# Codesport Column

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This month, as is our practice for every year-end column, we will discuss a bunch of programming questions.

Last month, we started discussing software bloat and its impact on the efficiency of our software. Every year, in the CodeSport column for December, I feature a bunch of programming questions that I have discussed in the column over the past 12 months. So let’s do the same this year and continue our discussion of software bloat next month.

(1)    You are given two sorted lists of size X and Y. How would you determine the kth smallest element in the union of the two lists? What would be the complexity of your algorithm?

(2)    You are given an array X of N integers. You are asked to write an algorithm which returns the hot element (one that appears more than N/3 times), if one exists. If no hot element exists, your algorithm should be able to determine that. The other constraint is that you can only compare the elements of the array for equality, i.e., you can only use comparisons of the form Is X[i] = X[j], which take constant time. What is the complexity of your algorithm?

(3)    You are given three containers of capacities 10 litres, 7 litres and 4 litres each. The 7 and 4 litre containers are full of water. The 10 litre container is empty. The only type of action allowed is pouring the contents of one container into another, and you can stop pouring only when either (a) the source container is completely empty; or (b) the receiving container is entirely full. Can you write an algorithm to determine whether there exists a sequence of actions that leaves exactly 2 litres in the 7 litre container?

(4)    Given a binary tree T = (V, E) represented in adjacency list format, along with a root node ‘r’, you are asked to design an algorithm such that it can pre-process the tree, and answer a bunch of queries of the form, Is vertex ‘u an ancestor of vertex ‘v’? in constant time. Note that only the query response needs to be in constant time, whereas you can spend time pre-processing the tree to build sufficient information to answer the queries in constant time. Note that a vertex ‘u’ is said to be an ancestor of vertex ‘v’ if the path from root node ‘r’ to vertex ‘v’ passes through vertex ‘u’.

(5)    You are given a directed graph G(V, E) in the form of an adjacency list. Give a linear-time algorithm for determining whether the graph contains an odd-length cycle. If you can show that it does not contain an odd-length cycle, can you infer anything about whether it is a bi-partite graph or not?

(6)    You are given a directed acyclic graph G(V, E). A Hamiltonian path is a path in a directed graph that touches each vertex exactly once. Write an algorithm to determine whether the directed graph contains a Hamiltonian path? What is the complexity of your algorithm? Instead of being given a DAG, if you are asked to determine whether a general directed graph contains a Hamiltonian path, what would be your solution?

(7)    We all know what a minimum spanning tree is. Now you are asked to find the maximum spanning tree of an undirected graph. The maximum spanning tree is the spanning tree with the largest total weight. What is the complexity of your algorithm? Can this be solved in polynomial time?
(8)    Your friend is organising a party for her classmates. There are N people in her class, excluding your friend. She also knows which pairs of her classmates know each other. She wants to invite as many people to the party as possible, subject to the following two constraints: (a) Every person invited should know at least 5 other people in the party; and (b) there should be 5 other people in the party whom a person invited to the party does not know. Write an algorithm that takes the list of class mates and the list of pairs who know each other, and outputs the best choice for the partys list of invitees. What is the complexity of your algorithm?

(9)    You are given a sequence of ‘n’ numbers a1, a2, .. aN. A subsequence is a subset of these numbers taken in order of the form ai1, ai2, aiK, where 1 = i1 = i2 =. ik = n. An increasing subsequence is one whose elements are strictly in increasing order, such that ai1 = ai2 = .. aiK. You are asked to write a program to find the longest increasing subsequence given a sequence of N integers. For example, if you are given 5, 2, 8, 6, 3, 6, 9, 7, then the longest increasing subsequence is 2, 3, 6, 9. What is the complexity of your algorithm?

(10)    You are given a string ‘s’ of N characters, which you believe to be the garbled text of a document. However, the garbling has resulted in all punctuation of the original text getting removed. You are also given a dictionary, which can tell whether a particular sub-string is a word. You can check whether a string is a valid word in the dictionary by calling the function ‘dict’ with a string; it returns true if the given string is a valid word in the dictionary, else it returns false. Write an algorithm to check whether the given sentence can be reconstructed into a valid sequence of words in the dictionary. Is it possible to write a greedy algorithm for solving this problem?

(11)    Given a graph G(V, E) where V is the set of vertices, and E is the set of edges in the form of an adjacency matrix, a subset S of V is an independent set if there are no edges among the vertices in S. Is it possible to write a polynomial time algorithm to determine the largest independent set in a given directed graph? Now instead of a directed graph, you are given a tree. Can you write an algorithm for determining the largest independent set in the tree?

(12)    A vertex cover of a graph G(V, E) is a subset of vertices V ? V such that each edge E is incident on at least one vertex in V. That is, each edge E is covered by at least one vertex in V, hence the name vertex cover. If G is a tree instead of an arbitrary graph, write an algorithm to determine the minimum-size vertex cover of G.

(13)    You are given a directed graph G(V, E) with positive edge weights. You are asked to write an algorithm that returns the length of the shortest cycle in the graph. Your algorithm should return 0 if the graph is acyclic. What is the complexity of your algorithm?

(14)    You are given a directed acyclic graph G(V, E) and two vertices in the DAG, namely x and y. Write an algorithm to count the number of distinct paths in DAG between x and y.

(15)    You are given a string ‘s’ of length N. You are asked to write an algorithm to output all the permutations of the given string. What is the complexity of your algorithm?

My must-read book for this month
This month’s must-read book suggestion comes from our reader Nethravathi. She suggests a classic computer science book: A Discipline of Programming by Edsger W Dijkstra. Nethravathi says that this is an essential book to understand programming from the mathematical formalism perspective. It builds a framework for building correct programs and how to reason about correct programs. While this is not an easy book to read, I would suggest it to anyone who is seriously interested in programming formally. Thank you,