Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
expand_more
expand_more
format_list_bulleted
Concept explainers
Question
Chapter 33.1, Problem 7E
Program Plan Intro
To compute whether a point p0 is in the interior of an n-vertex polygon P in
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Given the following set of vertices V and set of edges E.
V = {p, q, r, s, t, u}
E = {(p,q), (q,r), (r, s), (t,p), (q,t), (r, t), (t,u)}
Which of the following is correct?
Select one:
A.There are 7 vertices
B.There are 7 edges.
C.The degree at vertex p is 7
D.The total degree of all vertices is 7
Consider eight points on the Cartesian two-dimensional xx-yy plane.
For each pair of vertices uu and vv, the weight of edge uvuv is the Euclidean (Pythagorean) distance between those two points. For example, dist(a,h) = \sqrt{4^2 + 1^2} = \sqrt{17}dist(a,h)=42+12=17 and dist(a,b) = \sqrt{2^2 + 0^2} = 2dist(a,b)=22+02=2.
Using the algorithm of your choice, determine one possible minimum-weight spanning tree and compute its total distance, rounding your answer to one decimal place. Clearly show your steps.
In a country, there are N cities, and there are some undirected roads between them. For every city there is an integer value (may be positive, negative, or zero) on it. You want to know, if there exists a cycle (the cycle cannot visit a city or a road twice), and the sum of values of the cities on the cycle is equal to 0. Note, a single vertex is not a cycle. Prove this problem is NP-Complete. Use a reduction from the Hamiltonian cycle problem.
Write down the claim that the Hamiltonian cycle problem is polynomially reducible to the original problem.
Chapter 33 Solutions
Introduction to Algorithms
Ch. 33.1 - Prob. 1ECh. 33.1 - Prob. 2ECh. 33.1 - Prob. 3ECh. 33.1 - Prob. 4ECh. 33.1 - Prob. 5ECh. 33.1 - Prob. 6ECh. 33.1 - Prob. 7ECh. 33.1 - Prob. 8ECh. 33.2 - Prob. 1ECh. 33.2 - Prob. 2E
Ch. 33.2 - Prob. 3ECh. 33.2 - Prob. 4ECh. 33.2 - Prob. 5ECh. 33.2 - Prob. 6ECh. 33.2 - Prob. 7ECh. 33.2 - Prob. 8ECh. 33.2 - Prob. 9ECh. 33.3 - Prob. 1ECh. 33.3 - Prob. 2ECh. 33.3 - Prob. 3ECh. 33.3 - Prob. 4ECh. 33.3 - Prob. 5ECh. 33.3 - Prob. 6ECh. 33.4 - Prob. 1ECh. 33.4 - Prob. 2ECh. 33.4 - Prob. 3ECh. 33.4 - Prob. 4ECh. 33.4 - Prob. 5ECh. 33.4 - Prob. 6ECh. 33 - Prob. 1PCh. 33 - Prob. 2PCh. 33 - Prob. 3PCh. 33 - Prob. 4PCh. 33 - Prob. 5P
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.Similar questions
- In a country, there are N cities, and there are some undirected roads between them. For every city there is an integer value (may be positive, negative, or zero) on it. You want to know, if there exists a cycle (the cycle cannot visit a city or a road twice), and the sum of values of the cities on the cycle is equal to 0. Note, a single vertex is not a cycle. Prove this problem is NP-Complete. Use a reduction from the Hamiltonian cycle problem. Show that the problem belongs to NP.arrow_forwardIn a country, there are N cities, and there are some undirected roads between them. For every city there is an integer value (may be positive, negative, or zero) on it. You want to know, if there exists a cycle (the cycle cannot visit a city or a road twice), and the sum of values of the cities on the cycle is equal to 0. Note, a single vertex is not a cycle. Prove this problem is NP-Complete. Use a reduction from the Hamiltonian cycle problem. Prove the claim in the direction from the Hamiltonian cycle problem to the reduced problem.arrow_forwardIn a country, there are N cities, and there are some undirected roads between them. For every city there is an integer value (may be positive, negative, or zero) on it. You want to know, if there exists a cycle (the cycle cannot visit a city or a road twice), and the sum of values of the cities on the cycle is equal to 0. Note, a single vertex is not a cycle. Prove this problem is NP-Complete. Use a reduction from the Hamiltonian cycle problem. Show a polynomial time construction using a reduction from Hamiltonian cycle .arrow_forward
- The Triangle Vertex Deletion problem is defined as follows: Given: an undirected graph G = (V, E) , with IVI=n, and an integer k>= 0. Is there a set of at most k vertices in G whose deletion results in deleting all triangles in G? (a) Give a simple recursive backtracking algorithm that runs in O(3^k * ( p(n))) where p(n) is a low-degree polynomial corresponding to the time needed to determine whether a certain vertex belongs to a triangle in G. (b) Selecting a vertex that belong to two different triangles can result in a better algorithm. Using this idea, provide an improved algorithm whose running time is O((2.562^n) * p(n)) where 2.652 is the positive root of the equation x^2=x+4arrow_forwardPlease explain Give an algorithm that solves the Closest pair problem in 3D in Θ(n log n) time. (Closest pair problem in 3D: Given n points in the 3D space, find a pair with the smallest Euclidean distance between them.)arrow_forwardWe have the following directed graph G, where the number on each edge is the cost of the edge. 1. Step through Dijkstra’s Algorithm on the graph starting from vertex s, and complete the table below to show what the arrays d and p are at each step of the algorithm. For any vertex x, d[x] stores the current shortest distance from s to x, and p[x] stores the current parent vertex of x. 2. After you complete the table, provide the returned shortest path from s to t and the cost of the path.arrow_forward
- When we want to calculate the shortest paths from a vertex using the Bellman-Ford algorithm, it is possible to stop early and not do all |V| - 1 iterations on graphs without a negative cycle. How can we modify the Bellman-Ford Algorithm so that it stops early when all distances are correct?arrow_forwardLet H be a set of at least 3 half-planes. We call a half-plane h redundant if it doesn’tcontribute an edge to the intersection of all half planes in H. Prove that for any redundanthalf-plane h ∈ H, there are two other half-planes h′, h′′∈ H such that h′∩h′′contains h. Usethis fact to give an algorithm (as fast as possible) to compute all redundant half-planes.arrow_forwardProvide an eficient algorithm that given a directed graph G with n vertices and m edges as input, finds the outdegree of each vertex in G. Note that outdegree of a vertex u is the number of edges directed from u to some other vertex v. Discuss the running-time of your algorithm and Provide an algorithm that given a directed graph G with n vertices and m edges as input, nds the indegree of each vertex in G. Note that indegree of a vertex u is the number of edges directed into u from some other vertex v. Discuss the running-time of your algorithm.arrow_forward
- How many undirected graphs (not necessarily connected) can be constructed out of a given set V= {V 1, V 2,…V n} of n vertices ? Group of answer choices 2^(n(n-1)/2) 2^n n(n-l)/2 n! Which of the following is an advantage of adjacency list representation over adjacency matrix representation of a graph? Adding a vertex in adjacency list representation is easier than adjacency matrix representation. In adjacency list representation, space is saved for sparse graphs. DFS and BSF can be done in O(V + E) time for adjacency list representation. These operations take O(V^2) time in adjacency matrix representation. Here is V and E are number of vertices and edges respectively. All of the above Given the starting vertex A, what is the visit order of the graph shown in Fig. 1 under the DFS traversal algorithm. ABDCFE ACBDFE ABCDFE ADBCEF Assume you have the adjacency matrix representing a graph. 1 represents a connection while -1 represents a lack of one:[-1, 1, - 1][-1, -1, 1][1, -1,…arrow_forward3. From the graph above determine the vertex sequence of the shortest path connecting the following pairs of vertex and give each length: a. V & W b. U & Y c. U & X d. S & V e. S & Z 4. For each pair of vertex in no. 3 give the vertex sequence of the longest path connecting them that repeat no edges. Is there a longest path connecting them?arrow_forwardWrite a pseudocode to find all pairs shortest paths using the technique used in Bellman-Ford's algorithm so that it will produce the same matrices like Floyd-Warshall algorithm produces. Also provide the algorithm to print the paths for a source vertex and a destination vertex. For the pseudocode consider the following definition of the graph - Given a weighted directed graph, G = (V, E) with a weight function wthat maps edges to real-valued weights. w(u, v) denotes the weight of an edge (u, v). Assume vertices are labeled using numbers from1 to n if there are n vertices.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Database System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill Education
Starting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSON
Digital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSON
C How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSON
Database Systems: Design, Implementation, & Manag...Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage Learning
Programmable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education
Database System Concepts
Computer Science
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:9780134444321
Author:Tony Gaddis
Publisher:PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:9780132737968
Author:Thomas L. Floyd
Publisher:PEARSON
C How to Program (8th Edition)
Computer Science
ISBN:9780133976892
Author:Paul J. Deitel, Harvey Deitel
Publisher:PEARSON
Database Systems: Design, Implementation, & Manag...
Computer Science
ISBN:9781337627900
Author:Carlos Coronel, Steven Morris
Publisher:Cengage Learning
Programmable Logic Controllers
Computer Science
ISBN:9780073373843
Author:Frank D. Petruzella
Publisher:McGraw-Hill Education