Design and Analysis of Algorithms Lecture # 02 Muhammad Nasir Department of Computer Science COMSATS University Islamabad, Lahore Campus mnasir@cuilahore.edu.pk

Model of Computation Our analysis will be independent of the variations in: Machine hardware Operating system Compiler Programming language etc Algorithms need to understood by the people not machines

Model of Computation Ideally this model should be a reasonable abstraction of a standard generic single-processor machine We will call this machine as Random Access Machine

Random Access Machine (RAM) A Random Access Machine (RAM) is an ideal machine with Infinite large memory Executes one instruction at a time (no parallelism) Each instruction performs some basic operations on two values in memory

Basic RAM operations Basic operations include Assigning a value to a variable Performing basic arithmetic operation (+, -, *, /) on any size of integer values Accessing an element of an Array like array[100]

An Important Assumption Each basic operation takes same constant time to execute There is no difference between performing addition or multiplication on any size of the values

Running Time Analysis Criteria Worst case analysis Maximum running time over all (legal) inputs of size n Average case analysis average running time over all inputs of size n Best case analysis Running time over the best case input of size n We always look for the worst case running time

Theoretical versus empirical The difference is the theoretical analysis, as the name suggests, is based on theory and not actual events. Empirical analysis is based on real testing events. Empirical is what you KNOW, theoretical is what you THINK.

Example: 2-Dimension Maxima Suppose that you want to buy a computer You want to buy a fastest computer But fastest computer is expensive and you want the cheapest You are unable to decide which is more important: Speed or Price

Example: 2-Dimension Maxima Certainly, we don’t want a computer if there is another which is faster and cheaper We can say that the fast and cheap dominates the slow and expensive computer So, we want a list of computers that are not dominated by any other

Example: 2-Dimension Maxima This how we might model this as a formal problem Let a point p in 2-dimentional space be represented by its integer coordinate s, p = (p.x, p.y) A point p is said to be dominated by a point q if, p.x ≤ q.x and P.y ≤ q.y

Example: 2-Dimension Maxima Given a set of n points, P ={p1, p2, p3, … , pn}, a point p is said to be maximal if it is not dominated by any other point in the set P.

Example: 2-Dimension Maxima The computer selection problem can be modeled this way For each computer we associate (x, y) pair in such a way that x is the speed of computer y is the negation of price High y value means cheap computer and low y value means expensive computer Maximal point corresponds to fastest and cheapest computer

Example: 2-Dimension Maxima Input A set of n points i.e. P = {p1, p2, p3, …, pn} in 2 dimensional space Output The set of maximal points of P, the set of those points which are not dominated by the others

Example

Brute-Force Algorithm for 2-D Maxima Example

Analysis of 2-D Maxima To measure the running time of brute force maxima algorithm we can Count the number of steps of the pseudo code that are executed Or, count the number of times an element of set P is accessed Or, the number of comparisons that are performed

Running Time Analysis Depends upon the input size e.g. n Depends upon the structure of input For example, the breaking of inner loop in brute force algorithm depends upon the input size.

Analysis of 2-D Maxima Analysis of brute-force maxima algorithm Input size is n We are going to count the number of times an element of P is accessed

Analysis of 2-D Maxima (Cont…)

Analysis of 2-D Maxima (Cont…) The outer loop runs n times For each iteration of the outer loop, the inner loop runs n times P is accessed four times in the if statement P is accessed two times in output statement In worst case, every point is maximal so every point is output

Analysis of 2-D Maxima (Cont…)

Analysis of 2-D Maxima (Cont…) The worst case running time is

Some Important Summations

Some Important Summations

Analysis: A Harder Example