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.