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 25.3, Problem 4E
Program Plan Intro
To evaluate the drawback of the method proposed by the professor Greenstreet for reweight the edges in Johnson’s
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Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let P be the shortest path between two nodes s, t. Now, suppose we replace each edge weight ℓ(e) withℓ(e)^2, then P is still a shortest path between s and t.
True or false: For graphs with negative weights, one workaround to be able to use Dijkstra’s algorithm (instead of Bellman-Ford) would be to simply make all edge weights positive; for example, if the most negative weight in a graph is -8, then we can simply add +8 to all weights, compute the shortest path, then decrease all weights by -8 to return to the original graph.
Select one:
True
False
Let's say we have a graph with n nodes. If this graph has a 3-node cycle, write the brute-force algorithm step by step and calculate the time complexity?
Chapter 25 Solutions
Introduction to Algorithms
Ch. 25.1 - Prob. 1ECh. 25.1 - Prob. 2ECh. 25.1 - Prob. 3ECh. 25.1 - Prob. 4ECh. 25.1 - Prob. 5ECh. 25.1 - Prob. 6ECh. 25.1 - Prob. 7ECh. 25.1 - Prob. 8ECh. 25.1 - Prob. 9ECh. 25.1 - Prob. 10E
Ch. 25.2 - Prob. 1ECh. 25.2 - Prob. 2ECh. 25.2 - Prob. 3ECh. 25.2 - Prob. 4ECh. 25.2 - Prob. 5ECh. 25.2 - Prob. 6ECh. 25.2 - Prob. 7ECh. 25.2 - Prob. 8ECh. 25.2 - Prob. 9ECh. 25.3 - Prob. 1ECh. 25.3 - Prob. 2ECh. 25.3 - Prob. 3ECh. 25.3 - Prob. 4ECh. 25.3 - Prob. 5ECh. 25.3 - Prob. 6ECh. 25 - Prob. 1PCh. 25 - Prob. 2P
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- A Greedy AlgorithmA simple way to find MWM of a graph is to sort edges with respect to their weightsin non-increasing order and to select edges from this list in order, include the edgein matching and remove all adjacent edges from the list as shown below.1. Input: A weighted graph G = (V, E, w)2. Output: A maximal weighted matching M of G3. M ← Ø4. sort E and store the list in Q5. while Q = Ø6. remove first element e from Q7. M ← M ∪ e8. Q ← Q\ { all adjacent edges to e }Let us try to implement this algorithm in Python with the above steps.arrow_forwardNote: Your solution should have O(n) time complexity, where n is the number of elements in l, and O(1) additional space complexity, since this is what you would be asked to accomplish in an interview. Given a linked list l, reverse its nodes k at a time and return the modified list. k is a positive integer that is less than or equal to the length of l. If the number of nodes in the linked list is not a multiple of k, then the nodes that are left out at the end should remain as-is. You may not alter the values in the nodes - only the nodes themselves can be changed.arrow_forwardAn Unfair AlgorithmFinding the MWM of a graph is as easy as selecting edges from a list in order, including the edge in matching, and removing all neighbouring edges from the list. This is done by sorting edges according to their weights in a non-increasing sequence. 1. Input: A weighted graph G = (V, E, w)2. Output: A maximal weighted matching M of G 3. M ← Ø 4. sort E and store the list in Q 5. while Q = Ø6. remove first element e from Q7. M ← M ∪ e8. Q ← Q\ { all adjacent edges to e }Let's attempt to use the preceding methods to create this algorithm in Python.arrow_forward
- Let G = (X ∪ Y, E) be a bipartite graph such that the vertices are partitioned into two groups Xand Y , and each edge has one end point in X and one end point in Y .A 2-1 generalized matching is a set of edges S ⊂ E satisfying the following two conditions:1. Every vertex in X belongs to at most two edges in S.2. Every vertex in Y belongs to at most one edge in S.Give an algorithm to find the size (number of edges) of maximum 2-1 generalized matchingarrow_forward6. In Java create an algorithm for an undirected graph with n vertices and m edges that...- Takes as a parameter some integer k- Returns the maximum induced subgraph where each vertex of that subgraph has a degree greater than or equal to k (or returns null if no such subgraph exists)- Operates in O(n + m) timearrow_forwardCreate an algorithm that, given a directed graph g = (v e) and a distinguished vertex s v, finds the shortest path from s to v for each v v. If g contains n vertices and e edges, your method must execute in o(n + e) time.arrow_forward
- Provide an algorithm with the same time complexity as Bellman-Ford so that: it sets dist [w] to −∞for allvertices w for which there is a vertex that belongs to a negative-weighted cycle on some path from the sourcevertex to w, and outputs False if there is no such vertex w.arrow_forwardConsider 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.arrow_forwardFill in the blank Dijkstra's algorithm works because, on every shortest path p from a source vertex u to a target vertex v, there is a (predecessor) vertex w in p immediately before v such that removing v from p yields the shortest path from u to w. In other words, the path through the previous vertex is also the shortest path. Thus, choosing an edge from the previous vertex that brings us to v with the __ cost always yields the shortest path to v.arrow_forward
- The Python implementation updates the cost of reaching from the start vertex to each of the explored vertexes. In addition, when it decides on a route, A* considers the shortest path from the start to the target, passing by the current vertex, because it sums the estimate from the heuristic with the cost of the path computed to the current vertex. This process allows the algorithm to perform more computations than BFS when the heuristic is a proper estimate and to determine the best path possible.arrow_forwardWrite a Program to All-pairs shortest path on a line. Given a weighted line graph (undirected connected graph, all vertices of degree 2, except two endpoints which have degree 1), devise an algorithm that preprocesses the graph in linear time and can return the distance of the shortest path between any two vertices in constant timearrow_forwardWrite a polynomial time algorithm to check if an undirected graph G(V, E), whose maximum degree is 1000, has a clique size of at least k, where 1 ≤ k ≤ |V|.arrow_forward
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