Aufgaben:Exercise 2.4Z: LZW Coding and Decoding again - Revision history

Guenter at 09:34, 31 August 2021

2021-08-31T09:34:39Z

Guenter at 15:12, 30 August 2021

2021-08-30T15:12:37Z

Guenter at 12:12, 10 August 2021

2021-08-10T12:12:01Z

Noah at 20:05, 6 August 2021

2021-08-06T20:05:49Z

Guenter at 13:57, 17 July 2021

2021-07-17T13:57:27Z

Guenter at 12:44, 17 July 2021

2021-07-17T12:44:00Z

Guenter at 12:39, 17 July 2021

2021-07-17T12:39:28Z

Noah at 11:59, 15 July 2021

2021-07-15T11:59:00Z

Noah at 10:58, 15 July 2021

2021-07-15T10:58:27Z

Guenter at 13:37, 13 July 2021

2021-07-13T13:37:11Z

@@ Line 3: / Line 3: @@
 }}
-[[File:EN_Inf_Z2.4_vers2.png|right|frame|Two snapshots of the LZW dictionary &nbsp; &rArr; &nbsp; "Wörterbuch"; <br> "Inhalt" &nbsp; &rArr; &nbsp; "Content"]]
+[[File:EN_Inf_Z2.4_vers2.png|right|frame|Two snapshots of the LZW dictionary ]]
 The graph shows snapshots of the dictionary created during LZW coding of the input symbol sequence.
 *The upper graph is for input symbol sequence&nbsp; <b>ABABABBAA</b>.

@@ Line 3: / Line 3: @@
 }}
-[[File:EN_Inf_Z_2_4.png|right|frame|Two snapshots of the LZW dictionary &nbsp; &rArr; &nbsp; "Wörterbuch"; <br> "Inhalt" &nbsp; &rArr; &nbsp; "Content"]]
+[[File:EN_Inf_Z2.4_vers2.png|right|frame|Two snapshots of the LZW dictionary &nbsp; &rArr; &nbsp; "Wörterbuch"; <br> "Inhalt" &nbsp; &rArr; &nbsp; "Content"]]
 The graph shows snapshots of the dictionary created during LZW coding of the input symbol sequence.
 *The upper graph is for input symbol sequence&nbsp; <b>ABABABBAA</b>.

← Older revision		Revision as of 12:12, 10 August 2021
Line 1:		Line 1:

−	{{quiz-Header\|Buchseite=~~Informationstheorie~~/~~Komprimierung nach Lempel~~, ~~Ziv und Welch~~	+	{{quiz-Header\|Buchseite=Information_Theory/Compression_According_to_Lempel,_Ziv_and_Welch
	}}		}}

@@ Line 37: / Line 37: @@
-''Hints:''
+Hints:
 *The exercise belongs to the chapter&nbsp; [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch|Compression according to Lempel, Ziv and Welch]].
 *In particular, reference is made to the pages&nbsp;

@@ Line 3: / Line 3: @@
 }}
-[[File:EN_Inf_Z_2_4.png|right|frame|Snapshots of the <br>LZW dictionaries]]
+[[File:EN_Inf_Z_2_4.png|right|frame|Two snapshots of the LZW dictionary &nbsp; &rArr; &nbsp; "Wörterbuch"; <br> "Inhalt" &nbsp; &rArr; &nbsp; "Content"]]
 The graph shows snapshots of the dictionary created during LZW coding of the input symbol sequence.
 *The upper graph is for input symbol sequence&nbsp; <b>ABABABBAA</b>.
@@ Line 84: / Line 84: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; We use&nbsp; $W(I)$&nbsp; to denote a field (array), that describes the dictionary and whose elements contain characters or strings. &nbsp;
+'''(1)'''&nbsp; We use&nbsp; $W(I)$&nbsp; to denote a field (array),&nbsp; that describes the dictionary and whose elements contain characters or strings. &nbsp;
-*The encoding of&nbsp; <b>ABABABBAA</b>&nbsp; then proceeds as follows:
+*The encoding of&nbsp; <b>ABABABBAA</b>&nbsp; proceeds as follows:
 :: $i = 1$: &nbsp; &nbsp; <b>A</b> &nbsp;&nbsp;&nbsp;&#8594;&nbsp;&nbsp; $\underline{I=0}$, &nbsp; $W(I = 2) =$ <b>AB</b>,
 :: $i = 2$: &nbsp; &nbsp; <b>B</b> &nbsp;&nbsp;&nbsp;&#8594;&nbsp;&nbsp; $\underline{I=1}$, &nbsp; $W(I = 3) =$ <b>BA</b>,
@@ Line 100: / Line 100: @@
 *Afterwards, the following applies:
 :: $i = 4$: &nbsp; &nbsp; <b>ABA</b> &#8594;&nbsp;&nbsp; $\underline{I=4}$, &nbsp; $W(I = 5) =$ <b>ABAB</b>,
-:: $i = 5$: &nbsp; &nbsp; <b>BA</b> &#8594;&nbsp;&nbsp; $\underline{I=3}$, &nbsp; coding completed, no new dictionary entry possible.
+:: $i = 5$: &nbsp; &nbsp; <b>BA</b> &#8594;&nbsp;&nbsp; $\underline{I=3}$, &nbsp; encoding completed, no new dictionary entry possible.
 '''(3)'''&nbsp; The comparison with above results shows:
-*The dictionary of the <u>coder</u> has the entries shown exactly after&nbsp; $\underline{i=4}$&nbsp; coding steps.
+*The dictionary of the&nbsp; <u>encoder</u>&nbsp; has the entries shown exactly after&nbsp; $\underline{i=4}$&nbsp; encoding steps.
-*With the <u>decoder</u>, on the other hand, there is a time delay of one step:  &nbsp; $\underline{i=5}$.
+*With the&nbsp; <u>decoder</u>, on the other hand, there is a time delay of one step:  &nbsp; $\underline{i=5}$.
@@ Line 113: / Line 113: @@
 '''(4)'''&nbsp; <u>Solution suggestion 2</u> is correct:
 *The special case regulation of the decoding is necessary&nbsp; (in the present example)&nbsp; if the index&nbsp; $I =i$&nbsp; is output in the coding step&nbsp; $i$&nbsp;.
-*During decoding, it then does not find the required assignment&nbsp; "Index&nbsp; &#8594;&nbsp; String"&nbsp; since the generated dictionary at time&nbsp; $i$&nbsp; only contains entries with indices&nbsp; $I < i$&nbsp;.
+*During decoding, it then does not find the required assignment&nbsp; "Index&nbsp; &#8594;&nbsp; String"&nbsp; since the generated dictionary at time&nbsp; $i$&nbsp; only contains entries with&nbsp; $I < i$&nbsp;.
-*For the sequence&nbsp; <b>ABABABBAA</b>&nbsp; &nbsp; $I < i$ always applies according to subtask&nbsp; '''(1)'''&nbsp;.
+*For the sequence&nbsp; <b>ABABABBAA</b> &nbsp; &rArr; &nbsp; $I < i$ always applies according to subtask&nbsp; '''(1)'''&nbsp;.
 *In contrast, the following indices would result for the sequence&nbsp; <b>ABABABABA</b>&nbsp;:
 :$$i = 1\hspace{-0.15cm}: \hspace{0.15cm} I = 0\hspace{0.05cm},
@@ Line 122: / Line 122: @@
 \hspace{0.5cm}i = 5\hspace{-0.15cm}: \hspace{0.15cm}I = 3\hspace{0.05cm}.  $$
-Here is a summary of the entire decoding of&nbsp; <b>ABABABABA</b>:
+Here is a summary of the entire decoding process of&nbsp; <b>ABABABABA</b>:
 *The pre-assignment of the dictionary includes&nbsp; $(I=0$:&nbsp; <b>A</b>$)$ &nbsp; &nbsp; and&nbsp; $(I=1$:&nbsp; <b>B</b>$)$.
 *Then with the dictionary array&nbsp; $W(I)$:
-:: $i = 1$: &nbsp; &nbsp; decoding &nbsp; $I=0$ &nbsp; &#8594; &nbsp; <b>A</b>,
+:: $i = 1$: &nbsp; &nbsp; Decoding &nbsp; $I=0$ &nbsp; &#8594; &nbsp; <b>A</b>,
-:: $i = 2$: &nbsp; &nbsp; decoding&nbsp; $I=1$ &nbsp; &#8594; &nbsp; <b>B</b>, &nbsp; &nbsp; &nbsp; $W(I = 2) =$ <b>AB</b>,
+:: $i = 2$: &nbsp; &nbsp; Decoding&nbsp; $I=1$ &nbsp; &#8594; &nbsp; <b>B</b>, &nbsp; &nbsp; &nbsp; $W(I = 2) =$ <b>AB</b>,
-:: $i = 3$: &nbsp; &nbsp; decoding&nbsp; $I=2$ &nbsp; &#8594; &nbsp; <b>AB</b>, &nbsp;&nbsp;&nbsp; $W(I = 3) =$ <b>BA</b>,
+:: $i = 3$: &nbsp; &nbsp; Decoding&nbsp; $I=2$ &nbsp; &#8594; &nbsp; <b>AB</b>, &nbsp;&nbsp;&nbsp; $W(I = 3) =$ <b>BA</b>,
 :: $i = 4$: &nbsp; &nbsp; An entry with index&nbsp; $I = 4$&nbsp; is not present &nbsp; &#8658; &nbsp; '''Special case rule''': <br>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; One takes the last decoding result&nbsp; $($here&nbsp; <b>AB</b>$)$ and adds the first character of this sequence at the end &nbsp; &#8658; &nbsp; <b>ABA</b>. <br>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Then&nbsp; <b>ABA</b>&nbsp; is stored in the dictionary under&nbsp; index&nbsp; $I = 4$&nbsp;.
 :: $i = 5$: &nbsp; &nbsp; Decoding &nbsp; $I=3$ &nbsp; &#8594; &nbsp; <b>BA</b>. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  '''End of the decoder input sequence'''.

@@ Line 50: / Line 50: @@
 <quiz display=simple>
-{Code the input sequence&nbsp; <b>ABABABBAA</b>&nbsp; according to the <u>above diagram</u>.&nbsp; Which indices&nbsp; $I_i$&nbsp; result for the steps&nbsp; $i=1$, ... , $i=5$?
+{Encode the input sequence&nbsp; <b>ABABABBAA</b>&nbsp; according to the <u>above diagram</u>.&nbsp; Which indices&nbsp; $I_i$&nbsp; result for the steps&nbsp; $i=1$, ... , $i=5$?
 |type="{}"}
 $I_1 \ = \ $ { 0. }
@@ Line 60: / Line 60: @@
-{Now code the input sequence&nbsp; <b>ABABABABA</b>&nbsp; according to the <u>diagram below</u>.&nbsp; Specify the indices&nbsp; $I_i$&nbsp; for the steps&nbsp; $i=4$&nbsp; and&nbsp; $i=5$&nbsp;.
+{Now encode the input sequence&nbsp; <b>ABABABABA</b>&nbsp; according to the <u>diagram below</u>.&nbsp; Specify the indices&nbsp; $I_i$&nbsp; for the steps&nbsp; $i=4$&nbsp; and&nbsp; $i=5$&nbsp;.
 |type="{}"}
 $I_4 \ = \ $ { 4 }
@@ Line 66: / Line 66: @@
-{For which step&nbsp; $i$&nbsp; does the snapshot of the dictionary shown on the input page apply with respect to
+{For which step&nbsp; $i$&nbsp; does the snapshot of the dictionary shown on the exercise description page apply with respect to
 |type="{}"}
 $\text{Encoding:} \ \ i \ = \ $ { 4 }

@@ Line 3: / Line 3: @@
 }}
-[[File:EN_Inf_Z_2_4.png|right|frame|Snapshots of the LZW dictionaries]]
+[[File:EN_Inf_Z_2_4.png|right|frame|Snapshots of the <br>LZW dictionaries]]
 The graph shows snapshots of the dictionary created during LZW coding of the input symbol sequence.
 *The upper graph is for input symbol sequence&nbsp; <b>ABABABBAA</b>.
@@ Line 14: / Line 14: @@
-In <u>LZW coding</u>, an index $I$ is selected for each coding step $i$ and transmitted (binary).&nbsp; The character pair <b>AB</b> is represented by the index  $I = 2$ in the two dictionaries.&nbsp; Here we consider the index $I$ as a decimal number and leave the binary representation out of consideration for this task.
+In <u>LZW coding</u>, an index&nbsp; $I$&nbsp; is selected for each coding step&nbsp; $i$&nbsp; and transmitted (binary).&nbsp; The character pair&nbsp; <b>AB</b>&nbsp; is represented by the index&nbsp;  $I = 2$&nbsp; in the two dictionaries.&nbsp; Here we consider the index&nbsp; $I$&nbsp; as a decimal number and leave the binary representation out of consideration for this task.
 With <u>LZW decoding</u>, a character or character sequence is generated in the same way from each index&nbsp; $I$&nbsp; with the help of the dictionary, for example&nbsp; $I = 1$&nbsp; leads to the character&nbsp; <b>B</b>&nbsp; and&nbsp; $I = 2$&nbsp; to the character pair&nbsp; <b>AB</b>.
-If a dictionary entry with the desired index&nbsp; $I$&nbsp; is actually found, the decoding runs smoothly. However, this is not always the case:
+If a dictionary entry with the desired index&nbsp; $I$&nbsp; is actually found, the decoding runs smoothly.&nbsp; However, this is not always the case:
 * If a new index&nbsp; $I$&nbsp; is entered during encoding at step&nbsp; $i$&nbsp; and this&nbsp; $I$&nbsp; is at the same time the encoding result of the step, this index is not yet occupied in the dictionary at decoding step&nbsp; $i$&nbsp;.&nbsp; The reason for this is that with the decoder the entries are made one step later than with the coder.
-* In the case of a binary input sequence&nbsp; $($all characters are&nbsp; <b>A</b>&nbsp; or&nbsp; <b>B</b>$)$&nbsp; , a special rule must always be applied in LZW decoding if the entry with index&nbsp; $I = i$&nbsp; was made in coding step&nbsp; $i$&nbsp;.
+* In the case of a&nbsp; "binary"&nbsp; input sequence&nbsp; $($all characters are&nbsp; <b>A</b>&nbsp; or&nbsp; <b>B</b>$)$,&nbsp;  a special rule must always be applied in LZW decoding if the entry with index&nbsp; $I = i$&nbsp; was made in coding step&nbsp; $i$&nbsp;.

@@ Line 3: / Line 3: @@
 }}
-[[File:EN_Inf_Z_2_4.png|right|frame|Momentaufnahmen der LZW–Wörterbüchern]]
+[[File:EN_Inf_Z_2_4.png|right|frame|Snapshots of the LZW dictionaries]]
-Die Grafik zeigt Momentaufnahmen des Wörterbuchs, das während der&nbsp; <u>LZW&ndash;Codierung</u>&nbsp; der Eingangssymbolfolge entsteht.
+The graph shows snapshots of the dictionary created during LZW coding of the input symbol sequence.
-*Die obere Grafik gilt für Eingangssymbolfolge&nbsp; <b>ABABABBAA</b>.
+*The upper graph is for input symbol sequence&nbsp; <b>ABABABBAA</b>.
-*Das untere Wörterbuch entsteht bei der LZW&ndash;Codierung der Sequenz&nbsp; <b>ABABABABA</b>.
+*The lower dictionary is created during LZW coding of the sequence&nbsp; <b>ABABABABA</b>.
-In beiden Fällen wird vorausgesetzt, dass auch zu späteren Zeitpunkten keine anderen Zeichen als&nbsp; <b>A</b>&nbsp; und&nbsp; <b>B</b>&nbsp; vorkommen können.
+In both cases, it is assumed that no characters other than&nbsp; <b>A</b>&nbsp; and&nbsp; <b>B</b>&nbsp; can occur at later times.
-Bei der&nbsp; <u>LZW&ndash;Decodierung</u>&nbsp; entstehen gleiche Wörterbücher, doch erfolgen dann die Wörterbucheinträge erst einen Schritt später.&nbsp; In der Teilaufgabe&nbsp; '''(3)'''&nbsp; wird gefragt, für welchen Codierschritt bzw. für welchen Decodierschritt die beiden dargestellten Momentaufnahmen gültig sind.
+In <u>LZW decoding</u>, the same dictionaries are created, but then the dictionary entries occur one step later.&nbsp; In subtask&nbsp; '''(3)'''&nbsp; the question is asked for which coding step or for which decoding step the two snapshots shown are valid.
-Bei der <u>LZW&ndash;Codierung</u> wird zu jedem Codierschritt $i$ ein Index $I$ ausgewählt und (binär) übertragen.&nbsp; Das Zeichenpaar <b>AB</b> wird bei den beiden Wörterbüchern durch den Index $I = 2$ dargestellt.&nbsp; Wir betrachten hier den Index $I$ als Dezimalzahl und lassen bei dieser Aufgabe die Binärdarstellung außer Betracht.
+In <u>LZW coding</u>, an index $I$ is selected for each coding step $i$ and transmitted (binary).&nbsp; The character pair <b>AB</b> is represented by the index  $I = 2$ in the two dictionaries.&nbsp; Here we consider the index $I$ as a decimal number and leave the binary representation out of consideration for this task.
-Bei der <u>LZW&ndash;Decodierung</u> wird in gleicher Weise mit Hilfe des Wörterbuchs aus jedem Index&nbsp; $I$&nbsp; ein Zeichen bzw. eine Zeichenfolge generiert, zum Beispiel führt&nbsp; $I = 1$&nbsp; zum Zeichen&nbsp; <b>B</b>&nbsp; und&nbsp; $I = 2$&nbsp; zum Zeichenpaar&nbsp; <b>AB</b>.
+With <u>LZW decoding</u>, a character or character sequence is generated in the same way from each index&nbsp; $I$&nbsp; with the help of the dictionary, for example&nbsp; $I = 1$&nbsp; leads to the character&nbsp; <b>B</b>&nbsp; and&nbsp; $I = 2$&nbsp; to the character pair&nbsp; <b>AB</b>.
-Wird  tatsächlich ein Wörterbucheintrag mit dem gewünschten Index&nbsp; $I$&nbsp; gefunden, so läuft die Decodierung problemlos ab. Dies ist aber nicht immer so:
+If a dictionary entry with the desired index&nbsp; $I$&nbsp; is actually found, the decoding runs smoothly. However, this is not always the case:
-* Wird bei der Codierung beim Schritt&nbsp; $i$&nbsp; ein neuer Index&nbsp; $I$&nbsp; eingetragen und ist dieses&nbsp; $I$&nbsp; gleichzeitig das Codierergebnis des Schrittes, so ist dieser Index beim Decodierschritt&nbsp; $i$&nbsp; im Wörterbuch noch nicht belegt.&nbsp; Der Grund hierfür ist, dass  beim Decoder die Einträge um einen Schritt später erfolgen als beim Coder.
+* If a new index&nbsp; $I$&nbsp; is entered during encoding at step&nbsp; $i$&nbsp; and this&nbsp; $I$&nbsp; is at the same time the encoding result of the step, this index is not yet occupied in the dictionary at decoding step&nbsp; $i$&nbsp;.&nbsp; The reason for this is that with the decoder the entries are made one step later than with the coder.
-* Bei binärer Eingangsfolge&nbsp; $($alle Zeichen seien&nbsp; <b>A</b>&nbsp; oder&nbsp; <b>B</b>$)$&nbsp; ist bei der LZW&ndash;Decodierung genau immer dann eine Sonderregelung anzuwenden, wenn im Codierschritt&nbsp; $i$&nbsp; der Eintrag mit dem Index&nbsp; $I = i$&nbsp; vorgenommen wurde.
+* In the case of a binary input sequence&nbsp; $($all characters are&nbsp; <b>A</b>&nbsp; or&nbsp; <b>B</b>$)$&nbsp; , a special rule must always be applied in LZW decoding if the entry with index&nbsp; $I = i$&nbsp; was made in coding step&nbsp; $i$&nbsp;.
-Diese Sonderregelung soll an einem Beispiel veranschaulicht werden:
+This special rule shall be illustrated by an example:
-* Zum Schritt&nbsp; $i$&nbsp; gibt es keinen zum Index&nbsp; $I$&nbsp; passenden  Eintrag im Decoder&ndash;Wörterbuch.
+* For step&nbsp; $i$&nbsp; there is no entry in the decoder dictionary matching index&nbsp; $I$&nbsp;.
-* Wir nehmen an, dass beim vorherigen Schritt&nbsp; $(i- 1)$&nbsp; das Decodierergebnis&nbsp; <b>ABBABA</b>&nbsp; war.
+* We assume that in the previous step&nbsp; $(i- 1)$&nbsp; the decoding result was&nbsp; <b>ABBABA</b>&nbsp;.
-* Dann ergänzt man diese Zeichenfolge um das erste Zeichen der Folge.&nbsp; Hier:&nbsp; <b>ABBABAA</b>.
+* Then we add the first character of the sequence to this string. &nbsp; Here:&nbsp; <b>ABBABAA</b>.
-* Anschließend trägt man die Sequenz&nbsp; <b>ABBABAA</b>&nbsp; in das Wörterbuch unter dem Index&nbsp; $I$&nbsp; ein.
+* Then one enters the sequence&nbsp; <b>ABBABAA</b>&nbsp; into the dictionary under index&nbsp; $I$&nbsp;.
@@ Line 44: / Line 44: @@
 :: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Decoding_of_the_LZW_algorithm|Decoding of the LZW algorithm]].
-*Beachten Sie zudem bei der Lösung dieser Aufgabe, dass beim LZW&ndash;Algorithmus nicht von einem leeren Wörterbuch ausgegangen wird.
+*Also, when solving this task, note that the LZW algorithm does not assume an empty dictionary.
-*Vielmehr beinhalten die Indizes&nbsp; $I = 0$&nbsp; bis&nbsp; $I = M- 1$&nbsp; alle&nbsp; $M$&nbsp; zulässigen Zeichen des Alphabets.
+*Rather, the indices&nbsp; $I = 0$&nbsp; to&nbsp; $I = M- 1$&nbsp; contain all&nbsp; $M$&nbsp; permissible characters of the alphabet.
+===Questions===
 <quiz display=simple>
-{Codieren Sie die Eingangsfolge&nbsp; <b>ABABABBAA</b>&nbsp; entsprechend der <u>oberen Grafik</u>.&nbsp; Welche Indizes&nbsp; $I_i$&nbsp; ergeben sich zu den Schritten&nbsp; $i=1$, ... , $i=5$?
+{Code the input sequence&nbsp; <b>ABABABBAA</b>&nbsp; according to the <u>above diagram</u>.&nbsp; Which indices&nbsp; $I_i$&nbsp; result for the steps&nbsp; $i=1$, ... , $i=5$?
 |type="{}"}
 $I_1 \ = \ $ { 0. }
@@ Line 61: / Line 60: @@
-{Codieren Sie nun die Eingangsfolge&nbsp; <b>ABABABABA</b>&nbsp; entsprechend der <u>unteren Grafik</u>.&nbsp; Geben Sie die Indizes&nbsp; $I_i$&nbsp; zu den Schritten&nbsp; $i=4$&nbsp; und&nbsp; $i=5$&nbsp; an.
+{Now code the input sequence&nbsp; <b>ABABABABA</b>&nbsp; according to the <u>diagram below</u>.&nbsp; Specify the indices&nbsp; $I_i$&nbsp; for the steps&nbsp; $i=4$&nbsp; and&nbsp; $i=5$&nbsp;.
 |type="{}"}
 $I_4 \ = \ $ { 4 }
@@ Line 67: / Line 66: @@
-{Für welchen Schritt&nbsp; $i$&nbsp; gilt die Momentaufnahme des auf der Angabenseite dargestellten Wörterbuchs bezüglich
+{For which step&nbsp; $i$&nbsp; does the snapshot of the dictionary shown on the input page apply with respect to
 |type="{}"}
-$\text{Codierung:} \ \ i \ = \ $ { 4 }
+$\text{Encoding:} \ \ i \ = \ $ { 4 }
-$\text{Decodierung:} \ \ i \ = \ $ { 5 }
+$\text{Decoding:} \ \ i \ = \ $ { 5 }
-{Wann muss man auf die Decodier&ndash;Sonderfallregelung zurückgreifen?
+{When does one have to resort to the decoding special case rule?
 |type="[]"}
-- Bei der Decodierung von&nbsp; <b>ABABABBAA</b>&nbsp; im Schritt&nbsp; $i = 4$.
+- When decoding&nbsp; <b>ABABABBAA</b>&nbsp; in step&nbsp; $i = 4$.
-+ Bei der Decodierung von&nbsp; <b>ABABABABA</b>&nbsp; im Schritt&nbsp; $i = 4$.
++ When decoding&nbsp; <b>ABABABABA</b>&nbsp; in step&nbsp; $i = 4$.
-- Bei der Decodierung von&nbsp; <b>ABABABABA</b>&nbsp; im Schritt&nbsp; $i = 5$.
+- When decoding&nbsp; <b>ABABABABA</b>&nbsp; in step&nbsp; $i = 5$.
@@ Line 83: / Line 82: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
 '''(1)'''&nbsp; Wir bezeichnen mit&nbsp; $W(I)$&nbsp; ein Feld (Array), welches das Wörterbuch beschreibt und dessen Elemente Character oder Zeichenfolgen beinhalten.&nbsp;

@@ Line 37: / Line 37: @@
-''Hinweise:''
+''Hints:''
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch|Komprimierung nach Lempel, Ziv und Welch]].
+*The exercise belongs to the chapter&nbsp; [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch|Compression according to Lempel, Ziv and Welch]].
-*Insbesondere wird  Bezug genommen auf die Seiten
+*In particular, reference is made to the pages&nbsp;
-:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#LZ77_.E2.80.93_die_Grundform_der_Lempel.E2.80.93Ziv.E2.80.93Algorithmen|LZ77 &ndash; die Grundform der Lempel-Ziv-Algorithmen]],
+:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#LZ77_-_the_basic_form_of_the_Lempel-Ziv_algorithms|LZ77 - the basic form of Lempel-Ziv algorithms]],
-:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Lempel.E2.80.93Ziv.E2.80.93Codierung_mit_variabler_Indexbitl.C3.A4nge|Lempel-Ziv-Codierung mit variabler Indexbitlänge]],
+:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Lempel-Ziv_coding_with_variable_index_bit_length|Lempel-Ziv coding with variable index bit length]],
-:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Decodierung_des_LZW.E2.80.93Algorithmus|Decodierung des LZW-Agorithmus]].
+:: [[Information_Theory/Komprimierung_nach_Lempel,_Ziv_und_Welch#Decoding_of_the_LZW_algorithm|Decoding of the LZW algorithm]].
 *Beachten Sie zudem bei der Lösung dieser Aufgabe, dass beim LZW&ndash;Algorithmus nicht von einem leeren Wörterbuch ausgegangen wird.
 *Vielmehr beinhalten die Indizes&nbsp; $I = 0$&nbsp; bis&nbsp; $I = M- 1$&nbsp; alle&nbsp; $M$&nbsp; zulässigen Zeichen des Alphabets.