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The Growth of Functions

The Growth of Functions. M. S. Tame. The Growth of Functions. Introduction Q O W o w some notations and relations. Asymptotic notations. Introduction. Why are asymptotic notations important? They give a simple characterization of an algorithm’s efficiency.

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The Growth of Functions

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  1. The Growth of Functions M. S. Tame

  2. The Growth of Functions • Introduction • Q • O • W • o • w • some notations and relations Asymptotic notations

  3. Introduction Why are asymptotic notations important? • They give a simple characterization of an algorithm’s efficiency. • They allow the comparison of the performances of various algorithms. • For large values of components/inputs, the multiplicative constants and lower order terms of an exact running time are dominated by the effects of the input size (the number of components).

  4. Introduction Why are asymptotic notations important? • They give a simple characterization of an algorithm’s efficiency. • They allow the comparison of the performances of various algorithms. • For large values of components/inputs, the multiplicative constants and lower order terms of an exact running time are dominated by the effects of the input size (the number of components). - The running time of an algorithm on a particular input, is the number of primitive operations or “steps” executed. The input size depends on the problem being studied, But in most cases it is the number of items in the input. Eg. The total number of bits/qubits

  5. Introduction So in general, when looking at input sizes large enough to make the order of growth of the running time relevant for an algorithm, we are studying the asymptotic efficiency of an algorithm. And an algorithm that is asymptotically more efficient will be the best choice for all but very small inputs.

  6. Asymptotic notations • Q - Asymptotic tight bound • O - Asymptotic upper bound • W - Asymptotic lower bound • o - upper bound that is not asymptotically tight • w - lower bound that is not asymptotically tight It is also convenient to abuse these notations in certain circumstances: “It is important to understand the precise meaning of the notation so that when it is abused, it is not misused.” - T.H. Cormen, C.E. Leiserson, and R.L. Rivest, "Introduction to Algorithms",(MIT Press, Cambridge, MA, 1990).

  7. Q - Asymptotic tight bound Abuse:

  8. Q - Asymptotic tight bound Eg.

  9. Q - Asymptotic tight bound Eg.

  10. O - Asymptotic upper bound

  11. O - Asymptotic upper bound

  12. W - Asymptotic lower bound

  13. o - Upper bound that is not asymptotically tight

  14. w - Lower bound that is not asymptotically tight

  15. Asymptotic notation in equations

  16. Relations of Asymptotic Functions

  17. References References: T.H. Cormen, C.E. Leiserson, and R.L. Rivest, "Introduction to Algorithms", (MIT Press, Cambridge, MA, 1990).

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