Computer Science core problem solutions

Solution to Problem 1 – Resonant algorithm runtime#

The runtime is

$$ T = D_6 τ_r \log(X). $$

If $$τ_r$$ is halved:

$$ τ_r' = \frac{τ_r}{2}. $$

Thus,

$$ T' = D_6 \left(\frac{τ_r}{2}\right) \log(X) = \frac{T}{2}. $$

Answer: The runtime is cut in half.

Solution to Problem 2 – Data throughput#

Throughput is

$$ R = F_3 T_f D_3. $$

• $$D_3$$ increases by 2 → becomes $$3D_3$$.
• $$T_f$$ decreases by 10% → becomes $$0.9T_f$$.

Thus,

$$ R' = F_3 (0.9T_f)(3D_3) = 2.7R. $$

Answer: Throughput increases by a factor of 2.7 (a 170% increase).

Solution to Problem 3 – Error correction#

We have

$$ p = e^{-ΛΘ τ_r}. $$

We want

$$ p' = 0.5p. $$

Thus,

$$ e^{-ΛΘ τ_r'} = 0.5 e^{-ΛΘ τ_r}. $$

Divide both sides by $$e^{-ΛΘ τ_r}$$:

$$ e^{-ΛΘ (τ_r' - τ_r)} = 0.5. $$

Take logs:

$$ -ΛΘ (τ_r' - τ_r) = \ln(0.5) = -\ln 2. $$

Thus,

$$ τ_r' - τ_r = \frac{\ln 2}{ΛΘ}. $$

Answer: Increase $$τ_r$$ by $$\dfrac{\ln 2}{ΛΘ}$$.