Lecture 1 - Algorithms PDF

The Design and Analysis of Algorithms Lecture 1 CHAPTER 1: INTRODUCTION  What is an Algorithm ◼ Algorithm is more than a branch of computer science. It is the core of computer science, and, in all fairness, can be said to be relevant to most science, business, and...

The Design and Analysis of Algorithms Lecture 1 CHAPTER 1: INTRODUCTION  What is an Algorithm ◼ Algorithm is more than a branch of computer science. It is the core of computer science, and, in all fairness, can be said to be relevant to most science, business, and technology. ◼ An algorithm is a sequence of unambiguous instructions for solving a problem, i.e., for obtaining a required output for any legitimate input in a finite amount of time. has input data, and is expected to produce output data each instruction can be carried out in a finite amount of time in a deterministic way 2 ◼ What the Difference Between Algorithm and Program? 1. Writing an algorithm is simpler (we don’t need to worry about the detailed implementation, or the language syntax). 2. An algorithm is easier to read than a program written in, for instance, C. 3 Problems vs Algorithms vs Programs ◼ For each problem or class of problems, there may be many different algorithms. ◼ For each algorithm, there may be many different implementations (programs).  Difference between Algorithm, Pseudocode and Program ◼ Algorithm : Systematic logical approach which is a well-defined, step-by-step procedure that allows a computer to solve a problem. ◼ Pseudocode : It is a simpler version of a programming code in plain English which uses short phrases to write code for a program before it is implemented in a specific programming language. ◼ Program : It is exact code written for problem following all the rules of the programming language. 5 Algorithmic Representation of Computer Functions ◼ Input  Get information Get (input command) ◼ Storage  Store information Given/Result Intermediates/Set ◼ Process  Arithmetic Let (assignment command)  Repeat instructions Loop  Branch conditionals If ◼ Output  Give information Give (output command)  Properties of an Algorithm An algorithm must possess the following properties: ◼ finiteness: The algorithm must always terminate after a finite number of steps. ◼ definiteness: Each step must be precisely defined; the actions to be carried out must be rigorously and unambiguously specified for each case. ◼ input: An algorithm has zero or more inputs, taken from a specified set of objects. ◼ output: An algorithm has one or more outputs, which have a specified relation to the inputs. ◼ effectiveness: All operations to be performed must be sufficiently basic that they can be done exactly and in finite length  Expressing Algorithms An algorithm may be expressed in a number of ways, including: ◼ natural language: usually verbose and ambiguous ◼ flow charts: avoid most (if not all) issues of ambiguity; difficult to modify w/o specialized tools; largely standardized ◼ pseudo-code: also avoids most issues of ambiguity; vaguely resembles common elements of programming languages; no particular agreement on syntax ◼ programming language: tend to require expressing low- level details that are not necessary for a high-level understanding Common Elements of Algorithms ◼ acquire data (input): some means of reading values from an external source; most algorithms require data values to define the specific problem (e.g., coefficients of a polynomial) ◼ Computation: some means of performing arithmetic computations, comparisons, testing logical conditions, and so forth... ◼ Selection: some means of choosing among two or more possible courses of action, based upon initial data, user input and/or computed results ◼ Iteration: some means of repeatedly executing a collection of instructions, for a fixed number of times or until some logical condition holds ◼ report results (output): some means of reporting computed results to the user, or requesting additional data from the user Understand the problem Decide on computational means Exact vs approximate solution Data structures Algorithm design technique Design an algorithm Prove correctness Analyze the algorithm Code the algorithm 12 What does it mean to understand the problem? ◼ What are the problem objects? ◼ What are the operations applied to the objects? Deciding on computational means ◼ How the objects would be represented? ◼ How the operations would be implemented? 13 Design an algorithm Build a computational model of the solving process Prove correctness Correct output for every legitimate input in finite time Based on correct math formula By induction 14 Analyze the algorithm Efficiency: time and space Simplicity Generality: range of inputs, special cases Optimality: no other algorithm can do better Coding How the objects and operations in the algorithm are represented in the chosen programming language? 15 pseudo-Language ◼ See the posted notes on pseudo- language notation.  Testing Correctness ◼ How do we know whether an algorithm is actually correct? ◼ First, the logical analysis of the problem we performed in order to design the algorithm should give us confidence that we have identified a valid procedure for finding a solution. ◼ Second, we can test the algorithm by choosing different sets of input values, carrying out the algorithm, and checking to see if the resulting solution does, in fact, work.  Proving Correctness ◼ We can attempt to construct a formal, mathematical proof that, if the algorithm is given valid input values then the results obtained from the algorithm must be a solution to the problem. ◼ We should expect that such a proof be provided for every algorithm. ◼ In the absence of such a proof, we should view the purported algorithm as nothing more than a heuristic procedure, which may or may not yield the expected results  gcd(m,n) while n ≠0 do 1. t ← min (m ,n) r ← m mod n m←n 2. if m % t = 0 goto 3, n ←r else goto 4 return m 3. if n % t = 0 return t, else goto 4 4. t ← t - 1 5. goto 2 20 21 Figure 8-6 Three constructs Figure 8-7 Flowcharts for three constructs Figure 8-8 Pseudocode for three constructs Example 1 Write an algorithm in pseudocode that finds the average of two numbers Solution See Algorithm on the next slide.  Algorithm : Average of two AverageOfTwo Input: Two numbers 1. Add the two numbers 2. Divide the result by 2 3. Return the result by step 2 End Example 2 Write an algorithm to change a numeric grade to a pass/no pass grade. Solution See Algorithm on the next slide.  Algorithm : Pass/no pass Grade Pass/NoPassGrade Input: One number 1. if (the number is greater than or equal to 70) then 1.1 Set the grade to “pass” else 1.2 Set the grade to “no pass” End if 2. Return the grade End Example 3 Write an algorithm to change a numeric grade to a letter grade. Solution See Algorithm on the next slide. Algorithm : Letter grade LetterGrade Input: One number 1. if (the number is between 90 and 100, inclusive) then 1.1 Set the grade to “A” End if 2. if (the number is between 80 and 89, inclusive) then 2.1 Set the grade to “B” End if Continues on the next slide Algorithm Letter grade (continued) 3. if (the number is between 70 and 79, inclusive) then 3.1 Set the grade to “C” End if 4. if (the number is between 60 and 69, inclusive) then 4.1 Set the grade to “D” End if Continues on the next slide Algorithm Letter grade (continued) 5. If (the number is less than 60) then 5.1 Set the grade to “F” End if 6. Return the grade End  Example 4 Write an algorithm to find the largest of a set of numbers. You do not know the number of numbers. Solution See Algorithm on the next slide. Algorithm: Find largest FindLargest Input: A list of positive integers 1. Set Largest to 0 2. while (more integers) 2.1 if (the integer is greater than Largest) then 2.1.1 Set largest to the value of the integer End if End while 3. Return Largest End Example 5 Write an algorithm to find the largest of 1000 numbers. Solution See Algorithm on the next slide. Algorithm : Find largest of 1000 numbers FindLargest Input: 1000 positive integers 1. Set Largest to 0 2. Set Counter to 0 3. while (Counter less than 1000) 3.1 if (the integer is greater than Largest) then 3.1.1 Set Largest to the value of the integer End if 3.2 Increment Counter End while 4. Return Largest End  Important problem types ◼ Sorting ◼ Searching ◼ String processing ◼ Graph problems ◼ Combinatorial problems ◼ Geometric problems ◼ Numerical problems 37 Tasks 38