Publish on 28th October 2019
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Algorithms

Problems, Algorithms, Programs

Problem - a well defined task.Sort a list of numbers.Find a particular item in a list.Find a winning chess move.

Algorithms

A series of precise steps, known to stop eventually, that solve a problem.NOT necessarily tied to computers.There can be many algorithms to solve the same problem.

Characteristics of an Algorithm

Precise steps.Effective steps.Has an output.Terminates eventually.

Trivial Algorithm

Computing an average:Sum up all of the values.Divide the sum by the number of values.

Problems vs. Algorithms vs. Programs

There can be many algorithms that solve the same problem.There can be many programs that implement the same algorithm.We are concerned with:Analyzing the difficulty of problems.Finding good algorithms.Analyzing the efficiency of algorithms.

Example: Search

Search through a list of items for a particular value.Example:Search through an array of student records for the student with ID 12345.Search through an array of address records for the address of the person with last name Doe.

Linear Search

If we are searching in alist,start at the beginning and check each element until we find the one we want or reach the end.Bestcase?Worst case?Average case?

Binary Search

If we are searching in a sorted list, welook at the middle item and then choose which half to continue looking in.We continue to cut the area we are searching in half until we find thevalue,or there are no more values to check.Best case?Worstcase?Average case? (A little tricky)

BinarySearch: Worst Case

Let’s say the list has1024 items and the item is the last one we check.Check midpoint of 1024 items.Check midpoint of upper or lower half (512).Check midpoint of a half of that half (128).Successive ranges we are checking have lengths 64, 32, 16, 8, 4, 2, 1.How many checks was that? 10(log 1024 = 10)

BinarySearch

Aside: Note that binary search only works if the data in the list aresortedby the field on which we’re searching!

Classifying Problems

Problems fall into two categories.Computable problems can be solved using an algorithm.Non-computable problems have no algorithm to solve them.Historical note:Hilbert’s questions in 1900: complete? Consistent? Decidable?

Classifying Problems

Historical note:Hilbert posed the following questions in 1900: Is mathematics complete? Is mathematics consistent? Is every statement in mathematics decidable?In 1930, he thought the all 3 answers would be “yes.”Almost immediately, Gödel showed that no closed system can be both complete & consistent.By the mid-1930’s, Turing showed that the answer to the 3rdquestion is “no.”

Classifying Problems

Two categories of problems:ComputableNon-computableWouldn’t it be nice to know which category a problem falls into? (Topic for laterin class:this problem itself is non-computable.)

Classifying Computable Problems

TractableThere is an efficient algorithm to solve the problem.IntractableThere is an algorithm to solve the problem but there is no efficient algorithm. (This is difficult to prove.)

Examples

Sorting: tractable.The traveling salesperson problem: intractable. (we think…)Halting Problem: non-computable.(More on thisin a minute.)

Measuring Efficiency

We are (usually) concerned with the time an algorithm takes to complete.Weoften countthe number oftimes blocks of code are executed,as a function of the size of the input.Why not measure time directly?Why not count the number of instructions executed?

Example Code:

If the array has N elements, this function executes 4 + (2 * N) statements (i.e., 2N + 4).

defaFunction(array):statementA;statementB;statementC;forxin array:statementD;statementE;returnsomeValue;

Example: Computing an Average

The statement inside the for loop gets executedlen(array) times.If the length isn, we say this algorithm is “on the order ofn”, or,O(n).O(n)??? What’s this?

def average(array):sum = 0for x in array:sum += xreturn sum / len(array)

Big O

The worst case running time, discounting constants and lower order terms.Example:n3+ 2n is O(n3)

Exchange Sort

Let’s work out the big O running time…

def exchangeSort(array):for indx1 in range(len(array)):for indx2 in range(indx1, len(array)):if (array[indx1] > array[indx2]):swap(array, indx1, indx2)

Merge Sort

Given a list, split it into 2 equal piles.Then split each of these piles in half. Continue to do this until you are left with 1 element in each pile.Now merge piles back together in order.

Merge Sort

An example of how the merge works:Suppose the first half and second half of an array are sorted:5 9 10 12 17 1 8 11 20 32Merge these by taking a new element from either the first or second subarray, choosing the smallest of the remaining elements.Big O running time?

Big O Can Be Misleading

Big O analysis is concerned with worst case behavior.Sometimes we know that we are not dealing with the worst case.

Searching an Array

Worst case?Best case?

def search(array, key):for x in array:if x == key:return key

Algorithms Exercise…

Problem - Finding the Greatest Common Denominator

Examples:gcd(12, 2) = 2gcd(100, 95) = 5gcd(100, 75) = 25gcd(39, 26) = 13gcd(17, 8) = 1

Possible Algorithm #1

Assumption: A > B >= 0If A is a multiple of B, then gcd(A, B) = B.Otherwise, return an error.Works for gcd(12,2) = 2But what about gcd(100, 95)???

Possible Algorithm #2

Start with 1 and go up to B.If a number if a common divisor of both A and B, remember it.When we get to B, stop. The last number we remembered is the gcd.Works, but is there a better way?Think about gcd(100, 95)

Euclid’s Algorithm

Make use of the fact that:gcd(A, B) = gcd(B, A rem B)Note: A rem B refers to theremainderleft when A is divided by B.Examples:12 rem 2 = 0100 rem 95 = 5

Euclid’s Algorithm

If B = 0, then gcd(A, B) = A.Otherwise, gcd(A, B) = gcd (B, A rem B).Note – this algorithm isrecursive.Examples:gcd(12, 2) = gcd(2, 0) = 2gcd(100, 95) = gcd(95, 5) = gcd(5, 0) = 5

Why do we care?

Let’s say we want the gcd of 1,120,020,043,575,432 and 1,111,363,822,624,856Assume we can do 100,000,000 divisions per second.Algorithm #2 will take about three years.Euclid’s Algorithm will take less than a second.

Programs vs. Algorithms

Program: “A set of computer instructions written in a programming language”We writeProgramsthat implementAlgorithms

Algorithm vs. Program

def gcd(A, B):if B == 0:return Aelse:return gcd(B, A % B)

If B = 0, then gcd(A, B) = A.Otherwise, gcd(A, B) = gcd (B, A rem B).

Tractable vs. Intractable Problems

Problems with polynomial time algorithms are considered tractable.Problemswithoutpolynomial time algorithms are considered intractable.Eg. Exponential time algorithms.(More on Friday)

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Algorithms - cs.kzoo.edu