Chapter 2 Algorithms and Programming Languages

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Algorithms
• the central concept underlying all computation is that of the algorithm – an algorithm is a step-by-step sequence of instructions for carrying out some task • programming can be viewed as the process of designing and implementing algorithms that a computer can carry out – a programmer’s job is to: • create an algorithm for accomplishing a given objective, then • translate the individual steps of the algorithm into a programming language that the computer can understand 2

Algorithms in the Real World
• the use of algorithms is not limited to the domain of computing – e.g., recipes for baking cookies – e.g., directions to your house

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there are many unfamiliar tasks in life that we could not complete without the aid of instructions – in order for an algorithm to be effective, it must be stated in a manner that its intended executor can understand • a recipe written for a master chef will look different than a recipe written for a college student – as you have already experienced, computers are more demanding with regard to algorithm specifics than any human could be

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Designing & Analyzing Algorithms
• 4 steps to solving problems (George Polya) 1. understand the problem 2. devise a plan 3. carry out your plan 4. examine the solution

EXAMPLE: finding the oldest person in a room full of people
Understanding the problem initial condition, goal and assumptions – room full of people – identify the oldest person a person will give their real birthday if two people are born on the same day, they are the same age if there is more than one oldest person, finding any one of them is okay
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we will consider 2 different designs for solving this problem 4

Algorithm 1
• Finding the oldest person (algorithm 1)
1. 2. 3. line up all the people along one wall ask the first person to state his or her name and birthday, then write this information down on a piece of paper for each successive person in line: i. ask the person for his or her name and birthday ii. if the stated birthday is earlier than the birthday on the paper, cross out old information and write down the name and birthday of this person

when you reach the end of the line, the name and birthday of the oldest person will be written on the paper

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Algorithm 2
• Finding the oldest person (algorithm 2)
1. line up all the people along one wall 2. as long as there is more than one person in the line, repeatedly i. have the people pair up (1st with 2nd, 3rd with 4th, etc) – if there is an odd number of people, the last person will be without a partner ii. ask each pair of people to compare their birthdays iii. request that the younger of the two leave the line when there is only one person left in line, that person is the oldest

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Algorithm Analysis
• determining which algorithm is "better" is not always clear cut
– it depends upon what features are most important to you
• • if you want to be sure it works, choose the clearer algorithm if you care about the time or effort required, need to analyze performance

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algorithm 1 involves asking each person’s birthday and then comparing it to the birthday written on the page
– – the amount of time to find the oldest person is proportional to the (number of people – 1) if you double the amount of people, the time needed to find the oldest person will also double

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algorithm 2 allows you to perform multiple comparisons simultaneously
– the time needed to find the oldest person is proportional to the number of rounds it takes to shrink the line down to one person • which turns out to be the logarithm (base 2) of the number of people if you double the amount of people, the time needed to find the oldest person increases by the cost of one more comparison

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the words algorithm and logarithm are similar – do not be confused by this algorithm: a step-by-step sequence of instructions for carrying out a task logarithm: the exponent to which a base is raised to produce a number e.g., 210 = 1024, so log2(1024) = 10

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Algorithm Analysis (cont.)
• • when the problem size is large, performance differences can be dramatic for example, assume it takes 5 seconds to compare birthdays – for algorithm 1: • 101 people 5*100 = 500 seconds • 201 people 5*200 = 1000 seconds • 401 people 5*400 = 2000 seconds ... • 1,000,001 people 5*1,000,000 = 5,000,000 seconds – for algorithm 2: • 101 people 5* log2 100  = 35 seconds • 201 people 5* log2 200  = 40 seconds • 401 people 5* log2 400  = 45 seconds ... • 1,000,001 people 5* log2 1,000,000  = 100 seconds
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Round up to the next integer

Big-Oh Notation
• To represent an algorithm’s performance in relation to the size of the problem, computer scientists use what is known as Big-Oh notation – executing an O(N) algorithm requires time proportional to the size of problem • given an O(N) algorithm, doubling the problem size doubles the work – executing an O(log N) algorithm requires time proportional to the logarithm of the problem size • given an O(log N) algorithm, doubling the problem size adds a constant amount of work •based on our previous analysis: – algorithm 1 is classified as O(N) – algorithm 2 is O(log N)
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Another Algorithm Example
• SEARCHING: a common problem in computer science involves storing and maintaining large amounts of data, and then searching the data for particular values – data storage and retrieval are key to many industry applications – search algorithms are necessary to storing and retrieving data efficiently – e.g., consider searching a large payroll database for a particular record • if the computer selected entries at random, there is no assurance that the particular record will be found • even if the record is found, it is likely to take a large amount of time • a systematic approach assures that a given record will be found, and that it will be found more efficiently • There are two commonly used algorithms for searching a list of items – sequential search – general purpose, but relatively slow – binary search – restricted use, but fast
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Sequential Search
• sequential search is an algorithm that involves examining each list item in sequential order until the desired item is found sequential search for finding an item in a list 1. start at the beginning of the list 2. for each item in the list i. examine the item - if that item is the one you are seeking, then you are done ii. if it is not the item you are seeking, then go on to the next item in the list if you reach the end of the list and have not found the item, then it was not in the list • sequential search guarantees that you will find the item if it is in the list – but it is not very practical for very large databases – worst case: you may have to look at every entry in the list. Why? •

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Binary Search
• binary search involves continually cutting the desired search list in half until the item is found – the algorithm is only applicable if the list is ordered • e.g., a list of numbers in increasing order • e.g., a list of words in alphabetical order binary search for finding an item in an ordered list 1. initially, the potential range in which the item could occur is the entire list 2. as long as items remain in the potential range and the desired item has not been found, repeatedly i. examine at the middle entry in the potential range ii. if the middle entry is the item you are looking for, then you are done iii. if the middle entry is greater than the desired item, then reduce the potential range to those entries left of the middle iv. if the middle entry is less than the desired item, then reduce the potential range to those entries right of the middle by repeatedly cutting the potential range in half, binary search can done in on the value very quickly
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Binary Search Example suppose you have a sorted list of state names, and want to find MD
1. start by examining the middle entry (ND) since ND comes after MD alphabetically, can eliminate it and all entries that appear to the right

2. next, examine the middle of the remaining entries (IA) since IA comes before MD alphabetically, can eliminate it and all entries that appear to the left

3. next, examine the middle of the remaining entries (MD) the desired entry is found

Which state in the list could be located with the fewest number of checks (the easiest)? What about the largest number of checks (the hardest)?

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Search Analysis
• sequential search – in the worst case, the item you are looking for is in the last spot in the list (or not in the list at all) • as a result, you will have to inspect and compare every entry in the list – the amount of work required is proportional to the list size sequential search is an O(N) algorithm binary search – in the worst case, you will have to keep halving the list until it gets down to a single entry • each time you inspect/compare an entry, you rule out roughly half the remaining entries – the amount of work required is proportional to the logarithm of the list size binary search is an O(log N) algorithm

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imagine searching a phone book of the Malaysian (30 million people) sequential search requires at most 30 million inspections/comparisons binary search requires at most log2(30,000,000) = 25 inspections/comparisons
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Another Algorithm Example
• Newton’s Algorithm for finding the square root of N 1. start with an initial approximation of 1 2. as long as the approximation isn’t close enough, repeatedly refine the approximation using the formula: newApproximation = (oldApproximation + N/oldApproximation)/2

example: finding the square root of 1024

algorithm analysis: Newton's Algorithm does converge on the square root because each successive approximation is closer than the previous one generally, the difference between the given approximation and the actual square root is roughly cut in half by each successive refinement demonstrates O(log N) behavior What would happen if you tried to further refine the square root?
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Programming Languages
• programming is all about designing and coding algorithms for solving problems – the intended executor = computer or a program executing on that computer – instructions are written in programming languages which are extraordinarily specific the level of precision to write programs can be frustrating to beginner

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Machine Languages
• the first programming languages were known as machine languages – a machine language consists of instructions that correspond directly to the hardware operations of a particular machine • i.e., instructions deal directly with the computer’s physical components including main memory, registers, memory cells in CPU • programming in machine language is tedious and error prone • Eg. Add 1 to the value in register 3…0110011001000011…
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High-Level Languages
• in the early 1950’s, assembly languages evolved from machine languages
– – an assembly language substitutes words for binary codes much easier to remember and use words, but still a low level of abstraction (instructions correspond to hardware operations)

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in the late 1950's, high-level languages were introduced
– – – high-level languages allow the programmer to write code closer to the way humans think (as opposed to mimicking hardware operations) a much more natural way to solve problems plus, programs are machine independent

two high level languages that perform the same task (in JavaScript and C++)

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Program Translation
• using a high-level language, the programmer is able to reason at a high-level of abstraction
– but programs must still be translated into machine language that the computer hardware can understand/execute

• there are two standard approaches to program translation
– – interpretation compilation

•real-world analogy: translating a speech from one language to another
– an interpreter can be used provide a real-time translation
• • • the interpreter hears a phrase, translates, and immediately speaks the translation ADVANTAGE: the translation is immediate DISADVANTAGE: if you want to hear the speech again, must interpret all over again

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a translator (or compiler) translates the entire speech offline
• • • the translator takes a copy of the entire speech and translate ADVANTAGE: once translated, it can be read over and over very quickly DISADVANTAGE: must wait for the entire speech to be translated

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Speech Translation
•Interpreter:

Translator (compiler):

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Interpreters
• for program translation, the interpretation approach relies on a program known as an interpreter to translate and execute high-level statements
– – the interpreter reads one high-level statement at a time, immediately translating and executing the statement before processing the next one JavaScript is an interpreted language

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Compilers
• the compilation approach relies on a program known as a compiler to translate the entire highlevel language program into its equivalent machine-language instructions
– – the resulting machine-language program can be executed directly on the computer most languages used for the development of commercial software employ the compilation technique (C, C++)

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Interpreters and Compilers
• trade-offs between interpretation and compilation • interpreter – produces results almost immediately – particularly useful for dynamic, interactive features of web pages – program executes more slowly (slight delay between the execution of statements) • compiler – produces machine-language program that can run directly on the underlying hardware – program runs very fast after compilation – must compile the entire program before execution – used in large software applications when speed is of the utmost importance
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