# Recent questions tagged cmi2017

1
Consider the following function that takes as input a sequence $A$ of integers with n elements,$A[1],A[2], \dots ,A[n]$ and an integer $k$ and returns an integer value. The function length$(S)$ returns the length of the sequence $S$. Comments start with ... case complexity of this algorithm in terms of the length of the input sequence $A$? Give an example of a worst-case input for this algorithm.
1 vote
2
We are given a sequence of pairs of integers $(a_1, b_1),(a_2, b_2), \dots ,(a_n, b_n)$. We would like to compute the largest $k$ such that there is a sequence of numbers $c_{i_1} \leq c_{i_2} \leq \dots \leq c_{i_k}$ ... for each $j \leq k$, $c_{i_j}=a_{i_j}$ or $c_{i_j} = b_{i_j}$ . Describe an algorithm to solve this problem and explain its complexity.
1 vote
3
An undirected graph is $\text{connected}$ if, for any two vertices $\{u, v\}$ of the graph, there is a path in the graph starting at $u$ and ending at $v$. A tree is a connected, undirected graph that contains no cycle. A $\text{leaf}$ in a tree is a vertex that has degree ... $u \in V_1$ and v$\in V_2$ or vice versa. Prove that if $G$ is a tree with at least two vertices, then $G$ is bipartite.
4
In a party there are $2n$ participants, where $n$ is a positive integer. Some participants shake hands with other participants. It is known that there are no three participants who have shaken hands with each other. Prove that the total number of handshakes is not more than $n^2.$
5
Let $Σ = \{a, b\}$. Given words $u, v \in Σ*$ , we say that $v$ extends $u$ if $v$ is of the form $xuy$ for some $x, y ∈ Σ^*$ . Given a fixed word $u$, we are interested in identifying whether a finite state automaton accepts some word that extends $u$. ... reports Yes if some word in the language of $\mathcal{A}$ extends $u$ and No if no word in the language of $\mathcal{A}$ extends $u$.
6
There are a number of tourist spots in a city and a company GoMad runs shuttle services between them. Each shuttle plies between a designated origin and destination, and has a name. Due to lack of coordination, the same name may be allotted to multiple routes. To make matters worse, ... you are carrying a GoMad pass or a GoCrazy pass, you can start at $s$ and arrive at $t$ using the sequence $σ$.
7
Let $Σ = \{a, b, c\}$. Let Leven be the set of all even length strings in $Σ^*$ $(a)$ Construct a deterministic finite state automaton for $L_{even}$. $(b$) We consider an operation $Erase_{ab}$ that takes as input a string $w \in Σ^*$ and erases all occurrences of the pattern $ab$ ... in $L:$ $Erase_{ab}(L)$:= $\{ Erase_{ab}(w)\ |\ w\in L\}$ Show that $Erase_{ab}(L_{even}$) is a regular language.
8
We have constructed a polynomial time reduction from problem $A$ to problem $B$. Which of the following is a valid inference? If the best algorithm for $B$ takes exponential time, then there is no polynomial time algorithm for $A$ If the best algorithm for $A$ takes ... If we don't know whether there is a polynomial time algorithm for $B$, then there cannot be a polynomial time algorithm for $A$.
1 vote
9
Suppose we constructed the binary search tree shown below by starting with an empty tree and inserting one element at a time from an input sequence, without any rotations or other manipulations. Which of the following assertions about the order of elements in the input sequence $cannot$ be true? $8$ ... $1$ came after $12$ and $29$ came before $42$. $3$ came before $14$ and $16$ came before $28$.
10
A $\text{stable sort}$ preserves the order of values that are equal with respect to the comparison function. We have a list of three-dimensional points $[(7, 1, 8),(3, 5, 7),(6, 1, 4),(6, 5, 9),(0, 2, 5),(9, 0, 9)].$ We sort these in ascending order by the second coordinate. Which of the following corresponds to a ... $[(9, 0, 9),(6, 1, 4),(7, 1, 8),(0, 2, 5),(3, 5, 7),(6, 5, 9)]$
11
Consider the following functions f() and g(). f(){ g(){ w=5; z=w+1; w=2*z+2; z=3*z-w; } print(z); } We start with $w$ and $z$ set to $0$ and execute $f()$ and $g()$ in parallel—that is, at each step we either execute one statement from $f()$ or one statement from $g()$. Which of the following is not a possible value printed by $g()$ ? $-2$ $-1$ $2$ $4$
1 vote
12
What does the following function compute in terms of $n$ and $d$, for integer values of $d$ ? Note that the operation $/$ denotes floating point division, even if the arguments are both integers. function foo(n,d){ if (d == 0){ return 1; }else{ if (d < 0){ return foo(n,d+1)/n; }else{ return ... the values of $d$. $n \times d$ if $d>0$ ,$n\div d$ if $d<0$. $n \times d$ for all the values of $d$.
13
Let $G$ be an arbitrary graph on $n$ vertices with $4n − 16$ edges. Consider the following statements: There is a vertex of degree smaller than $8$ in $G$. There is a vertex such that there are less than $16$ vertices at a distance exactly $2$ from it. Which of the following is true: I Only II Only Both I and II Neither I nor II
14
City authorities are concerned about traffic accidents on major roads. They would like to have ambulances stationed at road intersections to quickly reach the scene of any accident along these roads. To minimize response time, ambulances are to be located at intersections with ... number of edges. Find a spanning tree with minimum cost. Find a minimal coloring. Find a minimum size vertex cover.
An FM radio channel has a repository of $10$ songs. Each day, the channel plays $3$ ...
The regular expression $(a^*+b)^*$ is equivalent to which of the following regular expressions: $a^*b^*$ $(a^*b+b)^*$ $(a+b^*)^*$ $(a^*b)^*$