What is an algorithm?

1. What is an algorithm? 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. problem algorithm “computer” input output

2. What is an algorithm? • is any well-defined computational procedure that takes some value, or set of values, as input and produces some value, or set of values, as output. • is thus a sequence of computational steps that transform the input into the output. • is a tool for solving a well - specified computational problem. • Any special method of solving a certain kind of problem (Webster Dictionary)

3. What is a problem? • Definition • A mapping/relation between a set of input instances (domain) and an output set (range) • Problem Specification • Specify what a typical input instance is • Specify what the output should be in terms of the input instance • Example: Sorting • Input: A sequence of N numbers a1…an • Output: the permutation (reordering) of the input sequence such that a1 a2 …  an .

4. Types of Problems • Search: find X in the input satisfying property Y • Structuring: Transform input X to satisfy property Y • Construction: Build X satisfying Y • Optimization: Find the best X satisfying property Y • Decision: Does X satisfy Y?

5. What do we analyze about algorithms • Correctness • Does the input/output relation match algorithm requirement? • Amount of work done (aka complexity) • Basic operations to do task • Amount of space used • Memory used

6. What do we analyze about algorithms • Simplicity, clarity • Verification and implementation. • Optimality • Is it impossible to do better?

7. Euclid’s Algorithm Problem: Find gcd(m,n), the greatest common divisor of two nonnegative, not both zero integers m and n Examples: gcd(60,24) = 12, gcd(60,0) = 60, gcd(0,0) = ? Euclid’s algorithm is based on repeated application of equality gcd(m,n) = gcd(n, m mod n) until the second number becomes 0, which makes the problem trivial. Example: gcd(60,24) = gcd(24,12) = gcd(12,0) = 12

8. Two descriptions of Euclid’s algorithm Step 1 If n = 0, return m and stop; otherwise go to Step 2 Step 2 Divide m by n and assign the value fo the remainder to r Step 3 Assign the value of n to m and the value of r to n. Go to Step 1. whilen ≠ 0 do r ← m mod n m← n n ← r returnm

9. Other methods for computing gcd(m,n) Consecutive integer checking algorithm Step 1 Assign the value of min{m,n} to t Step 2 Divide m by t. If the remainder is 0, go to Step 3; otherwise, go to Step 4 Step 3 Divide n by t. If the remainder is 0, return t and stop; otherwise, go to Step 4 Step 4 Decrease t by 1 and go to Step 2

10. Other methods for gcd(m,n) [cont.] Middle-school procedure Step 1 Find the prime factorization of m Step 2 Find the prime factorization of n Step 3 Find all the common prime factors Step 4 Compute the product of all the common prime factors and return it as gcd(m,n) Is this an algorithm?

11. Sieve of Eratosthenes Input: Integer n ≥ 2 Output: List of primes less than or equal to n for p ← 2 to n do A[p] ← p for p ← 2 to n do if A[p]  0 //p hasn’t been previously eliminated from the listj ← p*p while j ≤ ndo A[j] ← 0 //mark element as eliminated j ← j+ p Example: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

12. Algorithm design and analysis process.

13. Two main issues related to algorithms • How to design algorithms • How to analyze algorithm efficiency

14. Algorithm design techniques/strategies • An algorithm design technique (or “strategy” or “paradigm”) is a general approach to solving problems algorithmically that is applicable to a variety of problems from different areas of computing.

15. Brute force Divide and conquer Decrease and conquer Transform and conquer Space and time tradeoffs Algorithm design techniques/strategies • Greedy approach • Dynamic programming • Iterative improvement • Backtracking • Branch and bound

16. Algorithm design techniques/strategies • Brute force is a straightforward approach to solving a problem, usually directly based on the problem statement and definitions of the concepts involved. • The decrease-and-conquer technique is based on reducing the size of the input instance. • Divide-and-Conquer • A problem is divided into several subproblems of the same type, ideally of about equal size. • The subproblems are solved separately. • the solutions to the subproblems are combined to get a solution to the original problem.

17. Algorithm design techniques/strategies • Transform-and-Conquer: Firstly, the problem instance is modified to be more Appropriate to solution. Then, in the second or conquering stage, it is solved. • Dynamic programming is a technique for solving problems with overlapping subproblems. • The greedy approach suggests constructing a solution through a sequence of steps until a complete solution to the problem is reached. • Iterative improvement starts with some feasible solution and proceeds to improve it by repeated applications of some simple step.

18. Analysis of algorithms • How good is the algorithm? • time efficiency • space efficiency • Does there exist a better algorithm? • lower bounds • optimality

19. Properties as important as performance • Modularity • Maintainability • Functionality • Robustness and Reliability • User-friendliness

20. Important problem types • Sorting • rearrange the items of a given list in non-decreasing order. • Searching • deals with f inding a given value, called a search key, in a given set • string processing: Eg, string matching • graph problems • graph-traversal algorithms • shortest-path algorithms

21. Important problem types • Combinatorial problems: find a combinatorial object—such as a permutation, a combination, or a subset—that satisfies certain constraints. • Geometric problems: deal with geometric objects such as points, lines, and polygons. • Numerical problems: involve mathematical objects of continuous nature: • solving equations and systems of equations, • computing definite integrals, • evaluating functions, and • ….. so on.

22. list array linked list Fundamental data structures

23. Fundamental data structures • stack • queue • priority queue

24. Fundamental data structures • Graph Adjacency matrix Adjacency list

25. Fundamental data structures • Tree: is a connected acyclic graph • Rooted Trees: for every two vertices in a tree, there always exists exactly one simple path from one of these vertices to the other. • Set and dictionary: an unordered collection (possibly empty) of distinct items called elements of the set. • The operations we need to perform for a set often are searching for a given item, adding a new item, and deleting an item from the collection. • A data structure that implements these three operations is called the dictionary.