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Consider a directed acyclic graph G=(N,E) with a source node m∈N and target node n∈N. P2.1. Provide an algorithm that computes the number of paths starting at m and ending at n. P2.2. We say that the directed acyclic graph G has a bottleneck if there is a node b, distinct from source m and target n, such that all paths from m to n go through node b. Write an algorithm that returns true if and only if G has such a bottleneck. You may assume that there is at-least one path from m to n. For each question, explain why your algorithm is correct, what the complexity of your algorithm is, and which graph representation you use.

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Final answer:

To compute the number of paths starting at node m and ending at node n in a directed acyclic graph, we use a dynamic programming approach. We create an array pathCount where pathCount[i] represents the number of paths from m to i. To check for the presence of a bottleneck, we perform a topological sort and iterate through the nodes.

Step-by-step explanation:

To compute the number of paths starting at node m and ending at node n in a directed acyclic graph, we can use a dynamic programming approach. We create an array pathCount, where pathCount[i] represents the number of paths from node m to node i. We initialize pathCount[m] as 1 and the rest of the array as 0. Then, for each node j that is reachable from i, we update pathCount[j] by adding pathCount[i] to it. Finally, the value of pathCount[n] will give us the number of paths from m to n.

To check for the presence of a bottleneck in the graph, we can perform a topological sort on the graph and then iterate through the nodes in reverse order. For each node b, we can check if all the paths from m to n pass through b by checking if pathCount[b] is equal to pathCount[n]. If we find such a node, we return true. If we reach the end of the iteration without finding such a node, we return false.

The complexity of the algorithm is O(V + E), where V is the number of nodes and E is the number of edges in the graph. We can use an adjacency list representation of the graph to efficiently perform the required operations.

User Rob Monhemius
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