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Small fixes

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Antti H S Laaksonen 2017-04-18 20:08:21 +03:00
parent b4e89f13ec
commit e256200528
1 changed files with 5 additions and 5 deletions
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chapter03.tex
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@ -104,7 +104,7 @@ Whenever two consecutive elements are found
that are not in correct order, that are not in correct order,
the algorithm swaps them. the algorithm swaps them.
The algorithm can be implemented as follows The algorithm can be implemented as follows
for an array \texttt{x}: for an array \texttt{t}:
\begin{lstlisting} \begin{lstlisting}
for (int i = 0; i < n; i++) { for (int i = 0; i < n; i++) {
for (int j = 0; j < n-1; j++) { for (int j = 0; j < n-1; j++) {
@ -260,7 +260,7 @@ as follows:
\index{inversion} \index{inversion}
Bubble sort is an example of a sorting Bubble sort is an example of a sorting
algorithm that always swaps consecutive algorithm that always swaps \emph{consecutive}
elements in the array. elements in the array.
It turns out that the time complexity It turns out that the time complexity
of such an algorithm is \emph{always} of such an algorithm is \emph{always}
@ -600,7 +600,7 @@ corresponds to the following bookkeeping array:
For example, the value at position 3 For example, the value at position 3
in the bookkeeping array is 2, in the bookkeeping array is 2,
because the element 3 appears 2 times because the element 3 appears 2 times
in the original array (positions 2 and 6). in the original array.
Construction of the bookkeeping array Construction of the bookkeeping array
takes $O(n)$ time. After this, the sorted array takes $O(n)$ time. After this, the sorted array
@ -783,7 +783,7 @@ for (int i = 0; i < n; i++) {
The time complexity of this approach is $O(n)$, The time complexity of this approach is $O(n)$,
because in the worst case, it is needed to check because in the worst case, it is needed to check
all elements in the array. all elements in the array.
If the array may contain any elements, If the order of the elements is arbitrary,
this is also the best possible approach, because this is also the best possible approach, because
there is no additional information available where there is no additional information available where
in the array we should search for the element $x$. in the array we should search for the element $x$.
@ -825,7 +825,7 @@ while (a <= b) {
The algorithm maintains a range $a \ldots b$ The algorithm maintains a range $a \ldots b$
that corresponds to the active region of the array. that corresponds to the active region of the array.
Initially, the range is $1 \ldots n$, the whole array. Initially, the range is $0 \ldots n-1$, the whole array.
The algorithm halves the size of the range at each step, The algorithm halves the size of the range at each step,
so the time complexity is $O(\log n)$. so the time complexity is $O(\log n)$.