulation
functions
C. Contiguous memory used to hold an ordered
collection of fields of different types
•
•
(FYI, I think B is the best one of the above.)
Examples: ArrayList, HashSet, trees, tables,
stacks, queues
–
Is a PlayList an example of this?
Abstract Data Types
• Remember data abstraction?
• Important CS concept:
– a model of data items stored, and
– a set of operations that manipulate them
– A particular data structure implements an ADT
• Remember: “data structure” defines how it’s
implemented. A data structure is not abstract.
• Java’s design reflects the idea of ADTs
• Question: Which Java language-feature
most closely supports programming using
1.
2.
3.
4.
A superclass
An abstract class
An interface
A function object (e.g. Comparator)
Why do Java interfaces “match” ADTs?
1. Interfaces don’t have data fields,
so they’re not a good match!
2. Interfaces are good abstractions.
3. Inheritance of interface is
4. An interface can be used without
knowing how the concrete class is
implemented.
• ADTs define operations and a given data structure
implements these
– Think and design using ADTs, then code using data structures
• There may more than one data structure that
– Efficiency / performance is often a major consideration
• Question: how do we compare efficiency of
implementations?
• Answer: We compare the algorithms that implement the
operations.
– In Java: in the concrete class, how data is stored, how methods
are coded
Why Not Just Time Algorithms?
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 ---&gt; Exponential function
f(n) = 100n2 + 50n + 7 ---&gt; Quadratic function
f(n) = 30 n lg n – 10 ---&gt; Log-linear function
f(n) = 1000n ---&gt; 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)
250
f(n) = n
f(n) = log(n)
f(n) = n log(n)
f(n) = n^2
f(n) = n^3
f(n) = 2^n
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Comparison of Growth Rates (3)
1000
f(n) = n
f(n) = log(n)
f(n) = n log(n)
f(n) = n^2
f(n) = n^3
f(n) = 2^n
0
1
3
5
7
9
11
13
15
17
19
Order Classes
• For a given algorithm, we count something:
– f(n) = 100n2 + 50n + 7 ---&gt; 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
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!
Common 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,
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
Earlier we said this about Maps:
• “Lookup values for a key” -- an important
– Java has two concrete classes that implement
Map interface
– HashMap has O(1) lookup cost
– TreeMap is O(lg n) which is still very good
– But TreeMap can print in sorted order in O(n),
and HashMap can’t.
Maximum Sum Subvector
• Input: vector X of N floats (pos. and neg.)
• Output: maximum sum of any contigous
subvector X[i]…X[j]
31 -41 59 26 -53 58 97 -93 -23 84
9/1/02
24
One Solution
maxSoFar = 0
for i = 1 to N
for j = i to N {
sum = 0
for k = i to j {
sum += X[k]
}
maxSoFar = max(maxSoFar, sum)
}
• Complexity? O(n3) You can do better!
9/1/02
25
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
I understand how Binary Search works
1.
2.
3.
4.
5.
Strongly Agree
Agree
Neutral
Disagree
Strongly Disagree
I can explain why Bin Search is efficient
1.
2.
3.
4.
5.
Strongly Agree
Agree
Neutral
Disagree
Strongly Disagree
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
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 4
Index returned is?
Vote on indices (positions) compared to
target-value:
1.
2.
3.
4.
5.
-1 4
0 1
5 -1 4
2 0 1
other
Binary Search Example #2
Input: -5 -2 -1 4 5 11 13 14 17 18
and target 3
Index returned is?
Vote on indices compared below:
1.
2.
3.
4.
5234
4123
4234
other
Binary Search Example #3
Input: -1 4 5 11 13 and target 13
Index returned is?
Vote on indices compared below:
1.
2.
3.
4.
0 1 2 3 4
2 4
2 3 4
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
– You’ll see more of these in CS4102
• 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
Sorting in CS2110
• sort() in Collections and Arrays class
– “under the hood”, it’s something called
mergesort
– Θ(n lg n) worst-case -- as good as we can do
– Use it! But we won’t explain how it works
(yet)
• Later we’ll see how mergesort works
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?
Reminder Slide: Order Classes
• For a given algorithm, we count something:
– f(n) = 100n2 + 50n + 7 ---&gt; 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
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
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
What’s Next?
• Why this topic now?
• Reminder:
– Data structures are design choices that