4.4: Interpreting results
- Page ID
- 12746
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Based on our understanding of how ArrayList works, we expect the add method to take constant time when we add elements to the end. So the total time to add n elements should be linear.
Figure \(\PageIndex{1}\): Profiling results: run time versus problem size for adding n elements to the end of an ArrayList.
To test that theory, we could plot total run time versus problem size, and we should see a straight line, at least for problem sizes that are big enough to measure accurately. Mathematically, we can write the function for that line:
\[ runtime = a + bn \nonumber \]
where a is the intercept of the line and b is the slope.
On the other hand, if add is linear, the total time for n adds would be quadratic. If we plot run time versus problem size, we expect to see a parabola. Or mathematically, something like:
\[ runtime = a + bn + cn^2 \nonumber \]
With perfect data, we might be able to tell the difference between a straight line and a parabola, but if the measurements are noisy, it can be hard to tell. A better way to interpret noisy measurements is to plot run time and problem size on a log-log scale.
Why? Let’s suppose that run time is proportional to \( n^k \), but we don’t know what the exponent k is. We can write the relationship like this:
\[ runtime - a + bn + \dots + cn^k \nonumber \]
For large values of n, the term with the largest exponent is the most important, so:
\[ runtime \approx cn^k \nonumber \]
where \( \approx \) means “approximately equal”. Now, if we take the logarithm of both sides of this equation:
\[ \log(runtime) \approx \log(c) + k\log(n) \nonumber \]
This equation implies that if we plot runtime versus n on a log-log scale, we expect to see a straight line with intercept \( \log(c) \) and slope k. We don’t care much about the intercept, but the slope indicates the order of growth: if \( k = 1\), the algorithm is linear; if \( k = 2 \), it’s quadratic.
Looking at the figure in the previous section, you can estimate the slope by eye. But when you call plotResults it computes a least squares fit to the data and prints the estimated slope. In this example:
Estimated slope = 1.06194352346708
which is close to 1; and that suggests that the total time for n adds is linear, so each add is constant time, as expected.
One important point: if you see a straight line on a graph like this, that does not mean that the algorithm is linear. If the run time is proportional to \( n^k \) for any exponent k, we expect to see a straight line with slope k. If the slope is close to 1, that suggests the algorithm is linear. If it is close to 2, it’s probably quadratic.