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Computation of Minimum-Cost Paths and Spanning Trees |
The objectives of this exercise are
Students should read through and try to solve the problems on this exercise before attending the tutorial.
This is based on the lecture on the computation of minimum-cost paths, information about minimum-cost spanning trees, and an algorithm that can be used to find them, presented in the lecture on Prim's Algorithm that was presented in class. Chapter 24 of the text book, Cormen, Leiserson, Rivest and Stein’s Introduction to Algorithms, includes more information about these algorithms. The Wikpedia pages on problem of computing minimum-cost paths in graphs and on Dijkstra’s algorithm for this problem are also recommended.
It is expected that all of the questions in this first section can be answered by any student who has successfully completed the prerequisites for this course and attended the lecture on the computation of minimum-cost paths was introduced (reviewing the online notes, as needed).
These warmup problems will not be discussed in the tutorial; students who do have difficulty with them should contact the course instructor to get extra help.
Give definitions of each of the following.
Consider the following Min-Heap. The values that are stored are shown inside nodes; priorities are shown, in brackets, beside them.
Trace the execution of the Decrease-Priority algorithm if the array A represents the above heap, i=8, and the new priority p is 2.
Briefly say what the MCP algorithm does.
Then trace the execution of MCP(G, 0) on each of the following weighted graphs.
Give an asymptotic upper bound on the number of steps used by the MCP algorithm in the worst case, when it is run on inputs G and s, for a graph G = (V, E), and explain how that bound can be proved to be correct.
Define the cost of a spannning tree in a connected weighted undirected graph G. Then define a minimum-cost spanning tree in G.
Trace the execution of MST-Prim on the following weighted undirected graphs. In order to obtain the same as executions, you should choose 0 as the start vertex and visit neighbours of each vertex by increasing value for the label.
What would happen if you applied Prim’s algorithm to a weighted graph G that is not connected? What, if anything, could be stated and proved about the algorithm’s output in this case?
Prove the following claim, which was used in the proof of correctness of both Dijkstra’s and Prim”s algorithms.
Claim: Suppose that each vertex in an undirected graph is coloured either white, grey, or black. Suppose, as well, that every neighbour of a black node is black, as well. Then the only vertices that are reachable from any black nodes are also black.
Compare and contrast each of the following.
Consider the Decrease-Priority algorithm that is used to update information about costs of paths during the MCP algorithm.
Review the information about operations on binary heaps that has previously been studied, and find an algorithm that uses essentially the same technique update a binary heap. Which algorithm does this, and what problem does it solve?
Referring to the information about this other algorithm as needed, identify a loop invariant and a loop variant for the while loop that does most of the work in the Decrease-Priority algorithm.
Use this to sketch a proof of the correctness of this algorithm and to show that it uses Θ(log n) steps, in the worst case, when applied to a priority queue with n elements.
Consider the problem of finding minimim costs paths from a given start vertex to other vertices in a directed weighted graph.
Modify the MCP algorithm so that it can be used to solve this version of the problem.
Describe any and all modifications that must be made to the analysis of the algorithm, in order to prove that your algorithm for directed graphs is correct and efficient.
Then, if time permits, trace the execution of your program on the following weighted directed graph, using 0 as the start vertex.
For the analysis of the graph algorithms that have been studied during the last few weeks of classes, it has been assumed that it is possible to enumerate the set of neighbours of a given vertex v using time that is linear in the number of neighbours, that is, linear in the degree of this vertex.
One of the two common representations of graphs supports an enumerations of the neighbours of a vertex at this cost, while the other does not.
Consider either Dijkstra’s algorithm to compute minimum-cost paths, or Prim’s algorithm to compute a minimum-cost spanning tree. Suppose that either is applied to some weighted, connected undirected graph G = (V, E) and vertex s.
Try to find the best upper bound on the number of times that each of the following operations is performed on a Min-Heap, when either of these algorithms is applied.
Hint: You should find that one of these operations can be performed significantly more often than any of the rest.
A Fibonacci heap is another data structure that can be used to implement a priority queue. While the worst-case of operations is greater than those for a binary heap, the amortized cost of a sequence is actually better: The total cost of any sequence of operations beginning with an empty heap, that includes p Insert, Minimum and Decrease-Priority operations and q DeleteMin operations is in O(p + q log n), if n is the maximum size of the priority queue during this sequence of operations.
Use this information to give an asymptotic upper bound (in terms of |V| and |E|) on the number of steps used in the worst case, by either Dijkstra’s algorithm or Prim’s algorithm, if the binary heap used by the algorithm is replaced with a Fibonacci heap.
For which graphs (or kinds of graphs) would versions of the algorithms using Fibonacci heaps be asymptotically faster than versions using binary heaps?
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