Problem Description

The given problem presents a Directed Acyclic Graph (DAG) which means a graph that has directed edges and no cycles. It includes 'n' vertices which are labeled from 0 to n-1. We're also provided an array of edges where each edge is represented by a pair [from_i, to_i], indicating a directed edge that goes from vertex from_i to vertex to_i.

The task is to find the smallest set of vertices with the property that starting from those vertices, one can reach all other vertices in the graph. The important aspect of this problem is to understand that these vertices should be the starting points for traversing the entire graph. It is also given that there is a unique solution for this problem, which means there's only one smallest set of such vertices and the vertices can be returned in any sequence.

Intuition

To solve this problem, we take advantage of a key characteristic of DAGs — a node that has no incoming edges can be a starting point to reach all other nodes because there are no other nodes that need to be visited before it. Conversely, nodes with incoming edges have at least one other node that must be processed first.

The solution approach is as follows:

1. We initialize a counter cnt to keep track of the number of incoming edges each vertex has.
2. To fill this counter, we iterate over all directed edges. For each edge, we increase the count for the target vertex to_i because this edge represents an incoming edge to to_i.
3. Once we have the count of incoming edges for each vertex, the vertices with a count of zero are exactly the vertices we're looking for. They have no incoming edges and can serve as starting points.
4. We iterate through all vertices from 0 to n-1, and those with a cnt value of zero are added to our results list as they represent the smallest set of vertices that can be used to reach all other vertices in the graph.

By following this process, we efficiently identify the vertices that are not dependent on any other vertices to be visited first, which fulfills the requirement of reaching every vertex in the graph starting from the smallest set of initial vertices.

Learn more about Graph patterns.

Solution Approach

The solution is implemented in Python, and it uses the Counter class from the collections module to keep track of the number of incoming edges for each vertex in the graph.

Here's a step-by-step breakdown of the algorithm implemented in the solution code:

1. A Counter object named cnt is created to count the incoming edges of each vertex. This is constructed by iterating over the edges list and counting each target vertex t (the second element of each edge [_, t]).

2. A for-loop comprehensions construct:

   [i for i in range(n) if cnt[i] == 0]

   This is used to iterate over all vertex indices from 0 to n-1.

3. For each index i, the loop checks if cnt[i] == 0. This condition is true for the vertices that do not have any incoming edges (meaning they are not the target of any edge in the edges list).

4. The indices i that satisfy the condition are collected into a list. These form the smallest set of vertices from which all nodes in the graph are reachable.

5. The final list is returned as the output of the findSmallestSetOfVertices function.

The algorithm essentially uses a counting method to identify source nodes in the graph. A source node is a node with no incoming edges, which means that it can be a starting point, and you can reach every other node in the DAG via a path from these sources.

This algorithm runs in linear time relative to the number of edges, as it only makes a single pass through the edges list to build the counter and then another pass through the list of vertex indices to collect the result. Therefore, the time complexity is O(E + V), where E is the number of edges and V is the number of vertices.

Example Walkthrough

Let's illustrate the solution approach with a small example:

Suppose we have a graph with n = 4 vertices and the following directed edges: edges = [[0,1], [0,2], [1,3], [2,3]]. The edges indicate that there are directed paths from vertex 0 to vertex 1, from vertex 0 to vertex 2, from vertex 1 to vertex 3, and from vertex 2 to vertex 3.

Let's walk through the approach:

1. We set up a counter cnt to track the incoming edges to each vertex. In our graph representation, the incoming edges are described by the second element in each edge pair. So initially, the counter cnt for all vertices is set to zero: cnt = {0: 0, 1: 0, 2: 0, 3: 0}.

2. Next, we iterate over the list of edges to update these counts. After iterating, we get:

   • For edge [0,1], increase cnt[1] by 1;
   • For edge [0,2], increase cnt[2] by 1;
   • For edge [1,3], increase cnt[3] by 1;
   • For edge [2,3], increase cnt[3] by 1 again.

   After processing all edges, the updated counts of incoming edges for the vertices are: cnt = {0: 0, 1: 1, 2: 1, 3: 2}.

3. With these counts, we now identify vertices with zero incoming edges, as they are potential starting points to reach any other vertex. From cnt, we see that only vertex 0 has a count of zero.

4. Using a list comprehension, we extract vertices with zero counts into our result list:

   result = [i for i in range(n) if cnt[i] == 0] # result will be [0]

5. The final result list, which contains the smallest set of vertices from which all other vertices are reachable, is [0].

This is the end of the example walkthrough. By starting at vertex 0, we can reach all other vertices in the graph, fulfilling the task. The solution approach works efficiently, regardless of the graph's size, by focusing on finding vertices with no dependencies (no incoming edges).

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code is determined by two main operations. The first operation is the creation of the Counter, which involves iterating over all the edges. If m represents the number of edges, then this operation is O(m).

The second operation is the list comprehension, which checks each vertex to see if it has a count of zero in the Counter. Since there are n vertices, this operation is O(n).

Since these two operations happen sequentially and not nested, the overall time complexity is O(m + n).

Space Complexity

The space complexity is primarily impacted by the Counter that stores occurrence counts for the target vertices of each edge. In the worst case, all m edges might be pointing to different target vertices, so the Counter would store m key-value pairs.

Therefore, the space complexity is O(m) for the Counter. However, the output list's size is at most n in the case when no vertices are targets. This n sized list is separate from the Counter storage.

Thus, when considering both the Counter and the final output list, the space complexity is O(m + n).

Learn more about how to find time and space complexity quickly using problem constraints.