Find Eventual Safe States

Identifying Problem Isomorphism

“Find Eventual Safe States” falls into the category of graph theory problems, more specifically those that deal with identifying specific nodes based on their connectivity and cycle participation.

A simpler problem in this category is “Course Schedule”. In this problem, you are asked if it’s possible to finish all courses given a list of prerequisite pairs. This problem is simpler because it requires detection of a cycle in a directed graph, but does not require the identification of terminal or safe nodes.

A more complex problem could be “Critical Connections in a Network”. This problem involves finding all the critical connections in a network. A connection is critical if removing it will make some server unable to reach some other server. This problem is more complex because it requires not only understanding of the graph structure and cycles, but also the computation of low-link values in a depth-first search to identify critical connections.

The reasoning for these selections:

“Course Schedule” involves cycle detection in a directed graph, which is a part of “Find Eventual Safe States”. However, it doesn’t involve identifying terminal or safe nodes, making it simpler.
“Critical Connections in a Network” requires a deeper understanding of graph structures and graph traversal to identify specific connections. Like “Find Eventual Safe States”, it involves selecting and identifying specific nodes, but with the added complexity of identifying critical connections.

So, the order of problems from simpler to more complex, based on understanding of graph structures and graph traversal, would be:

“Course Schedule”
“Find Eventual Safe States”
“Critical Connections in a Network”

This mapping is an approximation. While these problems all deal with understanding graph structures and graph traversal, the specific constraints and details of each problem can significantly influence the complexity and solution approach.

10 Prerequisite LeetCode Problems

“Find Eventual Safe States” is a graph-related problem where we need to find the nodes (or states) that are not part of any cycle.

Here are ten simpler problems as preparation:

207. Course Schedule: This problem introduces the concept of detecting cycles in a directed graph, which is an important part of the “Find Eventual Safe States” problem.
210. Course Schedule II: This problem is a follow-up to the Course Schedule problem and introduces topological sorting, which could also be useful.
261. Graph Valid Tree: This problem requires determining if a graph is a valid undirected tree, i.e., it doesn’t have any cycles.
785. Is Graph Bipartite?: The problem asks to find out whether you can divide the nodes of a graph into two disjoint sets such that every edge of the graph crosses the partition.
133. Clone Graph: This problem provides practice on graph traversal.
200. Number of Islands: This is a simpler problem that requires understanding of how to identify connected components in a grid.
994. Rotting Oranges: A problem that requires breadth-first search in a grid, similar to graph traversal.
127. Word Ladder: This problem introduces the concept of a transformation sequence which is similar to paths in a graph.
279. Perfect Squares: A problem that requires breadth-first search to find the shortest path in a graph, which is a fundamental graph concept.
547. Number of Provinces: This problem is about finding connected components in a graph, which is a key part of understanding how nodes/vertices are connected.

These cover various graph concepts, including detecting cycles, finding connected components, and traversing graphs using both depth-first and breadth-first search, which are all important for solving the “Find Eventual Safe States” problem.

Problem Analysis and Key Insights

Here are some key insights we can glean from the problem statement:

Graph Traversal: The problem is fundamentally about traversing a directed graph. We need to follow paths from every node in the graph and determine where those paths lead.
Terminal Nodes: A node is considered “terminal” if there are no outgoing edges from it. This definition is straightforward and doesn’t depend on the rest of the graph’s structure.
Safe Nodes: A node is considered “safe” if every possible path starting from that node leads to a terminal node. This definition is more complex, as it depends not just on the node’s own edges, but also on the structure of the rest of the graph. In particular, it depends on which nodes are reachable from the given node.
Possible Cycles: The graph may contain cycles, i.e., paths that lead back to the same node. This introduces an additional layer of complexity, as we need to account for the possibility that some paths may not lead to any terminal node.
Sorting the Output: The output should be sorted in ascending order. This is a minor point compared to the others, but it’s something we’ll need to keep in mind when designing our solution.

From these insights, it seems clear that a key part of the problem will be designing an efficient graph traversal algorithm that can determine which nodes are “safe”. This will likely involve some form of depth-first or breadth-first search, and will need to account for the possibility of cycles in the graph. The sorting of the output should be relatively straightforward once we’ve identified the safe nodes.

Problem Boundary

The scope of this problem falls into the domain of graph theory, specifically focusing on traversing directed graphs. The main objectives are:

Understanding the Graph: The first part of the problem is understanding the structure of the graph. It is a directed graph represented by a 2D integer array, where each index corresponds to a node, and the array at that index represents the nodes to which it has directed edges.
Identifying Terminal Nodes: A terminal node is a node from which there are no outgoing edges. In the context of this problem, it refers to nodes that have an empty list of adjacent nodes.
Identifying Safe Nodes: A safe node is defined as a node from which every possible path leads to a terminal node. We need to explore all paths from each node to determine if it is safe or not.
Cycles Handling: The graph can contain cycles, which should be properly handled to avoid infinite looping during the graph traversal.
Result Presentation: After identifying all safe nodes, the result needs to be returned as an array sorted in ascending order.

Thus, the scope of this problem involves designing an efficient algorithm for traversing a directed graph, handling cycles, identifying certain node types (safe and terminal), and arranging the result according to the given specification.

The boundary of this problem can be established by clearly defining the inputs, expected outputs, and constraints.

Inputs:

A directed graph represented as a 2D integer array, with n nodes labeled from 0 to n - 1, and where graph[i] is an array of nodes to which node i has outgoing edges.

Expected Outputs:

An array of all safe nodes in the graph, sorted in ascending order. A safe node is defined as a node from which all paths lead to a terminal node or another safe node. A terminal node is a node with no outgoing edges.

Constraints:

n, the number of nodes, is between 1 and 10,000.
Each node’s outgoing edges, if any, are represented as a list that is sorted in strictly increasing order.
The graph may contain self-loops and cycles.
The total number of edges in the graph will be in the range [1, 4 * 10^4].

The problem’s boundary is defined within these inputs, outputs, and constraints. Any solution must operate within these parameters. Additionally, the problem statement does not specify any time or space complexity requirements, but a good solution should be efficient in terms of both. It should be able to handle the maximum size of the graph as described in the constraints.

One more important boundary condition to consider is handling cycles in the graph since they can lead to infinite loops in the traversal if not properly managed.

Problem Classification

This problem falls within the domain of Graph Theory and Depth-First Search/Breadth-First Search algorithms in Computer Science. It deals with the properties of nodes in a directed graph and involves traversing the graph to identify “safe” nodes.

What Components:

Input: The input consists of a 2D integer array graph which represents a directed graph. Each integer i in the array represents a node in the graph and graph[i] is a list of all nodes that have an edge from i.
Output: The output is an array containing all the “safe” nodes in the graph, sorted in ascending order. A node is considered “safe” if every possible path starting from that node leads to a terminal node.
Constraints: Constraints are provided on the size of the graph, the number of edges it can have, and the values the nodes can take. The graph may contain self-loops.
Examples/Scenarios: The problem provides examples of the input graph and the corresponding output, which helps illustrate the task at hand.

This problem is a classic example of graph traversal, specifically using depth-first or breadth-first search algorithms. This is due to the need to explore all possible paths from a given node to determine if it is “safe”.

The problem also has elements of sorting, as the output must be in ascending order. However, this is likely a secondary aspect of the problem, as the main challenge is identifying safe nodes in the first place.

The problem also contains an element of Dynamic Programming, as the safety of a node can be determined by the safety of the nodes it directly leads to. Hence, we can solve subproblems (i.e., is this node safe?) and use those results to solve the overall problem.

The constraint that the graph may contain self-loops introduces an additional challenge, as it means the traversal algorithm needs to account for the possibility of cycles within the graph.

Distilling the Problem to Its Core Elements

Fundamental Concept: This problem is fundamentally about graph traversal and identifying terminal nodes (those with no outgoing edges) and safe nodes (those which eventually lead to a terminal node).

Simple Description: Imagine a maze where you start at a point and can follow one or many paths. A point is considered ‘safe’ if all possible paths from it eventually lead to an exit. Given a map of the maze with all points and paths, identify all the ‘safe’ points.

Core Problem: The core problem is identifying and listing all the ‘safe’ nodes in a directed graph. A ‘safe’ node is defined as a node where every possible path starting from that node leads to a terminal node or another safe node.

Problem Breakdown:

Understanding the graph structure: Each node may have zero or more outgoing edges to other nodes.
Identifying terminal nodes: Terminal nodes have no outgoing edges.
Identifying safe nodes: Nodes that only lead to terminal nodes or other safe nodes through their paths.
Output the safe nodes in ascending order.

Minimal Operations:

Traverse the graph to identify all terminal nodes.
Starting from each terminal node, backtrack and mark all nodes that can reach these terminal nodes.
Finally, return a list of marked nodes (safe nodes), sorted in ascending order.

Visual Model of the Problem

This problem can be visualized as a directed graph. A graph contains nodes and edges, where nodes can represent points and edges can represent connections or paths between these points. The graph is represented by a 2D integer array where each index of the array represents a node, and the value at that index is another array that represents the nodes that can be reached from the current node.

Here’s an example:

Suppose we have graph = [[1,2],[2,3],[5],[0],[5],[],[]]. This graph can be visualized as below:

0 ---> 1 ---> 2 ---> 3
|                     ^
v                     |
4 ---> 5 <------------|
|
v
6

In this graph:

Nodes 5 and 6 are terminal nodes, as they have no outgoing edges.
Nodes 2, 4, 5, and 6 are safe nodes, because any path starting from these nodes either directly terminates or leads to another safe node.

This visualization can help us understand the problem more intuitively and is helpful for designing an algorithm to solve the problem.

Problem Restatement

We are given a graph with nodes numbered from 0 to n - 1, represented as a 2D integer array. Each index of this array symbolizes a node, and the array at that index represents the nodes that are directly reachable from the current node through a directed edge.

There are two kinds of nodes to consider in this graph:

Terminal nodes: These nodes do not have any outgoing edges. In other words, no other node can be reached directly from these nodes.
Safe nodes: From these nodes, every path we can take either directly leads to a terminal node or another safe node.

The goal is to find all the safe nodes in the given graph. We need to return these nodes as an array sorted in ascending order.

The constraints ensure that the number of nodes is between 1 and 10,000, and the number of edges is between 1 and 40,000. The graph can contain self-loops, meaning a node might have a direct edge to itself. Additionally, the adjacency list representing the graph is sorted in strictly increasing order.

Abstract Representation of the Problem

We are given a directed graph G, represented as an adjacency list, with N vertices and E edges, where each vertex is identified by an integer in the range [0, N-1]. The graph may contain self-loops.

A vertex is considered “terminal” if it does not have any outgoing edges and “safe” if all paths starting from it lead to a terminal vertex or another safe vertex. The task is to find all safe vertices in the graph.

The problem’s constraints ensure that N is between 1 and 10,000 and E is between 1 and 40,000. The adjacency list representing the graph is sorted in strictly increasing order.

This problem is a graph traversal problem and can be viewed as an abstract problem in the field of graph theory and algorithms, focusing on identifying special types of vertices in a directed graph based on their connection patterns.

Terminology

There are a few specialized terms and concepts related to graph theory that are crucial for understanding and solving this problem:

Directed Graph: A graph in which edges have orientations, i.e., they go from one vertex to another. In this problem, the graph is directed, and this impacts how we traverse it.
Vertex (Node): A fundamental part of a graph, which can have a name or key and may also store additional information, called the “payload”. In this problem, vertices represent the nodes of the graph.
Edge: A connection between two vertices. In a directed graph, the connection is from a start vertex to an end vertex. In this problem, an edge represents a directed connection between two nodes.
Adjacency List: This is a way to represent a graph. Each vertex has a list of which vertices it is connected to. In this problem, the graph is represented as an adjacency list.
Terminal Vertex (Node): In the context of this problem, a terminal vertex is a node that doesn’t have any outgoing edges.
Safe Vertex (Node): In the context of this problem, a safe vertex is a node from which every possible path leads to a terminal vertex or another safe vertex.
Graph Traversal: The process of visiting (checking or updating) each vertex in a graph. Popular algorithms for graph traversal include Depth-First Search (DFS) and Breadth-First Search (BFS).

Understanding these terms and concepts is important for interpreting the problem statement correctly, conceptualizing a solution, and implementing it effectively.

Problem Simplification and Explanation

The problem involves dealing with a graph. You can visualize a graph as a network of cities connected by one-way roads. Each city is a node, and each road is a directed edge that connects one city to another.

In this scenario, a “terminal city” is one that has no roads leading out of it - you can get in, but you can’t get out. This represents a terminal node in our graph.

A “safe city” is a city from which, no matter which road you take, you will eventually end up in a terminal city. That is, all roads from a safe city eventually lead to cities with no outgoing roads.

Our task is to identify all the “safe cities” in the network. We want to find all cities such that no matter which path you take from them, you will eventually reach a terminal city.

The problem asks for a list of all the safe nodes (cities) in ascending order of their labels (city names).

The key concepts involved in this problem are graph theory, depth-first search or breadth-first search for traversing the graph, and the idea of marking nodes as safe or unsafe based on whether all paths from them lead to terminal nodes.

Constraints

In the “Find Eventual Safe States” problem, here are some characteristics and constraints that can be exploited to our advantage:

Graph Structure: The problem deals with a directed graph. This structure allows us to apply graph traversal algorithms such as Depth-First Search (DFS) or Breadth-First Search (BFS), which are quite efficient for such problems.
Terminal Nodes: A terminal node is defined as a node with no outgoing edges. We can use these terminal nodes as starting points for our traversal since any node that can reach a terminal node is safe. Also, if during a DFS we ever end up on a node that we know is safe, we can immediately stop and mark the current path as safe.
Sorting Requirement: The output nodes need to be sorted in ascending order. If we use DFS, we can take advantage of this by starting our search from the node with the highest label, thus avoiding the need for additional sorting.
No Multi-edges or Self-loops: The problem statement mentions that the graph may contain self-loops, but there is no multiple edges between two nodes. This simplifies the problem a bit, as we do not need to account for complex scenarios where multiple paths might exist between two nodes.
Large Input Size: The input size can be quite large (up to 10^4 nodes), which suggests that an efficient solution will likely need to avoid quadratic or higher time complexities.

These characteristics and constraints provide clues towards the path to an efficient solution.

Analyzing the constraints of the problem often reveals important information about the nature of the problem and the type of solution required. Here are the key insights from analyzing the constraints for this problem:

Directed Graph: The problem deals with a directed graph, implying that we might need to utilize graph traversal algorithms such as Depth-First Search or Breadth-First Search.
No Multi-edges: As there are no multiple edges between two nodes, the complexity of the problem is significantly reduced, as we do not need to deal with multiple paths between the same two nodes.
Terminal Nodes: These are nodes with no outgoing edges. By the definition in the problem, these nodes are automatically safe, and this fact can be leveraged in the solution.
Graph Size: The graph can be large (up to 10^4 nodes). This implies that the solution needs to be efficient. Algorithms with quadratic or higher time complexities may not be feasible for this problem.
Output Order: The output nodes need to be in ascending order. This constraint could impact the choice of data structure used for storing the safe nodes, or might necessitate an additional sorting step at the end.

By understanding these constraints, we get a better idea of what to consider while devising a solution. It can help us rule out certain approaches that will not work due to the constraints, and guide us towards potential solutions that fit within these boundaries.

Case Analysis

Let’s look at a few different scenarios:

Small Graph with Single Path (Smallest size)

1
2
Input: graph = [[1],[]]
Output: [1]

In this case, the graph only has two nodes. The only path leads from node 0 to node 1, which is a terminal node. Hence, the safe nodes are [1].

Graph with Multiple Independent Paths

1
2
Input: graph = [[1],[2],[],[4],[]]
Output: [2, 4]

Here we have two independent paths - one from node 0 to 1 to 2 and another from node 3 to 4. Both paths end at terminal nodes 2 and 4, making these the safe nodes.

Graph with Cycles (Not Safe Nodes)

1
2
Input: graph = [[1],[0]]
Output: []

In this case, the graph forms a cycle between nodes 0 and 1. Since there are no terminal nodes, there are no safe nodes.

Graph with Self-Loops

1
2
Input: graph = [[0],[1],[2],[]]
Output: [3]

Here, each node from 0 to 2 is a self-loop, and node 3 is a terminal node. Therefore, the only safe node is node 3.

Large Graph with Random Structure

For larger graphs, the exact structure will depend on the specific problem instance. However, the key things to consider would be:

The presence of terminal nodes (which are always safe).
Paths from safe nodes to other safe nodes.
The absence of cycles in paths starting from safe nodes.

Edge cases to consider include the smallest and largest possible inputs (e.g., a graph with only one node, or a graph with 10^4 nodes), and graphs with specific structures (e.g., all nodes forming a single cycle, or a graph with no edges).

Analyzing the different cases reveals a few key insights that are essential to solve this problem:

Terminal Nodes: Terminal nodes, which are nodes with no outgoing edges, are always safe. This is because there’s no path to follow from these nodes, so any path that reaches them will stop.
Independent Paths: If there are independent paths (paths that do not share nodes) that lead to terminal nodes, all the nodes in those paths are safe.
Cycles: Any node that is part of a cycle is not safe. This is because a cycle means there’s a path that doesn’t end at a terminal node. Thus, if there’s a cycle in the path from a node, that node isn’t safe.
Paths to Safe Nodes: If a node has an outgoing edge only to nodes that are known to be safe, that node is also safe. In other words, a node is safe if all paths from it lead to other safe nodes.
Large Graphs: For large graphs, a more efficient way to identify safe nodes will be needed to avoid time-out errors. This will likely involve identifying terminal nodes first, and then progressively identifying safe nodes that lead to already-identified safe nodes.
Boundary Conditions: For the smallest graph (only one node), the node is safe because it is a terminal node. For the largest graphs, efficient processing and avoiding cycles will be key. For graphs with no edges, all nodes are terminal and thus safe.

By understanding these aspects, we can approach the problem by first finding terminal nodes, and then finding nodes that only connect to these safe nodes, repeating the process until no more safe nodes can be found.

Identification of Applicable Theoretical Concepts

The problem requires traversing a directed graph and determining safe nodes, which are nodes from which all paths lead to a terminal node. There are a few key mathematical and computer science concepts at play here:

Graph Theory: The problem involves the traversal of a directed graph. Concepts like nodes (vertices), edges, directed edges (arcs), and terminal nodes are all components of graph theory. Understanding how to traverse a graph and track visited nodes is critical to solving this problem.
Depth-First Search (DFS) or Breadth-First Search (BFS): Both of these are standard graph traversal techniques. DFS explores as far as possible along each branch before backtracking, which can be useful for finding the terminal nodes. BFS explores all the neighbors at the present depth before moving on to nodes at the next depth level, which can be useful for checking if a node only leads to safe nodes. DFS is typically more suited to this problem as it can handle cycles in the graph better.
Topological Sorting: Topological sorting for Directed Acyclic Graph (DAG) is a linear ordering of vertices such that for every directed edge uv, vertex u comes before v in the ordering. Although this problem does not require a topological sort because there may be cycles in the graph, the idea of processing nodes based on their dependencies is related to topological sorting.
Dynamic Programming (DP): DP can help in maintaining the states of each node (safe or unsafe) and avoid re-computation. If you know that a node is safe (all paths lead to a terminal node), you can store this information and use it when traversing from other nodes.

Understanding and applying these concepts can help break down the problem into manageable parts and identify the most efficient approach to solve it.

Simple Explanation

Let’s use a metaphor of a maze to explain this problem.

Imagine you are in a large maze with multiple rooms. Each room has doors that lead to other rooms, but some rooms are dead-ends with no doors at all - we can call these dead-end rooms “terminal rooms”. Now, some rooms are quite special - if you start from one of these special rooms and start exploring through any door, no matter which way you go, you will eventually end up in a terminal room. We can call these special rooms “safe rooms”.

The problem is like you being given a map of this maze, and your task is to identify all the safe rooms in this maze. But remember, a room is considered safe only if all possible paths starting from that room lead to a terminal room.

So, given a map (the directed graph), you have to find out and list all these safe rooms (nodes).

This is the concept of the problem in a nutshell, using a metaphor of a maze to simplify the ideas of nodes, edges, terminal nodes, and safe nodes.

Problem Breakdown and Solution Methodology

To solve the problem “Find Eventual Safe States”, you need to identify all nodes in the graph that, if started from, will eventually lead to a terminal node (a node without any outgoing edges). The nodes that fulfill this condition are deemed “safe”.

Here’s a detailed step-by-step breakdown of how you might approach this problem:

Define Safe Nodes: Start by defining what a safe node is. A node is considered safe if every path you can take from it eventually leads to a terminal node (a node with no outgoing edges).
Identify Terminal Nodes: The first step in your code will be to identify all terminal nodes. You can do this by traversing the graph and adding all nodes with no outgoing edges to a “safe” list.
Reverse Graph: Create a reversed graph where the edges point from destination to source instead of the other way round. This is helpful as it allows us to look “backward” from our terminal nodes towards nodes that can reach them.
Depth-First Search: From each terminal node, perform a depth-first search (DFS) on the reversed graph to identify all nodes that can reach terminal nodes. If a node is reached during this search, it means that starting from this node will eventually lead us to a terminal node, and therefore, it is safe. We can add this node to our “safe” list.
Sort Safe Nodes: Finally, you will return a sorted list of all safe nodes, as the problem requires the safe nodes to be sorted in ascending order.

In terms of complexity, this approach would take O(N + E) time and O(N) space, where N is the number of nodes and E is the number of edges, as each node and each edge is processed once.

The parameters that could change the solution include the number of nodes, the number of edges, and the direction of edges. An increase in the number of nodes or edges would increase the time it takes to find the solution.

For example, let’s take the example from the problem statement:

1
graph = [[1,2],[2,3],[5],[0],[5],[],[]]

First, we would identify that nodes 5 and 6 are terminal nodes, as they have no outgoing edges. We would then create the reversed graph and perform a depth-first search from these terminal nodes. We would find that we can reach nodes 2 and 4, in addition to nodes 5 and 6 themselves. Therefore, these nodes are safe, and the output would be [2,4,5,6].

Inference of Problem-Solving Approach from the Problem Statement

Here are some key terms and concepts that guide the problem-solving approach for this problem:

Directed Graph: A directed graph, or digraph, is a set of vertices and a collection of directed edges. Each directed edge has an initial vertex and a terminal vertex. In this problem, we’re given a directed graph, which informs us that we’ll likely need to use a graph traversal algorithm.
Node: A node is a basic unit of a data structure, such as a linked list or tree data structure. Nodes contain data and also may link to other nodes. In the context of this problem, nodes are the fundamental elements in our graph that we need to analyze.
Terminal Node: A terminal node is one that doesn’t have any outgoing edges. This concept is important in our problem, as we are trying to find “safe” nodes, which are nodes that can only lead to terminal nodes.
Edge: An edge is another fundamental part of a graph, representing a connection between two nodes. The direction of the edge matters in this problem because we have a directed graph.
Graph Traversal Algorithms: These are algorithms used to visit or check and/or update nodes (or vertices) of a graph. In this case, we’re using a depth-first search (DFS) algorithm as part of our solution to identify all nodes that can reach terminal nodes.
Depth-First Search (DFS): DFS is a method for exploring a tree or graph. In a DFS, you start from a root and go as far as possible along each branch before backtracking. It’s ideal for this problem because we need to find every possible path from a node to see if it leads to a terminal node.

These concepts inform our problem-solving approach in that we start by identifying terminal nodes, then use a DFS on a reversed graph from these terminal nodes to find all the “safe” nodes.

The use of Depth-First Search (DFS) can be inferred from the nature of the problem and the properties of DFS.

Directed Graphs: The problem involves dealing with a directed graph and finding specific paths within it. DFS is a standard way to traverse or search within graphs, allowing us to explore all the paths from a given node.
Safe Nodes and Terminal Nodes: The problem defines safe nodes as nodes from which every possible path leads to a terminal node (or another safe node). DFS is an ideal strategy to determine such paths because it allows us to visit each node exactly once and avoid infinite loops in potential cycles.
Path Exploration: DFS, by its nature, explores as far as possible along each branch before backtracking. This characteristic makes DFS well-suited to problems where we need to find or validate specific paths in a graph, as is the case here.

However, the problem could potentially be solved using other graph traversal methods too, depending on the implementation. DFS is not the only way, but it is an intuitive and common approach to such problems.

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

Stepwise Refinement:
Here’s a high-level view of our approach to solving this problem:
a. We begin by identifying the terminal nodes. These are nodes without any outgoing edges.
b. We start a Depth-First Search (DFS) from each of these terminal nodes.
c. We mark each node we visit during this DFS as “safe”, since all paths from these nodes lead to terminal nodes.
d. Finally, we return all the nodes that are marked as safe in increasing order.
Granular, Actionable Steps:
a. Initialize an empty list/set to keep track of safe nodes.
b. Iterate through the graph and for each node, if it does not have any outgoing edges, add it to the safe nodes list.
c. For each terminal node, start a DFS.
d. In the DFS, for each neighboring node that is not already marked as safe, recursively perform the DFS and mark the node as safe.
e. After the DFS, return the list of safe nodes in ascending order.
Parts That Can Be Solved Independently:
a. Identifying terminal nodes: We can independently traverse the graph once to identify all terminal nodes.
b. Marking safe nodes: For each terminal node, we can independently start a DFS to mark all nodes that lead to it as safe.
c. Sorting safe nodes: Once we have marked all safe nodes, sorting them can be done independently.
Repeatable Patterns within Our Solution:
a. DFS Traversal: The DFS process is a repeatable pattern that is performed on every terminal node.
b. Marking Safe Nodes: As we encounter each node during our DFS traversal, we repeat the process of marking it as safe.
c. Sorting: Once we have marked all the safe nodes, we can use any sorting algorithm to sort them, which is a repeatable pattern.

Solution Approach and Analysis

Let’s break down the problem-solving approach.

Identify terminal nodes: The first step is to identify all the terminal nodes. Terminal nodes are the nodes from which there are no outgoing edges. For example, in the graph [[1,2],[2,3],[5],[0],[5],[],[]], the terminal nodes are 5 and 6 because there are no outgoing edges from them. You can think of these terminal nodes as “safe houses” in a city from where every road leads out of the city.
Start Depth-First Search (DFS): Once we identify all terminal nodes, we start a DFS from each terminal node. DFS is a technique for traversing a tree or graph where we go as deep as possible along each branch before backtracking. Think of it like exploring a maze - you go down one path as far as you can, and when you hit a dead end, you backtrack to the previous junction and try another path.
Mark safe nodes: During DFS, we mark each node that we visit as “safe”. A safe node is defined as a node from where every possible path leads to a terminal node. It’s like marking all roads that lead to the “safe houses”. Any node we reach from a terminal node during DFS is marked as safe because we started our DFS from a terminal node and that guarantees that every path from these nodes will lead to a terminal node.
Return all safe nodes: After marking all the safe nodes, we return them in increasing order. The safe nodes are the nodes from where you can reach a “safe house” no matter which path you choose.

Let’s take an example to understand this process better. Consider the graph [[1,2],[2,3],[5],[0],[5],[],[]]. We start by identifying terminal nodes 5 and 6. Then, we start DFS from node 5 and mark nodes 2 and 4 as safe because there are paths from these nodes to node 5. Next, we start DFS from node 6 but there are no paths from any node to node 6. Thus, our list of safe nodes is [2,4,5,6]. We return this list as our answer.

The number of nodes and edges in the graph affects the solution’s complexity. As the number of nodes or edges increases, the DFS process can become computationally expensive. Hence, for very large graphs, it might be necessary to use optimization techniques or more efficient data structures to handle the problem within reasonable time limits.

Identify Invariant

An invariant is a condition that remains unchanged during the execution of an algorithm or a process.

In the context of this problem, the invariant is that once a node is marked as “safe”, it remains safe throughout the execution of the algorithm. This is because a node is marked as safe when it can be determined that all paths starting from this node eventually lead to a terminal node. Once this is established, it doesn’t change, regardless of the subsequent nodes and paths explored in the graph.

Therefore, the “safeness” of a node, once established, is the invariant in this problem.

Identify Loop Invariant

A loop invariant is a condition that is initially true and remains true after each iteration of a loop.

In the context of this problem, the algorithm iteratively processes the nodes in the graph. If we consider a loop that iterates over each node, the loop invariant can be the fact that at the beginning of each iteration, all nodes that have been processed so far and marked as safe indeed lead to a terminal node or another safe node, and those marked as unsafe form a cycle or lead to a cycle.

It’s important to note that identifying a loop invariant can depend on how you structure your specific algorithm to solve this problem. The invariant I mentioned fits a Depth-First Search approach where nodes are categorized as safe or unsafe. If a different method is used, the invariant might look different.

Thought Process

Here are the steps involved in solving this problem:

Analyze the Problem Statement: We’re given a directed graph and asked to identify safe nodes. Safe nodes are defined as nodes that only lead to terminal nodes (nodes with no outgoing edges) or other safe nodes. This is a graph traversal problem, and the fact that we’re considering paths from nodes suggests a depth-first search or breadth-first search approach.
Develop a High-Level Approach: We can use Depth-First Search (DFS) to solve this problem. Starting from each node, we traverse the graph. If we encounter a terminal node or a node already marked as safe, we know our current path only leads to safe nodes. If we encounter a node that we’re currently visiting, we know we’ve found a cycle, and therefore, our current path is not safe.
Translate the Approach into Steps:

a. Initialize an empty list to keep track of the state of each node. A node can be in one of three states: not visited, visited (in progress), and done (safe or unsafe).

b. Iterate over each node. If a node hasn’t been visited, call a DFS function on it.

c. In the DFS function, mark the current node as visited (in progress). Then, for each neighbor of the current node, if the neighbor hasn’t been visited, recursively call the DFS function on the neighbor. If the DFS function returns false, mark the current node as unsafe and return false.

d. If all neighbors have been processed and none returned false, mark the current node as safe and return true.

Translate the Steps into Code:

Let’s write this approach in Python:

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class Solution:
    def eventualSafeNodes(self, graph: List[List[int]]) -> List[int]:
        NOT_VISITED = 0
        VISITING = 1
        SAFE = 2

        n = len(graph)
        state = [NOT_VISITED] * n

        def dfs(node):
            if state[node] != NOT_VISITED:
                return state[node] == SAFE
            state[node] = VISITING
            for neighbor in graph[node]:
                if not dfs(neighbor):
                    return False
            state[node] = SAFE
            return True

        return [i for i in range(n) if dfs(i)]

This solution starts a DFS from each node. It marks nodes as ‘VISITING’ when it first visits them and ‘SAFE’ when it’s confirmed they only lead to safe nodes. If it encounters a node that’s currently ‘VISITING’ during DFS, it knows it has found a cycle and the path is not safe. Therefore, the DFS function returns false. The nodes are marked as safe only if all their neighbors are confirmed safe.

Please note that the graph traversal problems, especially those involving detection of cycles, often hint towards a depth-first or breadth-first search solution. The nature of the problem and sometimes the constraints can provide cues about the problem-solving approach. For example, here we’re looking at each path, and DFS is more suited to exploring paths in a graph. Furthermore, the categorization of nodes as safe or unsafe is a key insight that informs our approach to marking and traversing nodes.

Establishing Preconditions and Postconditions

Given the Python function eventualSafeNodes(self, graph: List[List[int]]) -> List[int]:

Parameters:
- The method takes in one input parameter: graph which is a list of lists of integers.
- This parameter represents a directed graph where the length of the list is the number of nodes and each sublist at index i represents the nodes that node i can reach.
Preconditions:
- The graph must be a valid representation of a directed graph with nodes labeled from 0 to n - 1.
- There is no explicit precondition for the state of the program.
- The input graph must follow these constraints:
  - n == graph.length
  - 1 <= n <= 104
  - 0 <= graph[i].length <= n
  - 0 <= graph[i][j] <= n - 1
  - The graph may contain self-loops.
  - The number of edges in the graph will be in the range [1, 4 * 104].
Method Functionality:
- The method is expected to identify all the safe nodes in the graph and return them in ascending order.
- A node is considered safe if every possible path starting from that node leads to a terminal node (a node with no outgoing edges) or another safe node.
Postconditions:
- The method returns a list of integers, sorted in ascending order, which represent the safe nodes in the input graph.
- The state of the program is not modified by the method as the graph is not being modified in place.
- The return value indicates the safe nodes in the graph based on the definition provided in the problem statement.
Error Handling:
- If the graph does not meet the constraints, Python will raise an error.
- If the graph is empty or null, the method would return an empty list since there are no nodes to process.
- The method does not have explicit error handling. This is typical for LeetCode problems as they generally assume well-formed inputs. In a real-world scenario, additional error checking might be necessary.

Problem Decomposition

Problem Understanding:
- In this problem, we are given a directed graph represented as a list of lists, where each node i is associated with a list of nodes to which it has directed edges. We need to identify the “safe” nodes, where a node is considered safe if every possible path starting from that node leads to a terminal node (a node with no outgoing edges) or another safe node. The output should be a list of these safe nodes in ascending order.
Initial Breakdown:
- The problem can be broadly divided into:
  - Identify terminal nodes in the graph.
  - Determine the safe nodes.
  - Sort the safe nodes and return the list.
Subproblem Refinement:
- Terminal Nodes Identification: This involves iterating over the graph and checking the nodes with no outgoing edges.
- Safe Nodes Determination: This involves traversing the graph from each node and checking if we eventually reach a terminal node or a previously determined safe node. This can be done using Depth-First Search (DFS).
- Sorting Safe Nodes: Once we have a list of safe nodes, sort them in ascending order.
Task Identification:
- The repeated task here is the traversal of nodes in the graph which can be generalized and implemented using DFS.
Task Abstraction:
- We abstract the task of DFS traversal as it forms the backbone of our solution. It’s abstract enough to make it clear and reusable, and still makes sense in the context of our problem as it helps in determining safe nodes.
Method Naming:
- identify_terminal_nodes for identifying terminal nodes.
- depth_first_search for the DFS traversal.
- find_safe_nodes for determining safe nodes using DFS.
Subproblem Interactions:
- Initially, terminal nodes are identified. Using this information, we proceed to find the safe nodes through DFS traversal starting from each node. The order of these tasks is important as we need the terminal nodes information to determine the safe nodes. Finally, we sort the determined safe nodes. The sorting of safe nodes is independent and can be done once we have the list of safe nodes.

From Brute Force to Optimal Solution

A brute-force approach to this problem would involve checking every single possible path from every node in the graph and seeing if they lead to a terminal node or not. Here are the steps:

Identify the terminal nodes (i.e., nodes with no outgoing edges).
For each node, perform a Depth-First Search (DFS) traversal to find all possible paths.
For each path, check if it ends in a terminal node. If it does, mark the starting node as safe.
Repeat this process for every node in the graph.
Return the sorted list of safe nodes.

Inefficiencies: The major issue with this brute force approach is that it does a lot of redundant work. In the worst-case scenario, for each node, we are performing a complete DFS traversal. If the same node is reached via different paths, the same traversal is repeated multiple times, leading to high time complexity.

The time complexity of this approach is O(n^2) where n is the number of nodes, as in the worst case, we might need to traverse through all nodes for each node.

Optimization: The key observation to optimize the solution is that if a node leads to a safe node, it is safe itself. So, instead of finding all paths, we can stop the search as soon as we find a safe node. We can use memoization (a technique of storing computed values to reuse later) to remember the safe nodes we’ve already found.

Start the DFS traversal from each node. If we encounter a node that we have already determined to be safe or terminal, we can stop further traversal and consider the current node to be safe as well.
Store the safe nodes in a data structure such as an array or a set.
After determining whether a node is safe or not, save this information to avoid repeating the same computation in the future. This is the memoization step.
Once we’ve checked all nodes, return the sorted list of safe nodes.

Impact on Time and Space Complexity: With this optimization, our time complexity is reduced to O(V + E), where V is the number of vertices and E is the number of edges in the graph, because each node and edge is processed only once. The space complexity is O(n) for the memoization and stack space in the DFS traversal.

This optimized solution is much more efficient than the brute force approach because it eliminates redundant work by storing and reusing previously computed results.

Code Explanation and Design Decisions

Given the problem at hand, let’s discuss these elements in the context of the optimized solution that uses Depth-First Search (DFS) and memoization.

Initial Parameters: The initial parameters to our function would be the adjacency list of the graph and the number of nodes. The adjacency list represents the graph where we need to identify the terminal nodes. It is significant because it contains all the information about which nodes are connected to each other and is the primary structure we will be traversing to find the solution.
Primary Loop or Iteration: The primary loop iterates over each node in the graph. Each iteration represents the DFS traversal starting from the current node. It is within this iteration that we explore all possible paths from the current node to the terminal nodes.
Conditions or Branches: Within the DFS traversal, we have a condition to check whether the current node is safe or not. If it’s safe (either a terminal node or leads to a terminal node), we stop further traversal from this node and mark it as safe. This branching is vital to avoid unnecessary traversals and respects the constraints of the problem by correctly identifying safe nodes.
Updates or Modifications: Each time we identify a safe node, we mark it as safe in our memoization structure. This reflects a change in the state of our solution: as we traverse the graph, our knowledge of which nodes are safe increases. This update helps us to avoid redundant work in subsequent iterations.
Invariant: An invariant in this context could be that once a node is marked as safe, it remains safe for the rest of the algorithm execution. This invariant is helpful as it allows us to stop further unnecessary DFS traversals whenever we encounter a safe node, making the solution more efficient.
Final Output: The final output is a list of all the safe nodes, sorted in ascending order. This list represents the nodes from which we can always reach a terminal node regardless of the path we choose. The output satisfies the problem’s requirements by correctly identifying all the safe nodes based on the problem’s constraints.

Coding Constructs

High-Level Problem-Solving Strategies: The solution primarily uses the Depth-First Search (DFS) strategy for traversing the graph. It also leverages memoization, a form of caching, to store and retrieve previously computed results to avoid redundant computation. This combination makes it a dynamic programming approach.
Purpose of the Code to a Non-Programmer: This program is like a guide that helps you find your way out of a maze. Imagine you are in a large maze (a network of interconnected points), and there are certain points from which you can exit the maze. The guide (our code) helps you find those points from which you can definitely reach an exit point, no matter which path you choose.
Logical Elements: The code mainly uses iteration, conditional branches, and recursion. Iteration is used to traverse each node in the graph, conditional branches are used to determine whether a node is safe or not, and recursion is used to perform DFS and explore all paths from a node.
Algorithmic Approach in Plain English: Start from each point in the network. From there, try to reach an exit point by exploring different paths, always moving forward, never backward. If you can reach an exit point, mark the starting point as safe. Remember these safe points. If you encounter a point that you already know is safe, there’s no need to explore from this point again. Continue this process until you’ve started from all points in the network. In the end, you’ll have a list of all points that are safe, i.e., from which you can reach an exit.
Key Steps or Operations: The main steps this code performs are:
- Initialize an empty list and a memoization structure.
- Loop through each node in the graph.
- For each node, perform a DFS to check whether we can reach a terminal node. During the DFS, if we find a safe node (already explored), we stop further traversal and return true.
- If the DFS returns true, we add the node to our list of safe nodes and mark it in our memoization structure.
Algorithmic Patterns: The main algorithmic pattern used in the code is Depth-First Search (DFS) for graph traversal. It is combined with dynamic programming, specifically top-down dynamic programming (memoization), to remember the results of previous computations and avoid re-computing them. This pattern is prevalent in problems where the same sub-problem needs to be solved multiple times, like in this case where we could encounter the same node during multiple DFS traversals.

Language Agnostic Coding Drills

Dissecting the Code into Distinct Concepts:
- Variable Initialization and Assignment: This is the basic concept of creating and assigning values to variables.
- Arrays/List Manipulation: Creating and manipulating arrays or lists are fundamental in managing a collection of elements.
- Loops (for loop): Iterating over elements of a list or an array.
- Recursion: This involves breaking a problem into smaller subproblems and solving them recursively.
- Conditional Statements (if-else): Branching logic based on certain conditions.
- Graph Traversal (DFS): Depth-first search is an algorithm for traversing or searching tree or graph data structures.
- Memoization: An optimization technique used primarily to speed up computer programs by storing the results of expensive function calls and reusing them when the same inputs occur again.
Concepts/Drills in Order of Increasing Difficulty:
- Variable Initialization and Assignment: This is the simplest concept. It involves assigning a value to a variable.
- Arrays/List Manipulation: A bit more complex, as one needs to know how to create, access, and manipulate data in a list or an array.
- Loops (for loop): Requires understanding of iteration and control structures.
- Conditional Statements (if-else): These introduce decision-making capabilities to the code and require a basic understanding of Boolean logic.
- Recursion: Recursion is a more advanced concept requiring a good understanding of function calls and stack memory.
- Graph Traversal (DFS): Understanding graph traversal requires knowledge of data structures, specifically trees/graphs, and can be recursive or iterative.
- Memoization: This is a more advanced technique used to optimize code performance, requiring a good understanding of data structures and algorithms to implement effectively.
Problem-Solving Approach:
- Understanding the Problem: Comprehend the problem statement and the requirements, identify the key components and the goal.
- Variable Initialization and Assignment: Create the necessary variables and data structures (like arrays/lists) that will be used to store and manipulate the data.
- Looping through Nodes: Use a for loop to iterate through all the nodes in the graph.
- Implementing DFS: For each node, perform a depth-first search to explore the paths from that node. This requires understanding recursion, as DFS is usually implemented recursively.
- Conditional Branching in DFS: During the DFS, use if-else conditions to decide whether to continue exploring a path or return the result.
- Memoization: Store the results of DFS explorations in a data structure to avoid repeated explorations of the same node.
- Formulating the Final Result: After all nodes have been explored, return the list of nodes that are safe.

Targeted Drills in Python

Variable Initialization and Assignment:

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a = 5  # Initialize a variable 'a' with the value 5
b = a + 2  # Create a variable 'b' and assign it the value of 'a' plus 2

Arrays/List Manipulation:

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arr = [1, 2, 3, 4, 5]  # Initialize an array 'arr' with five elements
arr.append(6)  # Add the value 6 to the end of the array
first_element = arr[0]  # Access the first element of the array

Loops (for loop):

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for i in range(5):  # Iterate over the numbers 0 through 4
    print(i)  # Print the current number

Conditional Statements (if-else):

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a = 5
if a > 3:  # If 'a' is greater than 3
    print("Greater")  # Print "Greater"
else:
    print("Not Greater")  # Print "Not Greater"

Recursion:

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def factorial(n):  # Define a function to calculate the factorial of a number
    if n == 0:  # Base case: if n is 0, its factorial is 1
        return 1
    else:
        return n * factorial(n-1)  # Recursive case: n times the factorial of n-1

Graph Traversal (DFS):

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graph = {1: [2, 3], 2: [4], 3: [], 4: []}  # A dictionary representing a graph

def dfs(node):
    print(node)
    for neighbor in graph[node]:
        dfs(neighbor)

dfs(1)

Memoization:

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memo = {}  # Dictionary to store previous results

def fib(n):
    if n in memo:
        return memo[n]
    if n <= 2:
        result = 1
    else:
        result = fib(n-1) + fib(n-2)
    memo[n] = result
    return result

print(fib(10))

Problem-specific concept:

Graph Initialization and Safe Node Identification: For the specific problem, we need to initialize a graph based on the given data and identify which nodes are “safe”. Here, a safe node means a node that eventually does not point to an unsafe node. The “drill” for this is more about understanding the problem and defining what a safe node means rather than a specific Python coding concept.

Once all drills have been coded and understood, we can integrate them into the final solution. The key is to perform depth-first search (DFS) for each node in the graph and use memoization to avoid repeated computation. During the DFS, we identify whether a node is safe or unsafe based on its neighbors. If any neighbor is unsafe or leads to an unsafe node, the current node is also unsafe. These results are stored in the memoization array, and at the end, we return all the safe nodes.