Better terms for sum arrays

chapter09.tex

@@ -80,14 +80,14 @@ the answer for any possible query.
 
 \subsubsection{Sum queries}
 
-\index{sum array}
+\index{prefix sum array}
 
 It turns out that we can easily process
 sum queries on a static array,
-because we can construct a data structure called
+because we can use a data structure called
-a \key{sum array}.
+a \key{prefix sum array}.
-Each element in a sum array corresponds to
+Each value in such an array equals
-the sum of elements in the original array up to that position.
+the sum of values in the original array up to that position.
 
 For example, consider the following array:
 \begin{center}
@@ -115,7 +115,7 @@ For example, consider the following array:
 \node at (7.5,1.4) {$8$};
 \end{tikzpicture}
 \end{center}
-The corresponding sum array is as follows:
+The corresponding prefix sum array is as follows:
 \begin{center}
 \begin{tikzpicture}[scale=0.7]
 %\fill[color=lightgray] (3,0) rectangle (7,1);
@@ -144,7 +144,7 @@ The corresponding sum array is as follows:
 \end{center}
 Let $\textrm{sum}(a,b)$ denote the sum of elements
 in the range $[a,b]$.
-Since the sum array contains all values
+Since the prefix sum array contains all values
 of the form $\textrm{sum}(1,k)$,
 we can calculate any value of
 $\textrm{sum}(a,b)$ in $O(1)$ time, because
@@ -180,7 +180,7 @@ For example, consider the range $[4,7]$:
 \end{center}
 The sum in the range is $8+6+1+4=19$.
 This sum can be calculated using
-two values in the sum array:
+two values in the prefix sum array:
 \begin{center}
 \begin{tikzpicture}[scale=0.7]
 \fill[color=lightgray] (2,0) rectangle (3,1);
@@ -213,7 +213,7 @@ Thus, the sum in the range $[4,7]$ is $27-8=19$.
 It is also possible to generalize this idea
 to higher dimensions.
 For example, we can construct a two-dimensional
-sum array that can be used for calculating
+prefix sum array that can be used for calculating
 the sum of any rectangular subarray in $O(1)$ time.
 Each value in such an array is the sum of a subarray
 that begins at the upper-left corner of the array.
@@ -441,7 +441,7 @@ we can conclude that $\textrm{rmq}(2,7)=1$.
 \index{Fenwick tree}
 
 A \key{binary indexed tree} or \key{Fenwick tree} \cite{fen94}
-can be seen as a dynamic variant of a sum array.
+can be seen as a dynamic version of a prefix sum array.
 This data structure supports two $O(\log n)$ time operations:
 calculating the sum of elements in a range
 and modifying the value of an element.
@@ -449,9 +449,9 @@ and modifying the value of an element.
 The advantage of a binary indexed tree is
 that it allows us to efficiently update
 the array elements between the sum queries.
-This would not be possible using a sum array,
+This would not be possible using a prefix sum array,
 because after each update, we should build the
-whole sum array again in $O(n)$ time.
+whole array again in $O(n)$ time.
 
 \subsubsection{Structure}
 
@@ -628,7 +628,7 @@ the following values:
 Hence, the sum of elements in the range $[1,7]$ is $16+7+4=27$.
 
 To calculate the sum of elements in any range $[a,b]$,
-we can use the same trick that we used with sum arrays:
+we can use the same trick that we used with prefix sum arrays:
 \[ \textrm{sum}(a,b) = \textrm{sum}(1,b) - \textrm{sum}(1,a-1).\]
 Also in this case, only $O(\log n)$ values are needed.
 
@@ -1316,10 +1316,14 @@ elements in a range $[a,b]$ by $x$.
 
 Surprisingly, we can use the data structures
 presented in this chapter also in this situation.
-To do this, we build an \key{inverse sum array}
+To do this, we build a \key{difference array}
 for the array.
-The idea is that the original array is the sum array of the
+In such an array, each value indicates
-inverse sum array.
+the difference between two consecutive values
+in the original array.
+Thus, the original array is the
+prefix sum array of the
+difference array.
 For example, consider the following array:
 
 \begin{center}
@@ -1348,7 +1352,7 @@ For example, consider the following array:
 \end{tikzpicture}
 \end{center}
 
-The inverse sum array for the above array is as follows:
+The difference array for the above array is as follows:
 \begin{center}
 \begin{tikzpicture}[scale=0.7]
 \draw (0,0) grid (8,1);
@@ -1378,10 +1382,10 @@ The inverse sum array for the above array is as follows:
 For example, the value 5 at position 6 in the original array
 corresponds to the sum $3-2+4=5$.
 
-The advantage of the inverse sum array is
+The advantage of the difference array is
 that we can update a range
 in the original array by changing just
-two elements in the inverse sum array.
+two elements in the difference array.
 For example, if we want to 
 increase the elements in the range $2 \ldots 5$ by 5,
 it suffices to increase the value at position 2 by 5