Theory of algorithms introduction
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Theory of Algorithms: Introduction. James Gain and Edwin Blake {jgain | edwin} @cs.uct.ac.za Department of Computer Science University of Cape Town August - October 2004. Objectives. To define an algorithm To introduce: Problem types The Process of Algorithm Design Solution strategies

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Theory of algorithms introduction

Theory of Algorithms:Introduction

James Gain and Edwin Blake

{jgain | edwin} @cs.uct.ac.za

Department of Computer Science

University of Cape Town

August - October 2004


Objectives

Objectives

  • To define an algorithm

  • To introduce:

    • Problem types

    • The Process of Algorithm Design

    • Solution strategies

    • Ways of Analysing Algorithms

  • To cover the structure of the course, including practicals


Definitions

Definitions

  • An algorithm is a sequence of unambiguous instructions for solving a problem

    • For obtaining a required output for any legitimate input in a finite amount of time

    • Does not require implementation in software

    • Not an answer but a method for deriving an answer

  • Historical Perspective:

    • Named after Muhammad ibn Musa al-Khwarizmi – 9th century mathematician

    • www.lib.virginia.edu/science/parshall/khwariz.html


Notion of algorithm

Notion of algorithm

problem

algorithm

“computer”

input

output

  • Each step of the algorithm must be unambiguous

  • The range of inputs must be specified carefully

  • The same algorithm can be represented in different ways

  • Several algorithms for solving the same problem may exist - with different properties


What is an algorithm

What is an algorithm?

  • Recipe, process, method, technique, procedure, routine,… with following requirements:

  • Finiteness

    • Terminates after a finite number of steps

  • Definiteness

    • Rigorously and unambiguously specified

  • Input

    • Valid inputs are clearly specified

  • Output

    • Can be proved to produce the correct output given a valid input

  • Effectiveness

    • Steps are sufficiently simple and basic


Example sorting

Example: Sorting

  • Statement of problem:

    • Input: A sequence of n numbers <a1, a2, …, an>

    • Output: A reordering of the input sequence <a´1, a´2, …, a´n> so that a´i≤ a´j whenever i < j

  • Instance: The sequence <5, 3, 2, 8, 3>

  • Algorithms:

    • Selection sort

    • Insertion sort

    • Merge sort

    • (many others)


Selection sort

for i=1 to n

swap a[i] with smallest of a[i],…,a[n]

Selection Sort

  • Input: array a[1],..,a[n]

  • Output: array a sorted in non-decreasing order

  • Algorithm:

  • for i1 to ndo

    • min  i

    • for j  i+1 to n do

      • if a[j] < a[min] min  j

    • swap a[i] and a[min]


Exercise bridge puzzle

Exercise: Bridge Puzzle

  • Problem:

    • 4 People want to cross a bridge. You have 17 minutes to get them across

  • Constraints:

    • It is night and you have 1 flashlight. Max of 2 on the bridge at one time. All start on the same side

    • Those crossing must have the flashlight with them. The flashlight must be walked back and forth (no throwing)

    • People walk at different speeds: person A = 1 minute to cross, person B = 2 minutes, person C = 5 minutes, person D = 10 minutes

    • A pair walks at the speed of the slower person’s pace

  • Rumour: this problem is given to Microsoft interviewees


Solution bridge puzzle

Solution: Bridge Puzzle

  • Start (0 min): A B C D

  • AB Across (2 min): A B C D

  • A Back (1 min): B A C D

  • CD Across (10 min): B C D A

  • B Back (2 min): C D A B

  • AB Across (2 min): A B C D

  • Total Time = 17 minutes


Extension exercise

Extension Exercise

  • This is an instance of a problem. How would you generalise it?

  • Can you derive an algorithm to solve this generalised problem?

    • Must show the sequence of moves

    • Must output the minimum time required for crossing

    • Are there any special cases to watch out for?

    • Are there any constraints on the input?


Extension solution

Extension Solution

  • Input: a list a of crossing times for n people, numbered 1, …, n

  • Output: total time to cross

  • Strategy: use 1 & 2 as shuttles and send the others across in pairs

  • for i 2 to n/2do

    • t  a[2] // 1 & 2 across

    • t  t + a[1] // 1 back

    • t  t + a[i*2] // i*2 & (i*2)-1 across

    • t  t + a[2] // 2 back

  • t  a[2] // 1 & 2 across

  • return t


Extension problems

Extension Problems

  • This is an inadequate solution

  • It falsely assumes certain inputs

  • List may not be sorted in ascending order

    • Sort a

  • n may not be even numbered

    • Alter final iteration of loop

  • n > 3 not guaranteed

    • Special case for n = 1, 2, 3

  • Is not optimal for all inputs, e.g. 1, 20, 21, 22

    • Can you quantify the nature of these inputs? Suggest an alternative.

  • Final solution is left as an exercise. Attempt to make your solution elegant


Fundamentals of algorithmic problem solving

Fundamentals of Algorithmic Problem Solving

  • Understanding the Problem

    • Make sure you are solving the correct problem and for all legitimate inputs

  • Ascertaining the Capabilities of a Computational Device

    • Sequential vs. Parallel.

    • What are the speed and memory limits?

  • Choosing between exact and approximate Problem Solving

    • Is absolute precision required? Sometimes this may not be possible

  • Deciding on Appropriate Data Structures

    • Algorithms often rely on carefully structuring the data

    • Fundamental Data Structures: array, linked list, stacks, queues, heaps, graphs, trees, sets


Fundamentals of algorithm design

Fundamentals of Algorithm Design

  • Applying an Algorithm Design Technique

    • Using a general approach to problem solving that is applicable to a variety of problems

  • Specifying the Algorithm

    • Pseudocode is a mixture of natural language and programming constructs that has replaced flowcharts

  • Proving an Algorithms Correctness

    • Prove that an algorithm yields a required result for legitimate inputs in finite time

  • Analyzing an Algorithm

    • Consider time efficiency, space efficiency, simplicity, generality, optimality

    • Analysis can be empirical or theoretical

  • Coding an Algorithm


Well known computational problems

Well known Computational Problems

  • Sorting

  • Searching

  • String Processing

    • String Matching

  • Graph Problems

    • Graph Traversal, Shortest Path, Graph Colouring

  • Combinatorial Problems

    • Find a combinatorial object - permutation, combination, subset - subject to constraints

  • Geometric Problems

    • Closest-Pair, Convex-Hull

  • Numerical Problems

    • Solving systems of equations, computing definite integrals, evaluating functions, etc.


Algorithm design strategies

Algorithm Design Strategies

  • Brute force

    • A straightforward approach to solving a problem, usually directly based on the problem’s statement

  • Divide and conquer

    • Divide a problem into smaller instances, solve smaller instances (perhaps recursively), combine

  • Decrease and conquer

    • Exploit relationship between the problem and a smaller instance reduced by some factor (often 1)

  • Transform and conquer

    • Transform the problem to a simpler instance, another representation or an instance with a known solution


More algorithm design strategies

More Algorithm Design Strategies

  • Greedy approach

    • Make locally optimal steps which (hopefully) lead to a globally optimal solution for an optimization problem

  • Dynamic programming

    • Technique for solving problems with overlapping sub-domains

  • Backtracking and Branch and bound

    • A way of tackling difficult optimization and combinatorial problems without exploring all state-space

  • Space and time tradeoffs

    • Preprocess the input and store additional information to accelerate solving the problem


How to solve it understanding the problem

How to Solve It: Understanding the Problem

  • Taken from G. Polya, “How to Solve It”, 2nd edition. A classic textbook on problem solving for mathematics

  • You have to understand the problem.

    • What is the unknown? What are the data? Is the problem statement sufficient, redundant, contradictory

    • Draw a figure. Introduce suitable notation

    • Separate the various parts of the problem. Can you write them down?


Devising a plan

Devising a Plan

  • Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.

  • Have you seen it before? Or have you seen the same problem in a slightly different form?

  • Do you know a related problem? Do you know a theorem that could be useful?

  • Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.

  • Could you restate the problem? Could you restate it still differently? Go back to definitions.

  • If you cannot solve the proposed problem try to solve first some related problem. Are the unknown and the new data nearer to each other?

  • Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?


Carrying it through

Carrying it Through

  • Carry out the Plan

    • Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

  • Looking Back

    • Can you check the result? Can you check the argument?

    • Can you derive the solution differently? Can you see it at a glance?

    • Can you use the result, or the method, for some other problem?


Analysis of algorithms

Analysis of Algorithms

  • How good is the algorithm?

    • Correctness

    • Time efficiency

    • Space efficiency

    • Simplicity

  • Does there exist a better algorithm?

    • Lower bounds

    • Optimality


Why study algorithms

Why Study Algorithms?

  • Theoretical importance

    • The core of computer science

  • Practical importance

    • A practitioner’s toolkit of known algorithms

    • Framework for designing and analyzing algorithms for new problems

    • Useful mindset


Course structure

Course Structure

  • Fundamentals of the Analysis of Algorithms (Ch. 2)

    • Asymptotic notations, analysis of recursive and non-recursive algorithms, empirical analysis

  • Algorithmic Strategies (Ch. 3-9)

    • Brute force, Divide-and-Conquer, Decrease-and-Conquer, Transform-and-Conquer, Space and Time Tradeoffs, Greedy Techniques, Biologically-inspired techniques, Dynamic Programming

  • Limitations of Algorithms (Ch. 10 + handouts)

    • Turing Machines, Computability, Problem Classification

  • Coping with Limitations on Algorithms (Ch. 11)

    • Backtracking and Branch and Bound

  • Anany Levitin, “Introduction to the Design and Analysis of Algorithms”, International Edition, Addison-Wesley, 2003


Practicals

Practicals

  • Weekly mini prac exams

  • Given a problem specification that is solvable using the algorithm design strategies presented in the course

    • Design Algorithm

    • Code it in C++

    • Submit it for automatic marking

  • After the 3-hour lab session will be asked to do a short analysis of the solution


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