Algorithms 1st Edition solutions

Algorithms 1
Sequence alignment. When a new gene is discovered, a standard approach to understanding its function is to look through a database of known genes and find close matches. The closeness of two genes is measured by the extent to which they are aligned. To formalize this, think of a gene as being a long

Algorithms 1
Suppose a CS curriculum consists of n courses, all of them mandatory. The prerequisite graph G has a node for each course, and an edge from course v to course w if and only if v is a prerequisite for w. Find an algorithm that works directly with this graph representation, and computes the minimum num

Algorithms 1
A kite is a graph on an even number of vertices, say \(2 n\), in which \(n\) of the vertices form a clique and the remaining \(n\) vertices are connected in a "tail" that consists of a path joined to one of the vertices of the clique. Given a graph and a goal \(g\), the KITE problem asks for a subgra

Algorithms 1
Let us define a multiplication operation on three symbols a, b, c according to the following table; thus ab = b, ba = c, and so on. Notice that the multiplication operation defined by the table is neither associative nor commutative. Find an efficient algorithm that examines a string of these symbols

Algorithms 1
The police department in the city of Computopia has made all streets one-way. The mayor contends that there is still a way to drive legally from any intersection in the city to any other intersection, but the opposition is not convinced. A computer program is needed to determine whether the mayor is

Algorithms 1
Run the strongly connected components algorithm on the following directed graphs G. When doing DFS on GR: whenever there is a choice of vertices to explore, always pick the one that is alphabetically first. (i) Image (ii) Image In each case answer the following questions. (a) In what order are the st

Algorithms 1
STINGY SAT is the following problem: given a set of clauses (each a disjunction of literals) and an integer k, find a satisfying assignment in which at most k variables are true, if such an assignment exists. Prove that STINGY SAT is NP-complete

Algorithms 1
You are given a string of n characters s[1 . . . n], which you believe to be a corrupted text document in which all punctuation has vanished (so that it looks something like itwasthebestoftimes...). You wish to reconstruct the document using a dictionary, which is available in the form of a Boolean f

Algorithms 1
Pouring water. We have three containers whose sizes are 10 pints, 7 pints, and 4 pints, respectively. The 7-pint and 4-pint containers start out full of water, but the 10-pint container is initially empty. We are allowed one type of operation: pouring the contents of one container into another, stopp

Algorithms 1
Consider the following game. A dealer produces a sequence s1 sn of cards, face up, where each card si has a value vi . Then two players take turns picking a card from the sequence, but can only pick the first or the last card of the (remaining) sequence. The goal is to collect cards of largest tot

Algorithms 1
Pebbling a checkerboard. We are given a checkerboard which has 4 rows and n columns, and has an integer written in each square. We are also given a set of 2n pebbles, and we want to place some or all of these on the checkerboard (each pebble can be placed on exactly one square) so as to maximize the

Algorithms 1
In each of the following situations, indicate whether \(f=O(g)\), or \(f=\Omega(g)\), or both (in which case \(f=\Theta(g)\)). \(f(n)\) \(g(n)\) (a) \(n-100\) \(n-200\) (b) \(n^{1 / 2}\) \(n^{2 / 3}\) (c) \(100 n+\log n\) \(n+(\log n)^{2}\) (d) \(n \log n\) \(10 n \log 10 n\) (e)

Algorithms 1
Yuckdonalds is considering opening a series of restaurants along Quaint Valley Highway (QVH). The n possible locations are along a straight line, and the distances of these locations from the start of QVH are, in miles and in increasing order, m1, m2, . . . , mn. The constraints are as follows: At e

Algorithms 1
You are going on a long trip. You start on the road at mile post 0. Along the way there are n hotels, at mile posts a1 < a2 < < an, where each ai is measured from the starting point. The only places you are allowed to stop are at these hotels, but you can choose which of the hotels you stop at. Yo

Algorithms 1
In the 2SAT problem, you are given a set of clauses, where each clause is the disjunction (OR) of two literals (a literal is a Boolean variable or the negation of a Boolean variable). You are looking for a way to assign a value true or false to each of the variables so that all clauses are satisfied

Algorithms 1
For each node u in an undirected graph, let twodegree[u] be the sum of the degrees of us neighbors. Show how to compute the entire array of twodegree[] values in linear time, given a graph in adjacency list format.

Algorithms 1
Squares. Design and analyze an algorithm that takes as input an undirected graph G = (V,E) and determines whether G contains a simple cycle (that is, a cycle which doesnt intersect itself) of length four. Its running time should be at most O(|V|3). You may assume that the input graph is represented e

Algorithms 1
A salad is any combination of the following ingredients: (1) tomato, (2) lettuce, (3) spinach, (4) carrot, and (5) oil. Each salad must contain: (A) at least 15 grams of protein, (B) at least 2 and at most 6 grams of fat, (C) at least 4 grams of carbohydrates, (D) at most 100 milligrams of sodium. Fu

Algorithms 1
A contiguous subsequence of a list S is a subsequence made up of consecutive elements of S. For instance, if S is 5, 15, 30, 10, 5, 40, 10, then 15, 30, 10 is a contiguous subsequence but 5, 15, 40 is not. Give a linear-time algorithm for the following task: Input: A list of numbers, a1, a2, . . . ,

Algorithms 1
Run the DFS-based topological ordering algorithm on the following graph. Whenever you have a choice of vertices to explore, always pick the one that is alphabetically first. (a) Indicate the pre and post numbers of the nodes. (b) What are the sources and sinks of the graph? (c) What topological order