Question

I am reading algorithm analysis topic. Here is the text snippet from the book

When n doubles, the running time goes up by a factor of 2 for linear
programs, 4 for quadratic programs, and 8 for cubic programs.
Programs that run in logarithmic time take only an additive constant
longer when n doubles, and programs that run in O(n log n) take
slightly more than twice as long to run under the same circumstances.

These increases can be hard to spot if the lower-order terms have
relatively large coefficients and n is not large enough.

My question is what does author mean lower-order terms have relatively large coefficients? Can any one explain with example

Thanks!

Answer 1

When using O notation, you specify the largest term of the function that is your performance bound. For example, if the performance was always bound by f = c₃n³ + c₂n² + c₁n + c₀, you would say that is is O(n³). The author is saying that when n is small, the coefficients may have a larger impact than n on the performance, for example if c₂ were very large and c₃ very small, the performance may appear to be O(n²) until the size of n dominates the coefficients if you only go by the relative performance for specific small instances of n.