Problem Description

In the realm of a kingdom, we have a continuous line of succession from the king down to the various members of the royal family. This order of inheritance begins with the king and is followed by his children, grandchildren, and so on, in order of their ages. The problem involves managing this order of inheritance, taking into account the birth of new royal members and the death of existing ones. The challenge is to maintain a dynamic list of the succession line, which can change when a new child is born or when someone dies.

Flowchart Walkthrough

To determine the suitable algorithm for LeetCode 1600. Throne Inheritance using the Flowchart, let's go through the process step by step:

Is it a graph?

• Yes: The problem involves a family tree where each person can be considered a node, and the edges represent parental relationships.

Is it a tree?

• Yes: The family tree is a specific type of graph, fundamentally a tree because it has hierarchical relationships without cycles.

Conclusion: The flowchart leads to using DFS for this problem, as it involves traversing a tree to maintain and update the order of succession according to a depth-first traversal pattern.

Intuition

The primary approach to solving this problem involves simulating the family tree structure with appropriate data structures and then conducting depth-first search (DFS) traversals to generate the current order of inheritance.

Let's break down the solution into the following parts:

1. Data Structure: We need to efficiently represent the family tree to allow for the addition of new children (via births) and to keep track of deaths. We use a graph represented by a dictionary (self.g) where every person is a key, and their children form a list (the values).

2. Births: When a child is born, we add the child to the graph by appending the child's name to the parent's list in the dictionary. This operation corresponds to adding edges to the family tree graph.

3. Deaths: We keep track of deaths in a set (self.dead). We don't remove the person from the graph because their children still need to be in the inheritance order.

4. Order of Inheritance: To find the current inheritance order, we use DFS starting from the king. In this traversal, we add a person to the current order only if they are not marked as dead. Through DFS, the children of each person are visited in the order they were added, maintaining the age-based priority.

5. Exclusion of Dead People: Since we want the order to contain only living members, we check the set of dead people during DFS and only include those who are not in this set.

By following this approach, we can dynamically adjust the inheritance order as births and deaths are recorded, thus maintaining an up-to-date succession line.

Learn more about Tree and Depth-First Search patterns.

Solution Approach

The solution to the inheritance problem is built around several key concepts in computer science, namely data structures, depth-first search, and class design.

Let's dive into each of these points:

1. Data Structures: We create three main attributes within the ThroneInheritance class:

   • A graph self.g, represented as a dictionary, which captures the parent-child relationships.
   • A set self.dead to keep track of the deceased family members.
   • A variable self.king to store the name of the king, which is the root of our graph.

2. Class Methods: Three methods alter the state of the inheritance line:

   • birth receives a parent name and a child name. The child's name is appended to the parent's list of children in the graph.
   • death marks a family member as deceased by adding their name to the self.dead set.
   • getInheritanceOrder invokes a DFS traversal method to calculate the current inheritance order.

3. Depth-First Search (DFS): We perform a DFS starting from the king every time getInheritanceOrder is called. Here's the breakdown of the DFS method used:

   • A helper function dfs is defined within getInheritanceOrder for recursive traversal.
   • Each time the DFS visits a person, it checks if they are alive (not in self.dead). If so, the person is added to the ans list, which represents the current inheritance order.
   • The DFS continues recursively through the children (in the order they were added to self.g), ensuring the age-based succession is preserved.

This implementation allows us to efficiently manage the order of inheritance dynamically as births add to the self.g graph and deaths are recorded in self.dead. The getInheritanceOrder function can be called at any time to provide a snapshot of the living members in the line of succession, excluding those that have passed away.

Example Walkthrough

Consider a kingdom with the following initial succession line: King (George), his children: oldest (John), and middle (Liam), and youngest (Claire).

In this structure, we have:

• King George at the root.
• John, Liam, and Claire as the children of George, listed in the order of their ages.

Let's set up the ThroneInheritance at this point:

1. self.g = { "George": ["John", "Liam", "Claire"] }
2. self.dead = set()
3. self.king = "George"

Now let's walk through a series of events and how the data structures would change:

Event 1: John has a child named William.

• We execute the birth method for John and William.
• self.g becomes { "George": ["John", "Liam", "Claire"], "John": ["William"] }.

Event 2: Claire gives birth to a child named Emma.

• We call the birth method for Claire and Emma.
• self.g is updated to { "George": ["John", "Liam", "Claire"], "John": ["William"], "Claire": ["Emma"] }.

Event 3: Liam passes away.

• death method is called for Liam; so Liam is added to self.dead.
• self.dead becomes { "Liam" }.
• Note that Liam is not removed from self.g because it will affect the children's structure (if he had any).

Current Inheritance Order: At this time, if we call getInheritanceOrder, the DFS would start from George, move to John (as he is the oldest child), and then to William (child of John), skip Liam because he is in the self.dead set, and continue with Claire, and then Emma.

• DFS order: George -> John -> William -> Liam (skipped, he's deceased) -> Claire -> Emma
• The ans list would be ["George", "John", "William", "Claire", "Emma"].

This way, we can see how the dynamic list of succession maintains the correct order throughout the births and deaths within the royal family, representing the living members in the correct order of precedence at all times.

Solution Implementation

Time and Space Complexity

__init__ method

The __init__ method has a time complexity of O(1) as it simply initializes the data structures: a graph (self.g as a defaultdict of lists), a set (self.dead), and assigns the self.king variable . The space complexity of the __init__ method is also O(1) assuming the input kingName size does not contribute to the space complexity as it's a single value.

birth method

The birth method has a time complexity of O(1) as it just appends the child to the parent's children list in the graph. The space complexity is O(1) as well because it just adds one more element to the existing data structure.

death method

The death method has a time complexity of O(1) as it adds a name to the self.dead set. The space complexity is O(1) due to a single insertion into the set.

getInheritanceOrder method

The getInheritanceOrder method has a time complexity of O(N) where N is the total number of people in the inheritance order (both alive and dead). This is because it performs a depth-first search (DFS) over the entire family tree, visiting each person once. The space complexity of the getInheritanceOrder method can be considered to be O(N) for the DFS recursion stack in the worst case (assuming the family tree is heavily unbalanced and resembles a linked list), plus O(N) for the ans list where N is the number of alive people. So, the total space complexity is O(N).

Overall, the main performance concern in the code is related to how the DFS might behave on a very unbalanced family tree — in that case, the recursion could be a problem, and iterative approaches might need to be considered to keep the stack depth under control.

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