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Some fixes

This commit is contained in:
Antti H S Laaksonen 2017-05-25 23:23:54 +03:00
parent 43312caa92
commit d92fc7ea20
1 changed files with 16 additions and 19 deletions
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chapter05.tex
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@ -56,7 +56,7 @@ void search(int k) {
When the function \texttt{search} When the function \texttt{search}
is called with parameter $k$, is called with parameter $k$,
it decides whether to include it decides whether to include the
element $k$ in the subset or not, element $k$ in the subset or not,
and in both cases, and in both cases,
then calls itself with parameter $k+1$ then calls itself with parameter $k+1$
@ -137,7 +137,7 @@ the last bit corresponds to element 0,
the second last bit corresponds to element 1, the second last bit corresponds to element 1,
and so on. and so on.
For example, the bit representation of 25 For example, the bit representation of 25
is 11001, that corresponds to the subset $\{0,3,4\}$. is 11001, which corresponds to the subset $\{0,3,4\}$.
The following code goes through the subsets The following code goes through the subsets
of a set of $n$ elements of a set of $n$ elements
@ -183,7 +183,7 @@ using recursion.
The following function \texttt{search} goes The following function \texttt{search} goes
through the permutations of the set $\{0,1,\ldots,n-1\}$. through the permutations of the set $\{0,1,\ldots,n-1\}$.
The function builds a vector \texttt{permutation} The function builds a vector \texttt{permutation}
that describes the permutation, that contains the permutation,
and the search begins when the function is and the search begins when the function is
called without parameters. called without parameters.
@ -349,10 +349,10 @@ void search(int y) {
return; return;
} }
for (int x = 0; x < n; x++) { for (int x = 0; x < n; x++) {
if (r1[x] || r2[x+y] || r3[x-y+n-1]) continue; if (column[x] || diag1[x+y] || diag2[x-y+n-1]) continue;
r1[x] = r2[x+y] = r3[x-y+n-1] = 1; column[x] = diag1[x+y] = diag2[x-y+n-1] = 1;
search(y+1); search(y+1);
r1[x] = r2[x+y] = r3[x-y+n-1] = 0; column[x] = diag1[x+y] = diag2[x-y+n-1] = 0;
} }
} }
\end{lstlisting} \end{lstlisting}
@ -368,13 +368,13 @@ When the function \texttt{search} is
called with parameter $y$, called with parameter $y$,
it places a queen to row $y$ it places a queen to row $y$
and then calls itself with parameter $y+1$. and then calls itself with parameter $y+1$.
However, if $y=n$, a solution has been found Then, if $y=n$, a solution has been found
and the variable \texttt{count} is increased by one. and the variable \texttt{count} is increased by one.
The array \texttt{r1} keeps track of the columns The array \texttt{column} keeps track of columns
that already contain a queen, that contain a queen,
and the arrays \texttt{r2} and \texttt{r3} and the arrays \texttt{diag1} and \texttt{diag2}
keep track of the diagonals. keep track of diagonals.
It is not allowed to add another queen to a It is not allowed to add another queen to a
column or diagonal that already contains a queen. column or diagonal that already contains a queen.
For example, the columns and diagonals of For example, the columns and diagonals of
@ -437,9 +437,9 @@ the $4 \times 4$ board are numbered as follows:
\node at (8.5,0.5) {$2$}; \node at (8.5,0.5) {$2$};
\node at (9.5,0.5) {$3$}; \node at (9.5,0.5) {$3$};
\node at (-4,-1) {\texttt{r1}}; \node at (-4,-1) {\texttt{column}};
\node at (2,-1) {\texttt{r2}}; \node at (2,-1) {\texttt{diag1}};
\node at (8,-1) {\texttt{r3}}; \node at (8,-1) {\texttt{diag2}};
\end{scope} \end{scope}
\end{tikzpicture} \end{tikzpicture}
@ -619,9 +619,6 @@ the path can turn either left or right:
(3.5,0.5) -- (3.5,1.5) -- (1.5,1.5) -- (3.5,0.5) -- (3.5,1.5) -- (1.5,1.5) --
(1.5,2.5) -- (4.5,2.5) -- (4.5,0.5) -- (1.5,2.5) -- (4.5,2.5) -- (4.5,0.5) --
(5.5,0.5) -- (5.5,6.5); (5.5,0.5) -- (5.5,6.5);
\node at (4.5,6.5) {$a$};
\node at (6.5,6.5) {$b$};
\end{scope} \end{scope}
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
@ -716,10 +713,10 @@ to choose some numbers from the list so that
their sum is $x$. their sum is $x$.
For example, given the list $[2,4,5,9]$ and $x=15$, For example, given the list $[2,4,5,9]$ and $x=15$,
we can choose the numbers $[2,4,9]$ to get $2+4+9=15$. we can choose the numbers $[2,4,9]$ to get $2+4+9=15$.
However, if $x=10$, However, if $x=10$ for the same list,
it is not possible to form the sum. it is not possible to form the sum.
An easy solution to the problem is to A simple algorithm to the problem is to
go through all subsets of the elements and go through all subsets of the elements and
check if the sum of any of the subsets is $x$. check if the sum of any of the subsets is $x$.
The running time of such an algorithm is $O(2^n)$, The running time of such an algorithm is $O(2^n)$,