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Antti H S Laaksonen 2017-02-18 19:13:37 +02:00
parent 439890f3d6
commit 3fb9b598ae
1 changed files with 12 additions and 13 deletions
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luku28.tex
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@ -36,10 +36,9 @@ Using this approach, the function becomes as follows:
\begin{lstlisting} \begin{lstlisting}
int sum(int a, int b, int k, int x, int y) { int sum(int a, int b, int k, int x, int y) {
if (b < x || a > y) return 0; if (b < x || a > y) return 0;
if (a == x && b == y) return p[k]; if (a <= x && y <= b) return p[k];
int d = (y-x+1)/2; int d = (y-x+1)/2;
return sum(a, min(x+d-1,b), 2*k, x, x+d-1) + return sum(a,b,2*k,x,x+d-1) + sum(a,b,2*k+1,x+d,y);
sum(max(x+d,a), b, 2*k+1, x+d, y);
} }
\end{lstlisting} \end{lstlisting}
Now we can calculate the sum of Now we can calculate the sum of
@ -57,11 +56,11 @@ at the root of the segment tree.
The range $[x,y]$ corresponds to $k$ The range $[x,y]$ corresponds to $k$
and is initially $[0,N-1]$. and is initially $[0,N-1]$.
When calculating the sum, When calculating the sum,
if $[a,b]$ is outside $[x,y]$, if $[x,y]$ is outside $[a,b]$,
the sum is 0, the sum is 0,
and if $[a,b]$ equals $[x,y]$, and if $[x,y]$ is completely inside $[a,b]$,
the sum can be found in \texttt{p}. the sum can be found in \texttt{p}.
If $[a,b]$ is completely or partially inside $[x,y]$, If $[x,y]$ is partially inside $[a,b]$,
the search continues recursively to the the search continues recursively to the
left and right half of $[x,y]$. left and right half of $[x,y]$.
The size of both halves is $d=\frac{1}{2}(y-x+1)$; The size of both halves is $d=\frac{1}{2}(y-x+1)$;
@ -166,7 +165,7 @@ stops and the sum can be found in \texttt{p}.
\node at (13.5,-0.75) {$b$}; \node at (13.5,-0.75) {$b$};
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
Also in this implementation, Also using this implementation,
operations take $O(\log n)$ time, operations take $O(\log n)$ time,
because the total number of visited nodes is $O(\log n)$. because the total number of visited nodes is $O(\log n)$.
@ -192,7 +191,7 @@ In addition, the node may contain information
related to lazy updates, which has not been related to lazy updates, which has not been
propagated to its children. propagated to its children.
There are two possible types of range updates: There are two types of range updates:
each element in the range is either each element in the range is either
\emph{increased} by some value \emph{increased} by some value
or \emph{assigned} some value. or \emph{assigned} some value.
@ -300,7 +299,7 @@ where $h$ is the size of the intersection of $[a,b]$
and $[x,y]$, and continue our walk recursively in the tree. and $[x,y]$, and continue our walk recursively in the tree.
For example, the following picture shows the tree after For example, the following picture shows the tree after
increasing the elements in the range $[a,b]$ by 2: increasing the elements in $[a,b]$ by 2:
\begin{center} \begin{center}
\begin{tikzpicture}[scale=0.7] \begin{tikzpicture}[scale=0.7]
\fill[color=gray!50] (5,0) rectangle (6,1); \fill[color=gray!50] (5,0) rectangle (6,1);
@ -518,7 +517,7 @@ in the range $[a,b]$ is $p(i-a)$.
For example, adding $p(u)=u+1$ For example, adding $p(u)=u+1$
to $[a,b]$ means that the element at position $a$ to $[a,b]$ means that the element at position $a$
is increased by 1, the element at position $a+1$ is increased by 1, the element at position $a+1$
is increased by 2, etc. is increased by 2, and so on.
To support polynomial updates, To support polynomial updates,
each node is assigned $k+2$ values, each node is assigned $k+2$ values,
@ -653,7 +652,7 @@ are generated during the algorithm.
Using a dynamic implementation, Using a dynamic implementation,
it is also possible to create a it is also possible to create a
\key{persistent segment tree} that stores \key{persistent segment tree} that stores
the \key{modification history} of the tree. the \emph{modification history} of the tree.
In such an implementation, we can In such an implementation, we can
efficiently access efficiently access
all versions of the tree that have all versions of the tree that have
@ -813,7 +812,7 @@ in the following range:
The idea is to build a segment tree The idea is to build a segment tree
where each node is assigned a data structure where each node is assigned a data structure
that can calculate the number of any element $x$ that gives the number of any element $x$
in the corresponding range. in the corresponding range.
Using such a segment tree, Using such a segment tree,
the answer to a query can be calculated the answer to a query can be calculated
@ -965,7 +964,7 @@ corresponds to the above array:
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
For example, we can build the tree so We can build the tree so
that each node contains a \texttt{map} structure. that each node contains a \texttt{map} structure.
In this case, the time needed for processing each In this case, the time needed for processing each
node is $O(\log n)$, so the total time complexity node is $O(\log n)$, so the total time complexity