Computer Science (GATE Exam) 2013 - Previous Question Paper Solution

In a k-way set associative cache, the cache is divided into v sets, each of which consists of k lines. The lines of a set are placed in sequence one after another. The lines in set s are sequenced before the lines in set (s+1). The main memory blocks are numbered from 0 onwards. The main memory block numbered j must be mapped to any one of the cache lines from

Which of the following statements are TRUE?
(1) The problem of determining whether there exists a cycle in an undirected graph is in P.
(2) The problem of determining whether there exists a cycle in an undirected graph is in NP.
(3) If a problem A is NP-complete, there exists a non-deterministic polynomial time algorithm to solve A.

Which of the following statements is/are FALSE?
(1) For every non-deterministic Turing machine, there exists an equivalent deterministic Turing machine.
(2) Turing recognisable languages are closed under union and complementation.
(3) Turing decidable languages are closed under intersection and complementation.
(4) Turing recognisable languages are closed under union and intersection.

Three concurrent processes X, Y and Z execute three different code segments that access and update certain shared variables. Process X executes the P operation (i.e. wait) on semaphores a, b and c; process Y executes the P operation on semaphores b, c and d; process Z executes the P operation on semaphores c, d, and a before entering the respective code segments. After completing the execution of its code segment, each process invokes the V operation (i.e. signal) on its three semaphores. All semaphores are binary semaphores initialised to one. Which one of the following represents a deadlock-free order of invoking the P operations by the processes?

Consider an instruction pipeline with five stages without any branch prediction.
Fetch Instruction (FI), Decode Instruction (DI), Fetch Operand (FO), Execute Instruction (EI) and Write Operand (WO). The stage delays for FI, DI, FO, EI and WO are 5 ns, 7 ns, 10 ns, 8 ns and 6 ns respectively. There are intermediate storage buffers after each stage and the delay of each buffer is 1 ns. A program consisting of 12 instructions I₁, I₂, I₃, ….. I₁₂ is executed in this pipelined processor. Instruction I₄ is the only branch instruction and its branch target is 9 I₉. If the branch is taken during the execution of this program, the time (in ns) needed to complete the program is

A shared variable x, initialised to zero, is operated on by four concurrent processes W, X, Y and Z as follows: Each of the processes W and X reads x from memory, increments by one, stores it to memory and then terminates. Each of the processes Y and Z reads x from memory, decrements by two, stores it to memory and then terminates. Each process before reading x invokes the P operation (i.e. wait) on a counting semaphore S and invokes the V operation (i.e. signal) on the semaphore S after storing x to memory. Semaphore S is initialised to two. What is the maximum possible value of x after all processes complete execution?

Consider the following two sets of LR (1) items of an LR (1) grammar
X $\rightarrow$ c. X c / d X $\rightarrow$ c.X, $
X $\rightarrow$ .Cx, c / d X $\rightarrow$.Cx, $
X $\rightarrow$ .d, c / d X $\rightarrow$.d, $
Which of the following statements related to merging of the two sets in the corresponding LALR parser is/are FALSE?

1. Cannot be merged since look aheads are different
2. Can be merged but will result in S–R conflict
3. Can be merged but will result in R–R conflict
4. Cannot be merged since goto on c will lead to two different sets

Consider the following relational schema.

Students(rollno: integer, sname: string)
Courses(courseno: integer, cname: string)
Registration(rollno: integer, courseno: integer, percent: real)

Which of the following queries are equivalent to this query in English?

“Find the distinct names of all students who score more than 90% in the course numbered 107”

(I)

SELECT DISTINCT S.sname 
FROM Students as S, Registration as R 
WHERE R.rollno=S.rollno AND R.courseno=107 AND R.percent >90

(II) $ \prod_{sname}(σ_{courseno=107 ∧ percent > 90} (Registration ⋈ Students)) $

(III) $\left\{T \mid ∃S \in Students, ∃R \in Registration ( S.rollno=R.rollno \land R.courseno=107 ∧ R.percent>90 ∧T.sname=S.sname)\right \}$

(IV) $\left\{ \langle S_N\rangle \mid ∃S_R∃R_P (\langle S_R,S_N\rangle ∈Students ∧ \ \langle S_R,107,R_P\rangle ∈Registration ∧ R_P>90) \right\}$

The line graph L(G) of a simple graph G is defined as follows:
• There is exactly one vertex v(e) in L(G) for each edge e in G.
• For any two edges e and e’ in G, L(G) has an edge between v(e) and v(e’), if and only if e and e’ are incident with the same vertex in G.
Which of the following statements is/are TRUE?
(P) The line graph of a cycle is a cycle.
(Q) The line graph of a clique is a clique.
(R) The line graph of a planar graph is planar.
(S) The line graph of a tree is a tree.

Consider a hard disk with 16 recording surfaces (0 – 15) having 16384 cylinders (0 − 16383) and each cylinder contains 64 sectors (0 − 63). Data storage capacity in each sector is 512 bytes. Data are organised cylinder–wise and the addressing format is <cylinder no., sector no.>. A file of size 42797 KB is stored in the disk and the starting disk location of the file is <1200, 9, 40>. What is the cylinder number of the last sector of the file, if it is stored in a contiguous manner?

A certain computation generates two arrays a and b such that a [i] = f (i) for 0 $\le$ i < n and b [i] = g (a [i]) for 0 $\le$ i < n. Suppose this computation is decomposed into two concurrent processes X and Y such that X computes the array a and Y computes the array b. The processes employ two binary semaphores R and S, both initialised to zero. The array a is shared by the two processes. The structures of the processes are shown below.

Process X; 
Process Y;
private i;
private i;
for (i = 0; i < n; i + +) {                            
  for (i = 0; i < n; i + +) {
         a [i] = f (i);
         EntryY (R, S);
         ExitX (R, S); 
         b [i] = g (a [i]);
    }
}

Which one of the following represents the CORRECT implementations of ExitX and EntryY?

The following figure represents access graphs of two modules M1 and M2. The filled circles represent methods and the unfilled circles represent attributes. If method m is moved to module M2 keeping the attributes where they are, what can we say about the average cohesion and coupling between modules in the system of two modules?

Direction: The following code segment is executed on a processor which allows only register operands in its instructions. Each instruction can have almost two source operands and one destination operand. Assume that all variables are dead after this code segment.
c = a + b;
d = c * a;
e = c + a;
x = c * c;
if (x > a) {
y = a* a;
}
else {
d = d * d;
e = e * e;
}

Suppose the instruction set architecture of the processor has only two registers. The only allowed compiler optimisation is code motion, which moves statements from one place to another while preserving correctness. What is the minimum number of spills to memory in the compiled code?

Direction: The following code segment is executed on a processor which allows only register operands in its instructions. Each instruction can have almost two source operands and one destination operand. Assume that all variables are dead after this code segment.

c = a + b;
d = c * a;
e = c + a;
x = c * c;
if (x > a) {
  y = a* a;
}
else {
   d = d * d;
   e = e * e;
}

What is the minimum number of registers needed in the instruction set architecture of the processor to compile this code segment without any spill to memory? Do not apply any optimisation other than optimising register allocation.

The procedure given below is required to find and replace certain characters inside an input character string supplied in array A. The characters to be replaced are supplied in array oldc, while their respective replacement characters are supplied in array newc. Array A has a fixed length of five characters, while arrays oldc and newc contain three characters each. However, the procedure is flawed.

void find_and_replace (char *A, char *oldc, char *newc) {
    for (int i=0; i<5; i++)
        for (int j=0; j<3; j++)
            if (A[i] == oldc[j])
                A[i] = newc[j];
} 

The procedure is tested with the following four test cases.
(1) oldc = “abc”, newc = “dab” (2) oldc = “cde”, newc = “bcd”
(3) oldc = “bca”, newc = “cda” (4) oldc = “abc”, newc = “bac”

The tester now tests the program on all input strings of length five consisting of characters ‘a’, ‘b’, ‘c’, ‘d’ and ‘e’ with duplicates allowed. If the tester carries out this testing with the four test cases given above, how many test cases will be able to capture the flaw?

Relation R has eight attributes ABCDEFGH. Fields of R contain only atomic values.
F = {CH $\leftarrow$ G, A $\rightarrow$ BC, B $\rightarrow$ CFH, E $\rightarrow$ A, F $\rightarrow$ EG} is a set of functional dependencies (FDs) so that F+ is exactly the set of FDs that hold for R.

How many candidate keys does the relation R have?

A computer uses 46–bit virtual address, 32–bit physical address and a three–level paged page table organisation. The page table base register stores the base address of the first–level table (T₁) which occupies exactly one page. Each entry of T₁ stores the base address of a page of the second–level table (T₂). Each entry of T₂ stores the base address of a page of the third–level table (T₃). Each entry of T₃ stores a page table entry (PTE). The PTE is 32 bits in size. The processor used in the computer has a 1 MB 16 way set associative virtually indexed physically tagged cache. The cache block size is 64 bytes.

What is the size of a page in KB in this computer?

After several defeats in wars, Robert Bruce went in exile and wanted to commit suicide. Just before committing suicide, he came across a spider attempting tirelessly to have its net. Time and again, the spider failed but that did not deter it to refrain from making attempts. Such attempts by the spider made Bruce curious. Thus, Bruce started observing the near-impossible goal of the spider to have the net. Ultimately, the spider succeeded in having its net despite several failures. Such act of the spider encouraged Bruce not to commit suicide. And then, Bruce went back again and won many battles and the rest is history.
Which one of the following assertions is best supported by the above information?

A computer uses 46–bit virtual address, 32–bit physical address, and a three–level paged page table organization. The page table base register stores the base address of the first–level table (T₁) which occupies exactly one page. Each entry of T₁ stores the base address of a page of the second–level table (T₂). Each entry of T₂ stores the base address of a page of the third–level table (T₃). Each entry of T₃ stores a page table entry (PTE). The PTE is 32 bits in size. The processor used in the computer has a 1 MB 16 way set associative virtually indexed physically tagged cache. The cache block size is 64 bytes.

What is the minimum number of page colours needed to guarantee that no two synonyms map to different sets in the processor cache of this computer?