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FIT2004 Algorithms and data structures - S2 2026

FIT2004 Algorithms and data structures - S2 2026

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Consider the following array:

arr = [7, 9, 14, 1, 8, 2, 4, 12, 5, 6, 10]

Determine which partitioning scheme, Naive or Hoare's, was used to end

up with the provided resulting array based on the given pivot. Assume

0-indexing and that Naive 3-way partitioning is being considered.

The selected pivot was arr[7] = 12, resulting in the following partition:

[7, 9, 1, 8, 2, 4, 5, 6, 10, 12, 14]

The selected pivot was arr[9] = 6, resulting in the following partition:

[2, 5, 4, 1, 6, 8, 14, 12, 9, 7, 10]

The selected pivot was arr[8] = 5, resulting in the following partition:

[1, 2, 4, 5, 7, 9, 14, 8, 12, 6, 10]

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True or False: the identity

E[X + Y] = E[X] + E[Y] (linearity of expectation) holds ONLY when

Y are independent.

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A fair coin is flipped

10 times. What is the EXPECTED number of heads?

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Two INDEPENDENT events have probabilities

0.4. What is the probability that BOTH occur?

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An event has probability

0.3. What is the probability that it does NOT occur?

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4-sided die (faces $1, \dots, 4$ 1, \dots, 4, each equally likely) is rolled once. What is the probability of rolling a

2? Give your answer as a decimal (2 d.p.).

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Solve, in big-Θ, the following recurrence relation.

T(n) = T(n-4) + n*c, where n >= 4
T(n) = b,            where n < 4
for constants b and c.

Θ(n^2)

Θ(4^n)

Θ(n^3)

Θ(n log n)

Θ(log n)

Θ(n)

Θ(n!)

Θ(1)

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Given the following pseudocode, derive the recurrence relation that describes its time complexity.

def power(x, n):
    if n == 0:
        return 1
    return x * power(x, n - 1)

Let b and c represent constant values. What are the base case and the recursive step?

T(n) = n * T(n-1) + c
T(0) = b

T(n) = T(n-1) + c
T(0) = b

T(x,n) = x * T(x,n-1) + c
T(0) = b

T(n) = T(n-1) + c
T(0) = n

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A sequence satisfies

a(1) = 1 and $a(n) = 3 \cdot a(n-1)$ a(n) = 3 \cdot a(n-1) for $n \ge 2$ n \ge 2. Which closed form matches it?

a(n) = n(n+1)/2

$a(n) = 2^{n-1}$ a(n) = 2^{n-1}

$a(n) = 3^{n-1}$ a(n) = 3^{n-1}

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Consider an algorithm that makes about $\log n$ \log n levels of recursion, doing

O(n) work in total at each level (this is merge sort). What is its overall time complexity?

$O(\log n)$ O(\log n)

$O(n \log n)$ O(n \log n)

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