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Design and Analysis of Algorithms

Design and Analysis of Algorithms. Text Book: Horowitz , S. Sahni , Fundamentals of Computer Algorithms Reference Book : Introduction to algorithms  By Thomas H. Cormen. problem. algorithm. “computer” . output. input. Notion: Algorithms.

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Design and Analysis of Algorithms

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  1. Design and Analysis of Algorithms Text Book: Horowitz, S. Sahni, Fundamentals of Computer Algorithms Reference Book: Introduction to algorithms  By Thomas H. Cormen

  2. problem algorithm “computer” output input Notion: Algorithms • An algorithm is a sequence of unambiguous instructions for solving a computational problem, i.e., for obtaining a required output for any legitimate input in a finite amount of time. • Algorithm is thus a sequence of computational steps that transform the input into the output.

  3. More precisely, an algorithm is a method or process to solve a problem satisfying the following properties: • Finiteness • terminates after a finite number of steps • Definiteness • Each step must be rigorously and unambiguously specified. • Input • Valid inputs must be clearly specified. • Output • can be proved to produce the correct output given a valid can be proved to produce the correct output given a valid input. • Effectiveness • Steps must be sufficiently simple and basic. • Can be carried out with pen and paper • These are also called characteristics of an algorithm

  4. Examples • Is the following a legitimate algorithm?

  5. Four key terms: • Natural Language • Algorithm • Program : A program is the expression of an algorithm in a programming language • Psuedocode-mix of algorithm and some programming language. Key Points: • Each step of an algorithm must be unambiguous. • The same algorithm can be represented in several different ways. • There might exists more than one algorithm for a certain problem. • Algorithms for the same problem can be based on very different ideas and can solve the problem with dramatically different speeds.

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

  7. Analysis of Algorithms • How good is the algorithm? (Determined by the complexity) • time efficiency • space efficiency • Does there exist a better algorithm? • lower bounds • optimality

  8. Importance of Analyze Algorithm • Need to recognize limitations of various algorithms for solving a problem • Need to understand relationship between problem size and running time • When is a running program not good enough? • Need to learn how to analyze an algorithm's running time without coding it • Need to learn techniques for writing more efficient code

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

  10. Models of ComputationReference: The design and analysis of computer algorithms by AhoUllman

  11. Three models of computation • Random Access Machines • RASP Machines(Stored program model) • Turing Machines

  12. Random Access Machine (RAM)

  13. RAM Assumptions • Each register holds an integer • Program can’t modify itself • Memory instructions involve simple arithmetic • Addition, subtraction • Multiplication, division and control states (got, if-then, etc.)

  14. RASP Machinestored program model • Same as RAM but allow program to change itself as it is now stored in the memory • Same power as RAM • Example Von Neumann architecture • Let RAM use memory registers to store modifiable program of RASP

  15. Turing Machine • A Turing machine includes • A (conceptual) tape that extends infinitely in both directions • Holds the input to the Turing machine • Serves as memory • Is divided into cells • A unit that reads one cell of the tape at a time and writes a symbol in that cell • It is controlled by finite automaton that has finite number of states.

  16. Exampleof a finite automaton containing two states is a light switch. In this example, the two states are on and off. If the state of the lightswitch is on, pushing the switch is the input that will cause a transition to the off state. • Each cell contains one symbol • Symbols must come from a finite set of symbols called the alphabet • Alphabet for a given Turing machine • Contains a special symbol b (for “blank”) • Usually contains the symbols 0 and 1 • Sometimes contains additional symbols

  17. Each operation involves • Writing a symbol in the cell (replacing the symbol already there) • Going into a new state (could be same state) • Moving one cell left or right

  18. Each instruction says something like if (you are in state i) and (you are reading symbol j) then write symbol k onto the tape go into state s move in direction d

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