Question

Find the shortest encoded bit string of the following source sequence: Source sequence: ABCCDEAFABCBEDFABCBBAEFFEBCEFF

(a) using extended Huffman coding to encode the string where k=2

(b) using arithmetic coding.
For this case, you can ignore the termination issue ina a decoder, and output the shortestbit string of the first 5 symbols only, while the probability distribution is based on the whole sequence.

(c) Using adaptive Huffman coding to encode the first 10 symbols of the string. The initial code assignment is given below

• New : 0

• A : 00001

• B : 00010

• C : 00011

• D : 00100

• E : 00101

• F : 00110

Answer 1

Answer: Following are the LT properties

a) \(P=\varnothing^{\emptyset}\)

b) \(\mathrm{P}=\left\{A O, A 1, A 2 \ldots \in 2 A P w / x O \in A O^{A_{0}}, A_{1}, A_{2} \ldots \in\left(2^{A P}\right)^{w} \mid x_{0} \in A_{0}\right\}\)

c) \(\mathrm{P}=\left\{A O, A 1, A 2 \ldots \in 2 A P w / x O \notin A C A_{0}, A_{1}, A_{2} \ldots \in\left(2^{A P}\right)^{w} \mid x_{0} \notin A_{0}\right\}\)

d) \(P=\left\{A O, A 1, A 2 \ldots \in 2 A P w / x O \notin A O \wedge \exists A_{0}, A_{1}, A_{2} \ldots \in\left(2^{A P}\right)^{w} \mid x_{0} \notin A_{0} \wedge \exists_{i}:(x>i) \in A_{i} \wedge i>0\right\}\)

e) \(\mathrm{P}=\left\{A O, A 1, A 2 \ldots \in 2 A P_{W} A_{0}, A_{1}, A_{2} \ldots \in\left(2^{A^{p}}\right)^{w} \mid \exists_{i} \geq \geq 0: \forall j \geq i,(x>i) \notin A_{j}\right\}\)

f) \(P=\left\{A O, A 1, A 2 \ldots \in 2 A P w^{A_{0}}, A_{1}, A_{2} \ldots . \in\left(2^{A P}\right)^{w} \mid \forall i \geq 0: \exists_{j} \geq 1,(x>i) \in A_{j}\right\}\)

g) \(P=\left\{A O, A 1, A 2 \ldots \in 2 A P w^{A}_{0}, A_{1}, A_{2} \ldots . \in\left(2^{A P}\right)^{w} \mid \forall(x=0)^{\forall}(x=0) \in^{\epsilon} A_{1} A_{i} \wedge^{\wedge}(x>1) \in^{\epsilon} A i+1^{A}_{i+1} \wedge^{\wedge} i m o d z^{m o d}_{2}\right.\)

\(=0\) ) \(v\)

h) \(\mathrm{P}=2 \mathrm{APW}^{\left(2^{A P}\right)^{w}}\)

Safety property is a behavior in which "nothing bad happens".

Here, \(d, g\) is a safety property, as a bad prefix is for instance.