Transcript Algorithms ( § 2.1)

Algorithms
CS/APMA 202
Rosen section 2.1
Aaron Bloomfield
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What is an algorithm?
An algorithm is “a finite set of precise
instructions for performing a computation or for
solving a problem”
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A program is one type of algorithm
All programs are algorithms
Not all algorithms are programs!
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Directions to somebody’s house is an algorithm
A recipe for cooking a cake is an algorithm
The steps to compute the cosine of 90° is an
algorithm
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Some algorithms are harder than
others
Some algorithms are easy
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Finding the largest (or smallest) value in a list
Finding a specific value in a list
Some algorithms are a bit harder
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Sorting a list
Some algorithms are very hard
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Finding the shortest path between Miami and Seattle
Some algorithms are essentially impossible
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Factoring large composite numbers
In section 2.2, we’ll see how to rate how “hard”
algorithms are
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Algorithm 1: Maximum element
Given a list, how do we find the maximum
element in the list?
To express
pseudocode
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the
algorithm,
we’ll
use
Pseudocode is kinda like a programming
language, but not really
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Algorithm 1: Maximum element
Algorithm for finding
element in a list:
the
maximum
procedure max (a1, a2, …, an: integers)
max := a1
for i := 2 to n
if max < ai then max := ai
{max is the largest element}
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Algorithm 1: Maximum element
procedure max (a1, a2, …, an: integers)
max := a1
for i := 2 to n
max
if max < ai then max := ai
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Maximum element running time
How long does this take?
If the list has n elements, worst case
scenario is that it takes n “steps”
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Here, a step is considered a single step
through the list
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Properties of algorithms
Algorithms generally share a set of properties:
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Input: what the algorithm takes in as input
Output: what the algorithm produces as output
Definiteness: the steps are defined precisely
Correctness: should produce the correct output
Finiteness: the steps required should be finite
Effectiveness: each step must be able to be
performed in a finite amount of time
Generality: the algorithm should be applicable to all
problems of a similar form
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Searching algorithms
Given a list, find a specific element in the
list
We will see two types
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Linear search
a.k.a. sequential search
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Binary search
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Algorithm 2: Linear search
Given a list, find a specific element in the list
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List does NOT have to be sorted!
procedure linear_search (x: integer; a1, a2, …, an:
integers)
i := 1
while ( i ≤ n and x ≠ ai )
i := i + 1
if i ≤ n then location := i
else location := 0
{location is the subscript of the term that equals x, or it
is 0 if x is not found}
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Algorithm 2: Linear search, take 1
procedure linear_search (x: integer; a1, a2, …, an: integers)
i := 1
while ( i ≤ n and x ≠ ai )
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i := i + 1
if i ≤ n then location := i
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else location := 0
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Algorithm 2: Linear search, take 2
procedure linear_search (x: integer; a1, a2, …, an: integers)
i := 1
while ( i ≤ n and x ≠ ai )
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i := i + 1
if i ≤ n then location := i
location 0
else location := 0
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Linear search running time
How long does this take?
If the list has n elements, worst case
scenario is that it takes n “steps”
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Here, a step is considered a single step
through the list
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Algorithm 3: Binary search
Given a list, find a specific element in the list
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List MUST be sorted!
Each time it iterates through, it cuts the list in half
procedure binary_search (x: integer; a1, a2, …, an: increasing integers)
i := 1
{ i is left endpoint of search interval }
j := n
{ j is right endpoint of search interval }
while i < j
begin
m := (i+j)/2
{ m is the point in the middle }
if x > am then i := m+1
else j := m
end
if x = ai then location := i
else location := 0
{location is the subscript of the term that equals x, or it is 0 if x is not found}
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Algorithm 3: Binary search, take 1
procedure binary_search (x: integer; a1, a2, …, an: increasing integers)
i := 1
j := n
while i < j
begin
m := (i+j)/2
if x > am then i := m+1
else j := m
end
if x = ai then location := i
else location := 0
x
location
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Algorithm 3: Binary search, take 2
procedure binary_search (x: integer; a1, a2, …, an: increasing integers)
i := 1
j := n
while i < j
begin
m := (i+j)/2
if x > am then i := m+1
else j := m
end
if x = ai then location := Ii
else location := 0
x
location
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Algorithm 3: Binary search
A somewhat alternative view of what a
binary search does…
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Quick survey
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a)
b)
c)
d)
I felt I understood the search algorithms…
Very well
With some review, I’ll be good
Not really
Not at all
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Today’s demotivators
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Binary search running time
How long does this take (worst case)?
If the list has 8 elements
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It takes 3 steps
If the list has 16 elements
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It takes 4 steps
If the list has 64 elements
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It takes 6 steps
If the list has n elements
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It takes log2 n steps
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Sorting algorithms
Given a list, put it into some order
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Numerical, lexicographic, etc.
We will see two types
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Bubble sort
Insertion sort
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Algorithm 4: Bubble sort
One of the most simple sorting algorithms
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Also one of the least efficient
It takes successive elements and “bubbles” them
up the list
procedure bubble_sort (a1, a2, …, an)
for i := 1 to n-1
for j := 1 to n-i
if aj > aj+1
then interchange aj and aj+1
{ a1, …, an are in increasing order }
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Algorithm 4: Bubble sort
An example using physical objects…
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Bubble sort running time
Bubble sort algorithm:
for i := 1 to n-1
for j := 1 to n-i
if aj > aj+1
then interchange aj and aj+1
Outer for loop does n-1 iterations
Inner for loop does
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n-1 iterations the first time
n-2 iterations the second time
…
1 iteration the last time
Total: (n-1) + (n-2) + (n-3) + … + 2 + 1 = (n2-n)/2
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We can say that’s “about” n2 time
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Algorithm 5: Insertion sort
Another simple (and inefficient) algorithm
It starts with a list with one element, and inserts new elements into
their proper place in the sorted part of the list
procedure insertion_sort (a1, a2, …, an)
for j := 2 to n take successive elements in the list
begin
i := 1
find where that element should be
while aj > ai
in the sorted portion of the list
i := i +1
m := aj
move all elements in the sorted
for k := 0 to j-i-1
portion of the list that are greater
aj-k := aj-k-1
than the current element up by one
ai := m
end { a1, a2, …, an are sorted }
put the current element into it’s proper place in the sorted portion of the list26
Insertion sort running time
for j := 2 to n begin
i := 1
while aj > ai
i := i +1
m := aj
for k := 0 to j-i-1
aj-k := aj-k-1
ai := m
end { a1, a2, …, an are sorted }
Outer for loop runs n-1 times
In the inner for loop:
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Worst case is when the while keeps i at 1, and the for loop runs lots of
times
If i is 1, the inner for loop runs 1 time (k goes from 0 to 0) on the first
iteration, 1 time on the second, up to n-2 times on the last iteration
Total is 1 + 2 + … + n-2 = (n-1)(n-2)/2
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We can say that’s “about” n2 time
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Quick survey
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a)
b)
c)
d)
I felt I understood the sort algorithms…
Very well
With some review, I’ll be good
Not really
Not at all
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Comparison of running times
Searches
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Linear: n steps
Binary: log2 n steps
Binary search is about as fast as you can get
Sorts
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Bubble: n2 steps
Insertion: n2 steps
There are other, more efficient, sorting techniques
In principle, the fastest are heap sort, quick sort, and merge
sort
These each take take n * log2 n steps
In practice, quick sort is the fastest, followed by merge sort
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Greedy algorithms
Ignore this part…
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Quick survey
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a)
b)
c)
d)
I felt I understood the material in this slide set…
Very well
With some review, I’ll be good
Not really
Not at all
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Quick survey
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a)
b)
c)
d)
The pace of the lecture for this slide set was…
Fast
About right
A little slow
Too slow
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Quick survey
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a)
b)
c)
d)
How interesting was the material in this slide
set? Be honest!
Wow! That was SOOOOOO cool!
Somewhat interesting
Rather borting
Zzzzzzzzzzz
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