Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 21.2, Problem 4E
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To give a tight asymptotic bound on the running time of the sequence of operations in Figure 21.3 by using linked list weighted-union heuristic.
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Consider the disadvantages of employing an adjacency list representation to express a weighted graph.
Discuss the disadvantages of adjacency list representation of a weighted graph representation.
Respond to the following in a minimum of 175 words:
Python implementation of algorithm has three nested f or loops and the adjacency matrix T passed to this procedure simply multiplies this matrix with itselfusing logical operators. A test is carried to check whether all elements ofT is 1 which means graph G is connected, otherwise G is not connected as returned to the caller along with the connectivity matrix.
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- Give a high-level analysis of the running time of depth-first-search, assuming that the graph G has n nodes and m edges.arrow_forwardLinked lists are employed in a specific manner to express adjacence lists on a graph. Give an illustration of your point with an example. Does coding not require any prior knowledge?arrow_forwardApply the Breadth first traversal algorithm to the undirected graph G given in Example and replicated here for convenience. The goal is to do a Breadth first search for the key 9.arrow_forward
- Now analyze the above execution and determine whether the deadlock hasoccurred or not.If occurs,then(i) Detect the deadlock using wait for graph(ii) Find out the solution to recover from this deadlockarrow_forwardGiven the definition in Haskell: fmap :: (a -> b) -> [a] -> [b] fmap g xs = reverse (map g xs) With Functor law: (fmap g . fmap h) xs = fmap (g . h) xs Using list xs :: [Int], prove the definition does not satisfy the law:arrow_forwardTake into consideration the downsides of a weighted graph representation that utilises an adjacency list representation.arrow_forward
- Take into consideration the limitations of employing an adjacency list representation for a weighted graph.arrow_forwardFor any given connected graph, G, if many different spanning trees can be obtained, is there any method or condition setting that allows the DFS spanning tree of G to only produce a unique appearance? can you give me some simple opinion?arrow_forwardThe transpose of a directed graph G = (V, E) is the graph GT = (V; ET), where ET = { (u, v) | (v, u) ϵ E}. Thus, GT is G with all its edges reversed. Describe efficient algorithms for computing GT from G, for both the adjacency-list and adjacency-matrix representations of G. Analyze the running times of your algorithms.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_forwardThe reverse of a directed graph G = (V,E) is another directed graph G^R = (V,E^R) on the same vertex set, but with all edges reversed; that is, E^R = {(v, u) : (u, v) ∈ E}.Give a linear-time algorithm for computing the reverse of a graph in adjacency list format.arrow_forwardComputer Science Frequently, a planar graph G=(V,E) is represented in the edgelist form, which for each vertex vi V contains the list of its incident edges, arranged in the order in which they appear as one proceeds counterclockwise around i v . Show that the edge-list representation of G can be transformed to the DCEL (DoublyConnected-Edge-List) representation in time O(|V|).arrow_forward
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