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Understanding Algorithm Efficiency: Concepts and Big-O Notation

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This guide provides an overview of algorithm efficiency, focusing on how it varies with the number of elements processed. It discusses different types of functions such as linear, logarithmic, and nested loops, and how their efficiency can be calculated. The Big-O notation is introduced as a way to simplify the measurement of algorithm efficiency by expressing growth rates. Key examples illustrate how to derive Big-O from the function f(n), emphasizing the importance of understanding these concepts for optimizing algorithm performance.

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Understanding Algorithm Efficiency: Concepts and Big-O Notation

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  1. Algorithm Efficiency • Algorithm efficiency • A function of the number of elements to be processed • f(n)=efficiency • If a function is linear – that is, if it contains no loop – then its efficiency is a functions of the number of instructions it contains. • On the other hand, functions that loop vary widely in their efficiency.

  2. Algorithm Efficiency • Linear loop n=1000 i=1 loop (i<=1000) application code i=i+1 end loop f(n)=n

  3. Algorithm Efficiency • Logarithmic loop • Multiply 2 iteration < 1000 i=1 loop (i<1000) application code i=i x 2 end loop f(n)= log2 n

  4. Algorithm Efficiency • Logarithmic loop • Divide 1000 /2 iteration >=1 i=1000 loop (i>=1) application code i=i / 2 end loop f(n)= log2 n

  5. Algorithm Efficiency • Nested loop • Iterations = inner loop iterations x outer loop iterations • Linear logarithmic i=1 loop (i<=10) j=1 loop (j<=10) application code j= j x 2 end loop i=i + 1 end loop 10 f(n)= n [log2 n] log2 10

  6. Algorithm Efficiency • Nested loop • Quadratic i=1 loop (i<=10) j=1 loop (j<=10) application code j= j +1 end loop i=i + 1 end loop 10 f(n)= n2 10

  7. Big-O Notation The simplification of efficiency is known as big-O analysis. We don’t need to determine the complete measure of efficiency, only the factor that determines the magnitude. This factor is the big-O, and expressed as O(n) – that is, on the order of n.

  8. Big-O Notation • The big O notation can be derived from f(n) using the following steps: • In each term, set the coefficient of the term to 1. • Keep the largest term in the function and discard the others. Terms are ranked from lowest to highest as shown below: Log2n n nlog2n n2 n3… nk 2n n!

  9. Big-O Notation • Example: • Calculate the big-O notation for: • f(n) = n [(n+1)/2] Solution: ½ n2 + ½ n (Remove the coefficient) n2 + n (keep the largest term) n2 So, the big-O notation is stated as O( f(n) ) = O(n2)

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