All Ancestors of a Node in a Directed Acyclic Graph

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class Solution:
    def getAncestors(self, n: int, edges: List[List[int]]) -> List[List[int]]:
        res = [[] for _ in range(n)]
        al = [[] for _ in range(n)]

        def dfs(i, anc):
            for j in al[i]:
                if not res[j] or res[j][-1] != anc:
                    res[j].append(anc)
                    dfs(j, anc)

        for e in edges:
            al[e[0]].append(e[1])

        for i in range(n):
            dfs(i, i)

        return res

Identifying Problem Isomorphism

“All Ancestors of a Node in a Directed Acyclic Graph” requires traversing a graph from a specified node, discovering and listing all of its ancestors.

A simpler problem with a similar strategy is “Number of Connected Components in an Undirected Graph”. In this problem, you are also required to traverse a graph, but instead of tracking ancestors, the goal is to identify the number of distinct connected components.

A more complex problem is “Course Schedule II”. Here, you are not just asked to traverse a graph, but you must also return a possible order of courses (nodes) to take while adhering to the prerequisite constraints (directed edges). This involves identifying all predecessors (ancestors) of a node before it can be “taken”, similar to the task of identifying ancestors in the original problem.

The reasoning behind this selection:

“Number of Connected Components in an Undirected Graph” forms a basis for the traversal of a graph and identification of distinct components or node groupings.
“All Ancestors of a Node in a Directed Acyclic Graph” builds on this by not only requiring a traversal, but also identification of ancestors for a specific node.
“Course Schedule II” adds further complexity by not only requiring identification of ancestors (predecessors in this case), but also forming a valid ordering of nodes based on these relationships.

So, in increasing order of complexity:

“Number of Connected Components in an Undirected Graph”
“All Ancestors of a Node in a Directed Acyclic Graph”
“Course Schedule II”

This mapping is approximate. The specific coding techniques required to solve each problem may differ, but there are similarities in the underlying concepts of graph traversal and understanding node relationships.

10 Prerequisite LeetCode Problems

For “2192. All Ancestors of a Node in a Directed Acyclic Graph”, the following are a good preparation:

“207. Course Schedule” - This problem introduces the concept of directed acyclic graphs (DAG) and helps to understand the basics of graph traversal techniques.
“210. Course Schedule II” - This problem is an extension of the above, where one has to return the order of courses. This will be helpful to understand how to keep track of the order.
“785. Is Graph Bipartite?” - This problem helps to understand how to classify or divide the nodes of a graph based on certain conditions.
“684. Redundant Connection” - This problem introduces the concept of identifying and removing certain edges based on the structure of the graph.
“399. Evaluate Division” - In this problem, a graph is constructed from a list of equations and evaluated. The construction of a graph from a list will be helpful.
“743. Network Delay Time” - This problem is about the shortest path in a directed graph which can help you understand how to traverse through a directed graph.
“133. Clone Graph” - This problem introduces the concept of cloning or creating a copy of a graph which will help to understand the structure of the graph.
“332. Reconstruct Itinerary” - This problem involves finding a path in a directed graph which can help to understand how to find paths in a directed graph.
“261. Graph Valid Tree” - This problem involves determining if a given graph is a valid tree. Understanding this can help in comprehending the tree-like structure within a graph.
“886. Possible Bipartition” - This problem involves dividing the nodes of a graph into two distinct groups, a useful technique for handling nodes.

These cover graph traversal or operations on graphs which are core to understanding and solving problem “2192. All Ancestors of a Node in a Directed Acyclic Graph”.

Problem Classification

This is a graph theory problem involving finding ancestors in a directed acyclic graph (DAG).

The key ‘What’ components are:

Directed Acyclic Graph (DAG)
Nodes numbered 0 to n-1
Edges represented as node pairs
Find ancestors for each node
Ancestors are nodes reachable via edges
Return sorted list of ancestors for each node

Based on this, I would categorize this problem as:

Domain: Graph theory Type: Graph traversal Sub-type: Ancestor search in DAG

This is because the core task involves traversing a DAG to find all reachable nodes for each node, which represent its ancestors. The constraints that it is a directed acyclic graph and the nodes are numbered 0 to n-1 do not change the core algorithm - these just characterize properties of the graph input.

The key graph traversal aspect categorizes this as a graph algorithm problem. Specifically ancestor search in a DAG is a common graph theory problem with well known solutions. So I would classify this as an ancestor DAG search problem in the domain of graph theory.

Clarification Questions

Here are some clarification questions that could be asked about this ancestor search in DAG problem:

On input:

Are the edges represented as an adjacency list or adjacency matrix? This could affect algorithm efficiency.
Can we assume the input DAG is valid, with no cycles or invalid edges?
Can there be multiple edges between the same pair of nodes?
Is the number of nodes n guaranteed to be <= 1000?

On output:

Should each node’s ancestor list include the node itself, or only strict ancestors?
If a node has no ancestors, should its list be empty or contain just itself?
Should the ancestor list for node 0 always be empty since it has no ancestors?
For a disconnected graph, how should singleton nodes with no edges be handled?

On performance:

What are the time and space complexity requirements?
Can we assume the DAG fits in memory or need to account for large graphs?
Is preprocessing of the graph allowed to improve later query speed?

These questions would help clarify ambiguities in the problem definition, understand performance constraints, and identify potential edge cases to handle. Getting these details early can prevent incorrect assumptions and lead to a more robust solution.

Problem Analysis and Key Insights

Here are some key insights from analyzing this ancestor search in DAG problem statement:

Topological ordering of nodes can be leveraged since it is a DAG. Traversing nodes in topological order will ensure we process ancestors before descendants.
Recursive depth-first search can find ancestors for a single node by recording visited nodes.
Processing nodes in topological order allows propagating ancestor information forward.
Can build up adjacency list representing graph edges instead of matrix to save space.
Can use a map like ancestorMap to store ancestor list for each node.
Need to handle case where input node has no ancestors gracefully.
Output ancestor lists must be sorted.
Deduplication needed if multiple paths to an ancestor.
Can improve efficiency by preprocessing DAG into intermediate representation.
Time complexity should be O(V+E) for DFS on graph with V vertices and E edges.

Key insights are using topological order for propagation, DFS for ancestor search, intermediate storage for ancestor lists, and handling edge cases like no ancestors. The problem constraints suggest optimization of space complexity is important on large graphs.

Problem Boundary

Based on the problem statement, here is how I would define the scope of this ancestor search in DAG problem:

Input Scope:

Directed acyclic graph with n nodes numbered 0 to n-1
Edges specified as node pairs representing direction
Graph size n up to 1000 nodes
Number of edges up to n*(n-1)/2
Guaranteed no duplicate edges or cycles

Processing Scope:

Find ancestors for each node
Ancestor defined as reachable via edges
Allowed to preprocess graph into efficient representation
Need to handle nodes with no ancestors

Output Scope:

Return ancestor list for every node 0 to n-1
Lists should be sorted
Contains only ancestors (no self-reference)
Empty list for nodes with no ancestors

Performance Scope:

Aim for efficient algorithm with low time complexity
Optimize for space complexity due to large graph size
Preprocessing time/space can be excluded from analysis

The scope focuses on ancestry relationships in the input DAG. It does not require analyzing edge weights, distances, or other properties. The output is ancestry lists for all nodes. Efficiency is valued over completeness. These boundaries help frame an optimal solution.

Here are some ways to establish the boundaries and scope for this ancestor search in DAG problem:

Input Boundaries:

Graph is directed and acyclic - algorithms can leverage topological ordering
Nodes numbered 0 to n-1 - implies contiguous integer node IDs
Edges as node pairs - represents graph connectivity
Max nodes 1000 - can fit graph in memory
Max edges n*(n-1)/2 - sparse graph

Output Boundaries:

Ancestor list for each node - well-defined output format
Sorted lists - adds constraint on output format
Empty list if no ancestors - need to handle this case

Performance Boundaries:

Unspecified time complexity - aim for most efficient algorithm
Large input size - optimize for space complexity
Preprocessing allowed - can trade space for speed

Outside Boundaries:

Only ancestry - do not analyze edge weights or node properties
No cycle checking needed - input DAG guaranteed valid
No querying of non-ancestor relationships
Ancestor list completeness not required

The key is focusing on topology and ancestry on a valid DAG. Graph properties like weights, costs, and non-ancestor relationships are outside the scope. Preprocessing space can be traded for speed. Reasonable assumptions can be made about input validity. Together these boundaries help frame the solution space.

Distilling the Problem to Its Core Elements

The fundamental concept this ancestry search in DAG problem is based on is graph traversal and topological ordering. At its core, it involves traversing a directed acyclic graph in topological order to find ancestors for each node.

The simplest way I would describe this problem is:

“Given a one-way hierarchy represented by pairs of nodes, find all ancestors for each node in the hierarchy.”

The core problem is to find all nodes reachable from a given node via the directed edges. We can simplify it to this ancestry/reachability determination based on the input graph structure.

The key components of this problem are:

Modeling the input graph
Determining topological order of nodes
Traversing graph to find ancestors
Storing ancestry lists
Handling special cases like no ancestors

The minimal set of operations we need to perform are:

Build graph from input edges
Topologically sort nodes
For each node, depth-first search to find ancestors
Record ancestor list for each node
Return final ancestor lists

By breaking the problem down into modeling the graph, topological ordering, graph traversal, and recording results, we can focus on the core techniques needed while simplifying away unnecessary details.

Visual Model of the Problem

Here is one way to visually represent the ancestry search in DAG problem statement:

The problem involves a directed acyclic graph (DAG) with n nodes numbered 0 to n-1. The edges represent a one-way relationship from parent to child node.

For example:

        (0)
       /   \
      /     \
     /       \
   (1)      (2)
    |        | \ 
    |        |  \
   (3)      (4) (7)
    | \      |
   (5)(6)  (6)

The goal is to find all ancestor nodes for each node in the graph. For example:

Ancestors(0) = [] Ancestors(1) = [0] Ancestors(2) = [0] Ancestors(3) = [0, 1] Ancestors(4) = [0, 2] Ancestors(5) = [0, 1, 3] Ancestors(6) = [0, 1, 2, 3, 4] Ancestors(7) = [0, 2]

We can visualize this as a tree-like hierarchy of nodes, with edges flowing upwards from children to parents. The output is the set of nodes reachable from a given node by following incoming edges upwards through ancestors.

This diagram captures the key aspects of the DAG structure and ancestry relationship. Visualizing the graph topology and example output helps build an insight for the problem.

Problem Restatement

Here is how I would paraphrase the ancestry search in DAG problem statement in my own words:

We are given a directed acyclic graph (DAG) consisting of n nodes numbered from 0 to n-1. The graph edges are specified as pairs of nodes representing a one-way relationship from parent to child node.

Our goal is to find the ancestor list for each node in the graph. The ancestors of a node are defined as all other nodes that can reach that node by following incoming edges back through the graph.

The input guarantees the graph is a valid DAG with no cycles or duplicate edges. The number of nodes n is at most 1000.

For each node, we need to output the list of its ancestors sorted in ascending order. If a node has no ancestors, we should return an empty list.

The key requirements seem to be:

Model the input DAG efficiently
Perform ancestry search for each node
Handle cases like no ancestors gracefully
Return sorted ancestry lists

Optimization goals are to minimize time and space complexity as the graph can be quite large. Some preprocessing of the graph representation is allowed to improve performance.

Does this capture the essence of the problem succinctly and accurately? Let me know if I am missing or misunderstanding any key details. I want to ensure I have aligned my interpretation with the actual problem statement before moving forward.

Abstract Representation of the Problem

Here is one way to formulate an abstract representation of the ancestry search in DAG problem:

We can model the input as a directed graph G = (V, E) where:

V is the set of vertices representing the nodes
E is the set of directed edges representing the parent-child relationships

We can assign each node v ∈ V a unique integer label from 0 to n-1 to match the node numbering in the problem.

The goal is to find the transitive closure of the ancestor relation R, where uRv if there is a path from vertex u to v following the directed edges in E.

For each vertex v, we want to find the set of all vertices u such that uRv. This ancestry set for each vertex forms the output.

Additional constraints:

The graph G must be a directed acyclic graph to ensure R is non-circular.
The output ancestry sets must be sorted by vertex label.
Vertices with empty ancestry sets must be handled properly.

By abstracting away the specifics into vertices, edges, ancestry relation, and constraints on the graph and output, we can focus on the key structures and operations needed. The vertex identifiers help map the abstraction back to the original problem. This representation emphasizes the dependencies and relationships without distraction from the problem domain specifics.

Terminology

Here are some key technical concepts relevant to this ancestry search in DAG problem:

Directed Acyclic Graph (DAG): A graph with directed edges and no cycles. The edges point from parent to child node forming a hierarchical structure. DAGs allow topological ordering of vertices.

Topological Ordering: An ordering of the vertices in a DAG such that for every edge u -> v, u comes before v in the ordering. Allows propagating information from ancestors to descendants.

Depth-First Search (DFS): A graph traversal algorithm that explores as far as possible down each branch before backtracking. Can be used to find connected components and determine reachability.

Transitive Closure: The full set ofreachability relationships between all pairs of vertices implied by the graph edges. Finding transitive closure allows determining all ancestors.

Adjacency List: A graph representation that stores for each vertex the list of adjacent vertices it connects to. More space efficient than adjacency matrix for sparse graphs.

These concepts play key roles in modeling the DAG input, traversing the graph, determining ancestry relationships, and representing the graph efficiently. Understanding terms like topological order, DFS, transitive closure, and adjacency list will be crucial for solving this problem optimally.

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Provide names by categorizing these cases

What are the edge cases?

How to visualize these cases?

What are the key insights from analyzing the different cases?

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Simple Explanation

Can you explain this problem in simple terms or like you would explain to a non-technical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method. How can I recognize these properties by drawing tables or diagrams?

How did you infer from the problem statement that this problem can be solved using ?

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?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Identify Invariant

What is the invariant in this problem?

Identify Loop Invariant

What is the loop invariant in this problem?

Is invariant and loop invariant the same for this problem?

Identify Recursion Invariant

Is there an invariant during recursion in this problem?

Is invariant and invariant during recursion the same for this problem?

Thought Process

Can you explain the basic thought process and steps involved in solving this type of problem?

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

Establishing Preconditions and Postconditions

Parameters:
- What are the inputs to the method?
- What types are these parameters?
- What do these parameters represent in the context of the problem?
Preconditions:
- Before this method is called, what must be true about the state of the program or the values of the parameters?
- Are there any constraints on the input parameters?
- Is there a specific state that the program or some part of it must be in?
Method Functionality:
- What is this method expected to do?
- How does it interact with the inputs and the current state of the program?
Postconditions:
- After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
- What does the return value represent or indicate?
- What side effects, if any, does the method have?
Error Handling:
- How does the method respond if the preconditions are not met?
- Does it throw an exception, return a special value, or do something else?

Problem Decomposition

Problem Understanding:
- Can you explain the problem in your own words? What are the key components and requirements?
Initial Breakdown:
- Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
Subproblem Refinement:
- For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?
Task Identification:
- Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
Task Abstraction:
- For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?
Method Naming:
- Can you give each task a simple, descriptive name that makes its purpose clear?
Subproblem Interactions:
- How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Code Explanation and Design Decisions

Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

Coding Constructs

Consider the code for the solution of this problem.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.