Prove or disprove: Prim’s MST algorithm will work correctly even if weights may be negative.
Prove or disprove: Prim’s MST algorithm will work correctly even if weights may be negative.
MST For an undirected graph G = (V, E) with weights w(e) > 0 for each edge e ∈ E, you are given a MST T. Unfortunately one of the edges e* = (u, z) which is in the MST T is deleted from the graph G (no other edges change). Give an algorithm to build a MST for the new graph. Your algorithm should start from T. Note: G is connected, and G − e* is also connected. Explain...
(1) Prove or disprove that if all the elements of a matrix A is even, the determinant of A is even. (2) Compute the following determinant (1) (4 pts) Prove or disprove that if all the elements of a matrix A is even, the determinant of A is even. (2) (2+2 pts) Compute the following determinant (123) (100 A= 1023 B=020 003 co c
#4. TSP a) Solve with 2 MST approx. algorithm. Note: you can assume weights of edges: (CE) = 36 and w(C,A)=33 А B 24 1) Find MST 2) Double MST 3) Find Eulerian cycle 4) Do shortcuts (show steps here) 10 11 С. 30 25 8 E 28 Report the resulting Hamiltonian cycle and its length:
a) Solve with 2 MST approx. algorithm. Note: you can assume weights of edges: w(C,E) = 36 and w(C,A)=33 A B 24 1) Find MST 2) Double MST 3) Find Eulerian cycle 4) Do shortcuts (show steps here) 9 lol 11 30 25 8 E 28 Report the resulting Hamiltonian cycle and its length:
How much work must Kruskal's MST algorithm do before it starts choosing edges for its MST? Assume the undirected graph has n vertices and m edges. Explain the necessary preliminary work and its big-O cost if done efficiently What are the best case and worst case for Kruskal's MST algorithm with parameters n and/or m? Explain your answer. How much work must Kruskal's MST algorithm do before it starts choosing edges for its MST? Assume the undirected graph has n...
Prove or Disprove the following Let x,y. If x + xy + 1 is even then x is odd
(30 points) Prove or disprove the following statement: There exists a comparison-based sorting algorithm whose running time is linear for at least a fraction of 1/2" of the n! possible input instances of length n.
Modify the statement of the cut property so that it is true, even for the case of edge weights that are not necessarily distinct. Prove that your modified statement is true when edge weights may be repeated. Modify the statement of the cut property so that it is true, even for the case of edge weights that are not necessarily distinct. Prove that your modified statement is true when edge weights may be repeated.
a) Prove that by correctly numbering the vertices of your graph, the greedy colouring algorithm will give you an optimal coloring. (Hint: start with an optimal coloring of G, and use it to number your vertices.) b) Does this coloring algorithm always yield an optimal result? c) What can happens when you apply this algorithm to a complete bipartite graph Knn.
Prove or disprove the following: For any (non-directed) graph, the number of odd-degree nodes is even. In a minimally connected graph of n>2 nodes with exactly k nodes of degree 1 , 1<k<n. I.e., you cannot have a minimally connected graph with 1 node of degree 1 or n nodes of degree 1.