Fix a notation clash and some typos.

33-hashfunc.tex

@@ -87,29 +87,30 @@ for different kinds of key sets.
 		would be ``1100101'' or ``10101`.
 
 		To define a hash code for this case, we first pick an arbitrary,
-		but fixed numbering $| \cdot |$ of the symbols in $\Sigma$ with
+		but fixed numbering $\| \cdot \|$ of the symbols in $\Sigma$ with
 		the numbers from $0$ to $|\Sigma| -1$.
 		For example, for the Latin alphabet, we could take 
-		$|A| = 0, |B| = 1, |C| = 2, \dots, |Z| = 25$, or for the
-		binary alphabet, we could take $|0| = 0, |1| = 1$. Now, 
+		$\|A\| = 0, \|B\| = 1, \|C\| = 2, \dots, \|Z\| = 25$, or for the
+		binary alphabet, we could take $\|0\| = 0, \|1\| = 1$. Now, 
 		given a string $\tau$ over an alphabet $\Sigma$, we write
 		$\tau$ as a sequence as symbols: $\tau = \sigma_0 \sigma_1 \dots \sigma_{\ell - 1}$,
 		where $\ell$ is the total number of symbols in $\tau$. 
 		For example, if $\tau = \text{HELLO}$,
 		we have $\ell = 5$ and  
-		$\sigma_0 = H, \sigma_1 = E, \sigma_2 = L, \sigma_3 = L, \sigma_4 = O$.
+		$\sigma_0 = \text{H}, \sigma_1 = \text{E}, \sigma_2 = \text{L}, 
+		\sigma_3 = \text{L}, \sigma_4 = \text{O}$.
 		Then we define
 		\[
-			\text{hc}(\tau) = \sum_{i = 0}^{\ell - 1} |\sigma_i| \cdot |\Sigma|^i.
+			\text{hc}(\tau) = \sum_{i = 0}^{\ell - 1} \|\sigma_i\| \cdot |\Sigma|^i.
 		\]
 		In other words, we interpret $\tau$ as a ``number`` to base $|\Sigma|$, where the
 		''digits'' are represented by the symbols of $\Sigma$. For example,
 		\begin{align*}
-			\text{hc}(HALLO) &= |H| \cdot 26^0 + 
-                                           |A| \cdot 26^1 +
-                                           |L| \cdot 26^2 +
-                                           |L| \cdot 26^3 +
-                                           |O| \cdot 26^4 \\
+			\text{hc}(\text{HALLO}) &= \|\text{H}\| \cdot 26^0 + 
+					   \|\text{A}\| \cdot 26^1 +
+					   \|\text{L}\| \cdot 26^2 +
+					   \|\text{L}\| \cdot 26^3 +
+					   \|\text{O}\| \cdot 26^4 \\
 					   &= 7 \cdot 1 + 0 \cdot 26 + 11 \cdot 26^2 + 11 \cdot 26^3
 					   + 14 \cdot 26^4\\ 
 					   &= 6.598.443.

44-stringsearch.tex

@@ -237,15 +237,15 @@ we described the following way for computing a hash function $h'$
 for a string 
 $a = \alpha_0 \alpha_1 \dots \alpha_{\ell-1}$: pick a prime number $p$ and interpret the individual
 symbols
-$\alpha_i$ as numbers between $0$ and $|\Sigma | - 1$. Then, set 
+$\alpha_i$ as numbers $\| \alpha_i\|$ between $0$ and $|\Sigma | - 1$. Then, set 
 \[
-  h'(a) = \left(\sum_{j=0}^{\ell-1} \alpha_j |\Sigma|^{j}\right) \bmod p.
+  h'(a) = \left(\sum_{j=0}^{\ell-1} \|\alpha_j\||\Sigma|^{j}\right) \bmod p.
 \]
 For Rabin-Karp, it turns out to be advantageous to define the hash function slightly
 differently:
 \[
 	h(\sigma_i\dots \sigma_{i+\ell-1}) = 
-	\left(\sum_{j=0}^{\ell-1} \sigma_{i+j} |\Sigma|^{\ell-1-j}\right) \bmod p.
+	\left(\sum_{j=0}^{\ell-1} \|\sigma_{i+j}\| |\Sigma|^{\ell-1-j}\right) \bmod p.
 \]
 There are two main differences between $h$ and $h'$:
 \begin{enumerate}
@@ -260,8 +260,8 @@ Now, the main point is that
 Indeed, we have
 \[
   h(\sigma_{i+1}\dots \sigma_{i+\ell}) = 
-  \left(|\Sigma| \cdot h(\sigma_i\dots \sigma_{i+\ell-1})- |\Sigma|^\ell \cdot \sigma_i +
-  \sigma_{i+\ell}\right) \bmod p.
+  \left(|\Sigma| \cdot h(\sigma_i\dots \sigma_{i+\ell-1})- |\Sigma|^\ell \cdot \|\sigma_i\| +
+  \|\sigma_{i+\ell}\|\right) \bmod p.
 \]
 (Note that we can precompute $|\Sigma|^\ell$  in advance and can reuse it every time
 we update the hash function). We call $h$ a \emph{rolling hash}.
@@ -327,7 +327,8 @@ It remains to explain how to find a random prime number between
 $2$ and $\ell^2 \log (|\Sigma|\ell)$. For this, we simply take a random
 number between $2$ and $\ell^2 \log (|\Sigma|\ell)$ and check if it is a prime number.
 If the test fails, we repeat. By the prime number theorem, the probability that we find a
-prime number of $\Omega(1/$. Thus, we need $O(\log(\ell \Sigma))$ attempts in expectation to 
+	prime number is $\Omega(1/\log(\ell |\Sigma|))$. 
+	Thus, we need $O(\log(\ell |\Sigma|))$ attempts in expectation to 
 find such a number. There are very efficient algorithms testing whether a given number is a prime
 number. Thus, the time for this step is negligible.
 \end{proof}