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# Theory of Algorithms: Introduction - PowerPoint PPT Presentation

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

James Gain and Edwin Blake

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

Department of Computer Science

University of Cape Town

August - October 2004

• 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

• 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

• Historical Perspective:

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

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

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

• 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

• 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)

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]

• 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

• 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

• 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?

• 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

• 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

• 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

• 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

• 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.

• 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

• 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

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

• 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?

• 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?

• How good is the algorithm?

• Correctness

• Time efficiency

• Space efficiency

• Simplicity

• Does there exist a better algorithm?

• Lower bounds

• Optimality

• 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

• 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

• 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