Analysis of AlgorithmsAnalysis of Algorithms(complexity)(complexity)

Search Example:Search Example:

Finding a Name in a ListFinding a Name in a ListConsider a list of N names ordered randomly.Consider a list of N names ordered randomly.

1.1.How many comparisons must be made in order to find if a given How many comparisons must be made in order to find if a given name is in the list?name is in the list?

2.2.What if the list is in alphabetic order?What if the list is in alphabetic order?

JohnSarahBobFrankSallyAnitaBarbaraSamFredWilmaAlanHaroldCarolMikeTomBeth

AlanAnitaBarbaraBethBobCarolFrankFredHaroldJohnMikeSallySamSarahTomWilma

Algorithm SequentialSearch (of an array)found=“not”

for i = 1 to number of items in the array

if array(i) = desired element

found=“”

i = number of items in array + 1

end if

next i

print “Element was ”; found; “ found”

JohnSarahBobFrankSallyAnitaBarbaraSamFredWilmaAlanHaroldCarolMikeTomBeth

Algorithm binarySearch (a, first, last, desiredItem)mid = (first + last)/2 // approximate midpoint of arrayif (first > last)

return falseelse if (desiredItem equals a[mid])

return trueelse if (desiredItem < a[mid])

return binarySearch (a, first, mid-1, desiredItem)else // desiredItem > a[mid]

return binarySearch (a, mid+1, last, desiredItem)

AlanAnitaBarbaraBethBobCarolFrankFredHaroldJohnMikeSallySamSarahTomWilma

Search Example:Search Example:

The Traveling Salesman ProblemThe Traveling Salesman Problem

What is the shortest route a salesman can take What is the shortest route a salesman can take to visit all of the cities in his territory and return to visit all of the cities in his territory and return home?home?

Search Example:Search Example:

The Traveling Salesman ProblemThe Traveling Salesman Problem

What is the shortest route a salesman can What is the shortest route a salesman can take to visit all of the cities in his territory take to visit all of the cities in his territory and return home?and return home?

The (Real) Traveling Salesman “Problem”The (Real) Traveling Salesman “Problem”

As the number of cities increases, the time it takes to As the number of cities increases, the time it takes to find an exact solution increases exponentially.find an exact solution increases exponentially.

Example:Example:

Number of citiesNumber of cities # paths# paths Time to solve (on a fast PC)Time to solve (on a fast PC)

88 25202520 almost instantaneouslyalmost instantaneously

1010 181,440181,440 1 second1 second

1212 20 million 20 million 20 seconds20 seconds

… … …… ……

2020 60,800,000,000,000,00060,800,000,000,000,000 ??

100100 4.67 x 104.67 x 10157157 ??

Intractable!Intractable!

Chess

Carrano (2006), Data Structures and Abstractions with Java

How Can We put a List in Alphabetic order?How Can We put a List in Alphabetic order?

JohnSarahBobFrankSallyAnitaBarbaraSamFredWilmaAlanHaroldCarolMikeTomBeth

How much effort is required?How much effort is required?

Eight Puzzle

Eight Puzzle

30 shuffle moves

Compare:best-first with depth factor: 26 moves to solution (7725 in CLOSED; 7498 left in OPEN)best-first without depth factor: 48 moves to solution (272 in CLOSED; 332 in OPEN)

Shown below is a screen shot of a breath-first search in process for the 15-puzzle. The optimal solution is known (by another method) to be located at depth 26. How long will it take the breadth-first algorithm to discover this?

Shown below is a screen shot of a breath-first search in process for the 15-puzzle. The optimal solution is known (by another method) to be located at depth 26. How long will it take the breadth-first algorithm to discover this?

Time ti spent at level i ≈4*ti-1 (e.g., 53278/13034=4.09, 13034/3171=4.11,

3171/782=4.05). The search needs to go 26-16 = 10 levels deeper. So, it will take approximately 53278*49 to 53278*410 seconds to find the optimal solution (depending upon whether it finds it at the beginning of the search of level 26 or at the end). Taking the lower limit, we get 53278*49 seconds/(3600sec/hr)/(24hrs/day)/ (365days/yr) = 443 years. A similar calculation for the upper limit (or just multiply by 4) gives 1772 years. So, between 443 and 1772 years!

Complexity ComparisonComplexity Comparison

Carrano (2006), Data Structures and Abstractions with Java

Let f and g be functions mapping nonnegative reals into nonnegative reals. Then f is BIG OH of g, written f = O(g), if there exist positive constants x0 and c such that for x ≥ x0, f(x) ≤ cg(x). In this case, an algorithm's time requirement f(x) is of order at most g(x).

http://upload.wikimedia.org/wikipedia/commons/8/89/Big-O-notation.png

Consider a pattern matching algorithm whose purpose is to find all occurrences of string A in string B where A < B. What is the complexity of the task?

Consider a pattern matching algorithm whose purpose is to find all occurrences of string A in string B where A < B. What is the complexity of the task?

Identify the factors that determine the complexity…

Consider a pattern matching algorithm whose purpose is to find all occurrences of string A in string B where A < B. What is the complexity of the task?

Identify the factors that determine the complexity…Length of each stringAlgorithm used to perform the match

Consider a pattern matching algorithm whose purpose is to find all occurrences of string A in string B where A < B. What is the complexity of the task?

Identify the factors that determine the complexity…Length of each stringAlgorithm used to perform the match

Suggest some techniques and evaluate each…

Data StructuresData StructuresListsLists

StacksStacks

QueuesQueues

TreesTrees

GraphsGraphs

AA BB CC DD

DD

CC

BB

AA

AA BB CC DD