Let G=(V, E) be a simple directed graph in which every vertex u has some (real-valued)...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Let G=(V, E) be a simple directed graph in which every vertex u has some (real-valued) fine, which we denote by fine(u). For each vertex u, let MinFine(u) be the minimum fine that is reachable from vertex u in graph G. In other words, MinFine(u) = min (fine(r): z E reach(u)}, where reach(u) is the set of vertices v EV reachable by a path in G from vertex u (that is, such that G has a (u, v)-path). (i) Show that if two vertices u, v V are in the same strongly connected component in G, then MinFine(u) = MinFine (v). (ii) Design an O(n + m)-time algorithm that for any directed acyclic graph G determines the minimum fine from each vertex in G (that is, computes MinFine(u) for all vertices u EV). Explain your answer, prove correctness of your algorithm and give arguments about its running time. (iii) Use (1) and (ii) to design an O(n+m)-time algorithm that for any simple directed graph G (not necessarily acyclic) determines the minimum fine reachable from each vertex in G (that is, computes MinFine(u) for all vertices u V). Explain your answer, prove correctness of your algorithm and give arguments about its running time. Let G=(V, E) be a simple directed graph in which every vertex u has some (real-valued) fine, which we denote by fine(u). For each vertex u, let MinFine(u) be the minimum fine that is reachable from vertex u in graph G. In other words, MinFine(u) = min (fine(r): z E reach(u)}, where reach(u) is the set of vertices v EV reachable by a path in G from vertex u (that is, such that G has a (u, v)-path). (i) Show that if two vertices u, v V are in the same strongly connected component in G, then MinFine(u) = MinFine (v). (ii) Design an O(n + m)-time algorithm that for any directed acyclic graph G determines the minimum fine from each vertex in G (that is, computes MinFine(u) for all vertices u EV). Explain your answer, prove correctness of your algorithm and give arguments about its running time. (iii) Use (1) and (ii) to design an O(n+m)-time algorithm that for any simple directed graph G (not necessarily acyclic) determines the minimum fine reachable from each vertex in G (that is, computes MinFine(u) for all vertices u V). Explain your answer, prove correctness of your algorithm and give arguments about its running time.
Expert Answer:
Related Book For
Introduction to Algorithms
ISBN: 978-0262033848
3rd edition
Authors: Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest
Posted Date:
Students also viewed these operating system questions
-
CANMNMM January of this year. (a) Each item will be held in a record. Describe all the data structures that must refer to these records to implement the required functionality. Describe all the...
-
Q1. You have identified a market opportunity for home media players that would cater for older members of the population. Many older people have difficulty in understanding the operating principles...
-
The supplies account had a beginning balance of $1,523. Supplies purchased during the period totaled $4,928. At the end of the period before adjustment, $382 of supplies were hand. Prepare the...
-
Beasley, Inc. is going to elect nine board members next month. Betty Brown owns 12.4 percent of the total shares outstanding. How confident can she be of having one of her candidate friends elected...
-
In Problem write the resulting set using the listing method. {x|x 2 = 25}
-
Consider the in-line tube bank of Problem \(7.69(D=\) \(10 \mathrm{~mm}, L=1 \mathrm{~m}\), and \(S_{T}=S_{L}=15 \mathrm{~mm}\) ), with condensing steam used to heat atmospheric air entering the tube...
-
Montana Cycles started July with 25 bicycles that cost $36 each. On July 16, Montana bought 35 bicycles at $60 each. On July 31, Montana sold 33 bicycles for $105 each. Requirements 1. Prepare...
-
W24x94 W16x31 GIVEN: The steel-framed Roof structure as shown below, to be used as a Greenhouses building. The construction is indicated, and dead loads can be found from the table ASCE 7-10. Use...
-
You just retired and qualify for your employers pension plan which promises to pay you $4,200 per month until you die. Assume youre 60 years old in good health and estimate that you will live another...
-
What do these code snippets print? a. b. c. d. int result = 0; for (int i = 1; i
-
In this exercise, we will examine several loops and analyze their potential for parallelization. a. Does the following loop have a loop-carried dependency? b. In the following loop, find all the true...
-
A boat floats in a two-dimensional ocean. It has a position and a direction. It can move by a given distance in its current direction, and it can turn by a given angle. Provide methods: public double...
-
For the SGEMM code developed above for the i7 processor, include the use of AVX2 intrinsics to improve the performance. In particular, try to vectorize your code to better utilize the AVX hardware....
-
In problem 5.9, it was assumed that all transactions on the system were serially executed, which is both unrealistic and inefficient in a DSM multicore. We now relax this condition. We will require...
-
Problem 2: Butterworth Van Dyke (BVD) Equivalent Circuit (25 points) The circuit in figure below is used to represent a MEMS resonator in the electrical domain. Let's do some circuit analysis. V I HE...
-
The rate at which the temperature of an object changes is proportional to the difference between its own temperature and the temperature of the surrounding medium. Express this rate as a function of...
-
This problem develops properties of the Fibonacci numbers, which are defined by recurrence (3.22). We shall use the technique of generating functions to solve the Fibonacci recurrence. Define the...
-
Show that if HAM-CYCLE P, then the problem of listing the vertices of a hamiltonian cycle, in order, is polynomial-time solvable.
-
In the depth-determination problem, we maintain a forest F = {T i } of rooted trees under three operations: MAKE-TREE () creates a tree whose only node is . FIND-DEPTH () returns the depth of node ...
-
How is financial accounting different from management accounting?
-
As the new controller, reply to the following comment made by your plant manager: When I employ a proper accounting software, which can process all my daily accounting records and provide me with all...
-
Describe the five-step decision-making process.
Study smarter with the SolutionInn App