Consider a table $T$, where the elements $T[i][j]$, $0\le i,j\le n$, represent the cost of the optimal solutions of different subproblems of a problem that is being solved using a dynamic programming algorithm. The recursive formulation to compute the table entries is as follows:

$$T[0][k]=T[k][0]=1\text{for }k=0,1,2,\dots ,n$$
$$T[i][j]=2T[i-1][j]+3T[i][j-1]\text{for }1\le i,j\le n$$
Consider the following two algorithms to compute entries of $T$. Assume that for both the algorithms, for all $0\le i,j\le n$, $T[i][j]$ has been initialized to $1$.

Algorithm $B_1$: For $i=1,2,\dots ,n$
    For $j=1,2,\dots ,n$
        $T[i][j]=2T[i-1][j]+3T[i][j-1]$

Algorithm $B_2$: For $s=2,3,\dots ,2n$
    For $i=1,2,\dots ,n$
        For $j=1,2,\dots ,n$
            If $(i+j==s)$
                $T[i][j]=2T[i-1][j]+3T[i][j-1]$

Algorithm $B_k$, $k\in \{1,2\}$ is said to be correct if and only if it calculates the correct values of $T[i][j]$, for all $0\le i,j\le n$, (as per the recursive formulation) at the end of the execution of the algorithm $B_k$.

Which one of the following statements is true?

In a directed acyclic graph with a source vertex $s$, the quality-score of a directed path is defined to be the product of the weights of the edges on the path. Further, for a vertex $v$ other than $s$, the quality-score of $v$ is defined to be the maximum among the quality-scores of all the paths from $s$ to $v$. The quality-score of $s$ is assumed to be 1.


The sum of the quality-scores of all the vertices in the graph shown above is __________.

Consider the string abbccddeee. Each letter in the string must be assigned a binary code satisfying the following properties:

1. For any two letters, the code assigned to one letter must not be a prefix of the code assigned to the other letter.
2. For any two letters of the same frequency, the letter which occurs earlier in the dictionary order is assigned a code whose length is at most the length of the code assigned to the other letter.

Among the set of all binary code assignments which satisfy the above two properties, what is the minimum length of the encoded string?

Define $R_n$ to be the maximum amount earned by cutting a rod of length $n$ meters into one or more pieces of integer length and selling them. For $i>0$, let $p[i]$ denote the selling price of a rod whose length is $i$ meters. Consider the array of prices:

$$p[1]=1,p[2]=5,p[3]=8,p[4]=9,p[5]=10,p[6]=17,p[7]=18$$
Which of the following statements is/are correct about $R_7$?