Greedy

Let $G$ be a simple, finite, undirected graph with vertex set $\{v_1, \ldots, v_n\}$. Let $\Delta(G)$ denote the maximum degree of $G$ and let $\mathbb{N} = \{1, 2, \ldots\}$ denote the set of all possible colors. Color the vertices of $G$ using the following greedy strategy: for $i = 1, \ldots, n$ color$(v_i) \leftarrow \min\{j \in \mathbb{N} : \text{no neighbour of } v_i \text{ is colored } j\}$ Which of the following statements is/are TRUE?

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?

Consider the weights and values of items listed below. Note that there is only one unit of each item. $$\begin{array}{|c|c|c|} \hline \textbf{Item number} & \textbf{Weight (in Kgs)} & \textbf{Value (in Rupees)} \\ \hline 1 & 10 & 60 \\ \hline 2 & 7 & 28 \\ \hline 3 & 4 & 20 \\ \hline 4 & 2 & 24 \\ \hline \end{array}$$ The task is to pick a subset of these items such that their total weight is no more than 11 Kgs and their total value is maximized. Moreover, no item may be split. The total value of items picked by an optimal algorithm is denoted by $V_{\text{opt}}$. A greedy algorithm sorts the items by their value-to-weight ratios in descending order and packs them greedily, starting from the first item in the ordered list. The total value of items picked by the greedy algorithm is denoted by $V_{\text{greedy}}$. The value of $V_{\text{opt}} - V_{\text{greedy}}$ is _____.

A message is made up entirely of characters from the set $X = \{P, Q, R, S, T\}$. The table of probabilities for each of the characters is shown below: $$\begin{array}{|c|c|} \hline \text{Character} & \text{Probability} \\ \hline P & 0.22 \\ Q & 0.34 \\ R & 0.17 \\ S & 0.19 \\ T & 0.08 \\ \text{Total} & 1.00 \\ \hline \end{array}$$ If a message of 100 characters over $X$ is encoded using Huffman coding, then the expected length of the encoded message in bits is ________.

Given below are some algorithms, and some algorithm design paradigms. 1. Dijkstra's Shortest Path 2. Floyd-Warshall algorithm to compute all pairs shortest path 3. Binary search on a sorted array 4. Backtracking search on a graph i. Divide and Conquer ii. Dynamic Programming iii. Greedy design iv. Depth-first search v. Breadth-first search Match the above algorithms on the left to the corresponding design paradigm they follow.

Suppose P, Q, R, S, T are sorted sequences having lengths 20, 24, 30, 35, 50 respectively. They are to be merged into a single sequence by merging together two sequences at a time. The number of comparisons that will be needed in the worst case by the optimal algorithm for doing this is _____.