C960 Discrete Mathematics II
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Free C960 Discrete Mathematics II Questions
How many comparisons does Merge Sort make in the worst case for n=8 elements?
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8
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12
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17
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20
Explanation
Explanation:
Merge Sort is a divide-and-conquer algorithm that splits the array into two halves, recursively sorts each half, and then merges the sorted halves.
The worst-case number of comparisons can be computed using the recursive formula for comparisons C(n):
where n−1 comparisons are made during the merge step.
For n=8n
1.Split 8 into two halves of 4:C(8)=C(4)+C(4)+7
2.For C(4): split into 2 + 2:C(4)=C(2)+C(2)+3
3.For C(2): split into 1 + 1:C(2)=C(1)+C(1)+1=0+0+1=1
Now compute step by step:
C(4)=1+1+3=5
C(8)=5+5+7=17
How many leaves does a full ternary tree with 1093 internal nodes have?
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2187
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3279
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3280
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3281
Explanation
In a full mmm-ary tree, every internal node has exactly children. For a full ternary tree , the relationship between the number of leaves and the number of internal nodes I is:
Substitute :
=2187
Decrypt the ciphertext using the RSA private key .
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12
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3
-
4
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5
Explanation
Using Miller-Rabin, is 91 a prime number?
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Yes
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No, it is 7×13
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No, it is 3×31
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No, it is 7×11
Explanation
The number 91 is not prime, as it can be factored into integers greater than 1. In fact, 91=7×13. Applying the Miller-Rabin test would also identify it as composite. Therefore, the correct statement specifies its factorization as 7 times 13.
Which of the following languages requires a linear bounded automaton (LBA)?
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The set of palindromes over {0,1}
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The set of strings of the form
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The set of all binary strings
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The set of all strings with an even number of 0s
Explanation
A linear bounded automaton (LBA) is a type of Turing machine whose tape usage is limited to a linear function of the input length. LBAs recognize context-sensitive languages, which are strictly more powerful than context-free languages but less powerful than general Turing machines.
Palindromes over {0,1} are context-free, so a pushdown automaton can handle them.
Strings of the form are context-sensitive but not context-free, so they require an LBA.
Using Chinese Remainder Theorem, solve the system:
x ≡ 3 (mod 5)
x ≡ 2 (mod 7)
x ≡ 1 (mod 11)
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23
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48
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193
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308
Explanation
What is the expected number of coin flips until you get two heads in a row?
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4
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6
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8
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10
Explanation
To find the expected number of flips to get two consecutive heads, we can use a state-based approach. Let be the expected flips starting from no heads, and the expected flips starting with one head.
How many spanning trees does the wheel graph W₇ have?
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1024
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2048
-
4096
-
8192
Explanation
What is the minimal number of colors needed to color the vertices of the Mycielski graph with chromatic number 5?
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3
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4
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5
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6
Explanation
By definition, the chromatic number of a graph is the smallest number of colors needed to color its vertices so that no two adjacent vertices share the same color. If a particular Mycielski graph is given as having chromatic number 5, that directly means its minimal (i.e., least) number of colors required is 5 — the Mycielski construction produces graphs with a specified chromatic number while controlling other properties, but the chromatic number itself tells you the minimum coloring size.
How many ways can you arrange the letters in “MISSISSIPPI”?
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34,650
-
36,300
-
37,800
-
40,000
Explanation
The word MISSISSIPPI has 11 letters in total. The counts of each letter are:
M = 1
I = 4
S = 4
P = 2
The total number of distinct arrangements is given by dividing the factorial of the total letters by the factorial of the frequency of each repeated letter:
Number of arrangements=
so,
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