1 Algorithms CS 202 Epp section ??? Aaron Bloomfield.

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AlgorithmsAlgorithms

CS 202CS 202

Epp section ???Epp section ???

Aaron BloomfieldAaron Bloomfield

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What is an algorithm?What is an algorithm?

An algorithm is “a finite set of precise An algorithm is “a finite set of precise instructions for performing a computation or for instructions for performing a computation or for solving a problem”solving a problem” A program is one type of algorithmA program is one type of algorithm

All programs are algorithmsAll programs are algorithms

Not all algorithms are programs!Not all algorithms are programs! Directions to somebody’s house is an algorithmDirections to somebody’s house is an algorithm A recipe for cooking a cake is an algorithmA recipe for cooking a cake is an algorithm The steps to compute the cosine of 90The steps to compute the cosine of 90°° is an is an

algorithmalgorithm

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Some algorithms are harder than Some algorithms are harder than othersothers

Some algorithms are easySome algorithms are easy Finding the largest (or smallest) value in a listFinding the largest (or smallest) value in a list Finding a specific value in a listFinding a specific value in a list

Some algorithms are a bit harderSome algorithms are a bit harder Sorting a listSorting a list

Some algorithms are very hardSome algorithms are very hard Finding the shortest path between Miami and SeattleFinding the shortest path between Miami and Seattle

Some algorithms are essentially impossibleSome algorithms are essentially impossible Factoring large composite numbersFactoring large composite numbers

In section 2.2, we’ll see how to rate how “hard” In section 2.2, we’ll see how to rate how “hard” algorithms arealgorithms are

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Algorithm 1: Maximum elementAlgorithm 1: Maximum element

Given a list, how do we find the maximum Given a list, how do we find the maximum element in the list?element in the list?

To express the algorithm, we’ll use To express the algorithm, we’ll use pseudocodepseudocode Pseudocode is kinda like a programming Pseudocode is kinda like a programming

language, but not reallylanguage, but not really

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Algorithm for finding the maximum Algorithm for finding the maximum element in a list:element in a list:

procedureprocedure max ( max (aa11, , aa22, …, , …, aann: integers): integers)

maxmax := := aa11

forfor i := 2 i := 2 toto n n

ifif max < max < aaii thenthen maxmax := := aaii

{{maxmax is the largest element} is the largest element}

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procedureprocedure max ( max (aa11, , aa22, …, , …, aann: integers): integers)

maxmax := := aa11

forfor ii := 2 := 2 toto n n


maxmax := := aa11

forfor ii := 2 := 2 toto n n



44 11 77 00 55 22 99 33 66 88

aa11 aa22 aa33 aa44 aa55 aa66 aa77 aa88 aa99 aa1010

maxmax

ii 2345678910

479

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Maximum element running timeMaximum element running time

How long does this take?How long does this take?

If the list has If the list has nn elements, worst case elements, worst case scenario is that it takes scenario is that it takes nn “steps” “steps” Here, a step is considered a single step Here, a step is considered a single step

through the listthrough the list

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Properties of algorithmsProperties of algorithms

Algorithms generally share a set of properties:Algorithms generally share a set of properties: Input: what the algorithm takes in as inputInput: what the algorithm takes in as input Output: what the algorithm produces as outputOutput: what the algorithm produces as output Definiteness: the steps are defined preciselyDefiniteness: the steps are defined precisely Correctness: should produce the correct outputCorrectness: should produce the correct output Finiteness: the steps required should be finiteFiniteness: the steps required should be finite Effectiveness: each step must be able to be Effectiveness: each step must be able to be

performed in a finite amount of timeperformed in a finite amount of time Generality: the algorithm Generality: the algorithm shouldshould be applicable to all be applicable to all

problems of a similar formproblems of a similar form

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Searching algorithmsSearching algorithms

Given a list, find a specific element in the Given a list, find a specific element in the listlist

We will see two typesWe will see two types Linear searchLinear search

a.k.a. sequential searcha.k.a. sequential search Binary searchBinary search

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Algorithm 2: Linear searchAlgorithm 2: Linear search

Given a list, find a specific element in the listGiven a list, find a specific element in the list List does NOT have to be sorted!List does NOT have to be sorted!

procedureprocedure linear_search ( linear_search (xx: integer; : integer; aa11, , aa22, …, , …, aann: : integers)integers)

ii := 1 := 1whilewhile ( ( ii ≤ ≤ nn and and xx ≠ ≠ aaii ) )

ii := := ii + 1 + 1ifif ii ≤≤ nn thenthen locationlocation := i := ielseelse locationlocation := 0 := 0

{{locationlocation is the subscript of the term that equals x, or it is the subscript of the term that equals x, or it is 0 if x is not found}is 0 if x is not found}

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procedureprocedure linear_search ( linear_search (xx: integer; : integer; aa11, , aa22, …, , …, aann: integers): integers)

ii := 1 := 1

whilewhile ( ( ii ≤ ≤ nn and and xx ≠ ≠ aaii ) )

ii := := ii + 1 + 1

ifif ii ≤≤ nn thenthen locationlocation := i := i

elseelse locationlocation := 0 := 0

ii := 1 := 1


ii := := ii + 1 + 1



Algorithm 2: Linear search, take 1Algorithm 2: Linear search, take 1

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ii 23456781

xx 3

locationlocation 8

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procedureprocedure linear_search ( linear_search (xx: integer; : integer; aa11, , aa22, …, , …, aann: integers): integers)

ii := 1 := 1


ii := := ii + 1 + 1



ii := 1 := 1


ii := := ii + 1 + 1



Algorithm 2: Linear search, take 2Algorithm 2: Linear search, take 2

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ii 23456789101

xx 11

locationlocation 0

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Linear search running timeLinear search running time

How long does this take?How long does this take?

If the list has If the list has nn elements, worst case elements, worst case scenario is that it takes scenario is that it takes nn “steps” “steps” Here, a step is considered a single step Here, a step is considered a single step

through the listthrough the list

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Algorithm 3: Binary searchAlgorithm 3: Binary search

Given a list, find a specific element in the listGiven a list, find a specific element in the list List MUST be sorted!List MUST be sorted!

Each time it iterates through, it cuts the list in halfEach time it iterates through, it cuts the list in half

procedureprocedure binary_search ( binary_search (xx: integer; : integer; aa11, , aa22, …, , …, aann: increasing integers): increasing integers)ii := 1 := 1 { { ii is left endpoint of search interval } is left endpoint of search interval }jj := := nn { { jj is right endpoint of search interval } is right endpoint of search interval }whilewhile ii < < jjbeginbegin

m := m := ((ii++jj)/2)/2 { { mm is the point in the middle } is the point in the middle }if if x > x > aamm thenthen ii := := mm+1+1elseelse j j := := mm

endendifif xx = = aaii thenthen locationlocation := := iielseelse locationlocation := 0 := 0

{{locationlocation is the subscript of the term that equals x, or it is 0 if x is not found} 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 1Algorithm 3: Binary search, take 1

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ii jjmm

ii := 1 := 1

jj := := nn

procedureprocedure binary_search ( binary_search (xx: integer; : integer; aa11, , aa22, …, , …, aann: increasing integers): increasing integers)

whilewhile ii < < jj

beginbegin

m := m := ((ii++jj)/2)/2if if x > x > aamm thenthen ii := := mm+1+1

elseelse j j := := mm

endend

ifif xx = = aaii thenthen locationlocation := := ii


ii := 1 := 1

jj := := nn


beginbegin



endend


1

xx 14

1056 8 87 767

locationlocation 7

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Algorithm 3: Binary search, take 2Algorithm 3: Binary search, take 2

22 44 66 88 1010 1212 1414 1616 1818 2020


ii jjmm

ii := 1 := 1

jj := := nn

procedureprocedure binary_search ( binary_search (xx: integer; : integer; aa11, , aa22, …, , …, aann: increasing integers): increasing integers)


beginbegin



endend



ii := 1 := 1

jj := := nn


beginbegin



endend

ifif xx = = aaii thenthen locationlocation := := II


1

xx 15

1056 8 87

locationlocation 0

8

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Algorithm 3: Binary searchAlgorithm 3: Binary search

A somewhat alternative view of what a A somewhat alternative view of what a binary search does…binary search does…

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Binary search running timeBinary search running time

How long does this take (worst case)?How long does this take (worst case)?

If the list has 8 elementsIf the list has 8 elements It takes 3 stepsIt takes 3 steps



If the list has If the list has nn elements elements It takes logIt takes log22 nn steps steps

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Sorting algorithmsSorting algorithms

Given a list, put it into some orderGiven a list, put it into some order Numerical, lexicographic, etc.Numerical, lexicographic, etc.

We will see two typesWe will see two types Bubble sortBubble sort Insertion sortInsertion sort

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Algorithm 4: Bubble sortAlgorithm 4: Bubble sort

One of the most simple sorting algorithmsOne of the most simple sorting algorithms Also one of the least efficientAlso one of the least efficient

It takes successive elements and “bubbles” them It takes successive elements and “bubbles” them up the listup the list

procedureprocedure bubble_sort ( bubble_sort (aa11, , aa22, …, , …, aann))

for for ii := 1 := 1 to to nn-1-1forfor jj := 1 := 1 toto nn--ii

ifif aajj > > aajj+1+1

then then interchange interchange aajj and and aajj+1+1

{ { aa11, …, , …, aann are in increasing order } are in increasing order }

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Algorithm 4: Bubble sortAlgorithm 4: Bubble sort

An example using physical objects…An example using physical objects…

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Bubble sort running timeBubble sort running time

Bubble sort algorithm:Bubble sort algorithm:for for ii := 1 := 1 to to nn-1-1

forfor jj := 1 := 1 toto nn--ii

ifif aajj > > aajj+1+1

then then interchange interchange aajj and and aajj+1+1

Outer for loop does n-1 iterationsOuter for loop does n-1 iterationsInner for loop does Inner for loop does

nn-1 iterations the first time-1 iterations the first time nn-2 iterations the second time-2 iterations the second time …… 1 iteration the last time1 iteration the last time

Total: (Total: (nn-1) + (-1) + (nn-2) + (-2) + (nn-3) + … + 2 + 1 = (-3) + … + 2 + 1 = (nn22--nn)/2)/2 We can say that’s “about” We can say that’s “about” nn22 time time

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Algorithm 5: Insertion sortAlgorithm 5: Insertion sort

Another simple (and inefficient) algorithmAnother simple (and inefficient) algorithmIt starts with a list with one element, and inserts new elements into It starts with a list with one element, and inserts new elements into their proper place in the sorted part of the listtheir proper place in the sorted part of the list

procedureprocedure insertion_sort ( insertion_sort (aa11, , aa22, …, , …, aann))forfor jj := 2 := 2 toto nnbeginbegin

ii := 1 := 1whilewhile aajj > > aaii

ii := := ii +1 +1mm := := aajj

forfor kk := 0 := 0 toto jj--ii-1-1aajj--kk := := aajj--kk-1-1

aaii := := mmendend { { aa11, , aa22, …, , …, aann are sorted } are sorted }

take successive elements in the list

find where that element should be in the sorted portion of the list

move all elements in the sorted portion of the list that are greater

than the current element up by one

put the current element into it’s proper place in the sorted portion of the list

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Insertion sort running timeInsertion sort running time

forfor jj := 2 := 2 toto n n beginbeginii := 1 := 1whilewhile aajj > > aaii

ii := := ii +1 +1mm := := aajj

forfor kk := 0 := 0 toto jj--ii-1-1aajj--kk := := aajj--kk-1-1

aaii := := mmendend { { aa11, , aa22, …, , …, aann are sorted } are sorted }Outer for loop runs Outer for loop runs nn-1 times-1 timesIn the inner for loop:In the inner for loop:

Worst case is when the while keeps Worst case is when the while keeps ii at 1, and the for loop runs lots of at 1, and the for loop runs lots of timestimes

If If ii is 1, the inner for loop runs 1 time ( is 1, the inner for loop runs 1 time (kk goes from 0 to 0) on the first goes from 0 to 0) on the first iteration, 1 time on the second, up to iteration, 1 time on the second, up to nn-2 times on the last iteration-2 times on the last iteration

Total is 1 + 2 + … + Total is 1 + 2 + … + nn-2 = (-2 = (nn-1)(-1)(nn-2)/2-2)/2 We can say that’s “about” We can say that’s “about” nn22 time time

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Comparison of running timesComparison of running times

SearchesSearches Linear: Linear: nn steps steps Binary: logBinary: log22 nn steps steps Binary search is about as fast as you can getBinary search is about as fast as you can get

SortsSorts Bubble: Bubble: nn22 steps steps Insertion: Insertion: nn22 steps steps There are other, more efficient, sorting techniquesThere are other, more efficient, sorting techniques

In principle, the fastest are heap sort, quick sort, and merge In principle, the fastest are heap sort, quick sort, and merge sortsortThese each take take These each take take nn * log * log22 nn steps stepsIn practice, quick sort is the fastest, followed by merge sortIn practice, quick sort is the fastest, followed by merge sort

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