1600. Throne Inheritance

Description

A kingdom consists of a king, his children, his grandchildren, and so on. Every once in a while, someone in the family dies or a child is born.

The kingdom has a well-defined order of inheritance that consists of the king as the first member. Let's define the recursive function Successor(x, curOrder), which given a person x and the inheritance order so far, returns who should be the next person after x in the order of inheritance.

Successor(x, curOrder):
    if x has no children or all of x's children are in curOrder:
        if x is the king return null
        else return Successor(x's parent, curOrder)
    else return x's oldest child who's not in curOrder

For example, assume we have a kingdom that consists of the king, his children Alice and Bob (Alice is older than Bob), and finally Alice's son Jack.

1. In the beginning, curOrder will be ["king"].
2. Calling Successor(king, curOrder) will return Alice, so we append to curOrder to get ["king", "Alice"].
3. Calling Successor(Alice, curOrder) will return Jack, so we append to curOrder to get ["king", "Alice", "Jack"].
4. Calling Successor(Jack, curOrder) will return Bob, so we append to curOrder to get ["king", "Alice", "Jack", "Bob"].
5. Calling Successor(Bob, curOrder) will return null. Thus the order of inheritance will be ["king", "Alice", "Jack", "Bob"].

Using the above function, we can always obtain a unique order of inheritance.

Implement the ThroneInheritance class:

• ThroneInheritance(string kingName) Initializes an object of the ThroneInheritance class. The name of the king is given as part of the constructor.
• void birth(string parentName, string childName) Indicates that parentName gave birth to childName.
• void death(string name) Indicates the death of name. The death of the person doesn't affect the Successor function nor the current inheritance order. You can treat it as just marking the person as dead.
• string[] getInheritanceOrder() Returns a list representing the current order of inheritance excluding dead people.

 

Example 1:

Input
["ThroneInheritance", "birth", "birth", "birth", "birth", "birth", "birth", "getInheritanceOrder", "death", "getInheritanceOrder"]
[["king"], ["king", "andy"], ["king", "bob"], ["king", "catherine"], ["andy", "matthew"], ["bob", "alex"], ["bob", "asha"], [null], ["bob"], [null]]
Output
[null, null, null, null, null, null, null, ["king", "andy", "matthew", "bob", "alex", "asha", "catherine"], null, ["king", "andy", "matthew", "alex", "asha", "catherine"]]

Explanation
ThroneInheritance t= new ThroneInheritance("king"); // order: king
t.birth("king", "andy"); // order: king > andy
t.birth("king", "bob"); // order: king > andy > bob
t.birth("king", "catherine"); // order: king > andy > bob > catherine
t.birth("andy", "matthew"); // order: king > andy > matthew > bob > catherine
t.birth("bob", "alex"); // order: king > andy > matthew > bob > alex > catherine
t.birth("bob", "asha"); // order: king > andy > matthew > bob > alex > asha > catherine
t.getInheritanceOrder(); // return ["king", "andy", "matthew", "bob", "alex", "asha", "catherine"]
t.death("bob"); // order: king > andy > matthew > bob > alex > asha > catherine
t.getInheritanceOrder(); // return ["king", "andy", "matthew", "alex", "asha", "catherine"]

 

Constraints:

• 1 <= kingName.length, parentName.length, childName.length, name.length <= 15
• kingName, parentName, childName, and name consist of lowercase English letters only.
• All arguments childName and kingName are distinct.
• All name arguments of death will be passed to either the constructor or as childName to birth first.
• For each call to birth(parentName, childName), it is guaranteed that parentName is alive.
• At most 105 calls will be made to birth and death.
• At most 10 calls will be made to getInheritanceOrder.

Solutions

Solution 1: Preorder Traversal of a Multi-branch Tree

According to the problem description, we can find that the order of throne inheritance is actually a preorder traversal of a multi-branch tree. We can use a hash table $g$ to store the children of each person, and a set $dead$ to store the people who have died.

• When calling birth(parentName, childName), we add childName to the child list of parentName.
• When calling death(name), we add name to the dead set.
• When calling getInheritanceOrder(), we start a depth-first search from the king. If the current node x is not dead, we add x to the answer list, and then recursively traverse all children of x.

In terms of time complexity, both birth and death have a time complexity of $O(1)$, and getInheritanceOrder has a time complexity of $O(n)$. The space complexity is $O(n)$, where $n$ is the number of nodes.

Python3JavaC++GoTypeScriptC#

class ThroneInheritance:

    def __init__(self, kingName: str):
        self.king = kingName
        self.dead = set()
        self.g = defaultdict(list)

    def birth(self, parentName: str, childName: str) -> None:
        self.g[parentName].append(childName)

    def death(self, name: str) -> None:
        self.dead.add(name)

    def getInheritanceOrder(self) -> List[str]:
        def dfs(x: str):
            x not in self.dead and ans.append(x)
            for y in self.g[x]:
                dfs(y)

        ans = []
        dfs(self.king)
        return ans


# Your ThroneInheritance object will be instantiated and called as such:
# obj = ThroneInheritance(kingName)
# obj.birth(parentName,childName)
# obj.death(name)
# param_3 = obj.getInheritanceOrder()

class ThroneInheritance {
    private String king;
    private Set<String> dead = new HashSet<>();
    private Map<String, List<String>> g = new HashMap<>();
    private List<String> ans = new ArrayList<>();

    public ThroneInheritance(String kingName) {
        king = kingName;
    }

    public void birth(String parentName, String childName) {
        g.computeIfAbsent(parentName, k -> new ArrayList<>()).add(childName);
    }

    public void death(String name) {
        dead.add(name);
    }

    public List<String> getInheritanceOrder() {
        ans.clear();
        dfs(king);
        return ans;
    }

    private void dfs(String x) {
        if (!dead.contains(x)) {
            ans.add(x);
        }
        for (String y : g.getOrDefault(x, List.of())) {
            dfs(y);
        }
    }
}

/**
 * Your ThroneInheritance object will be instantiated and called as such:
 * ThroneInheritance obj = new ThroneInheritance(kingName);
 * obj.birth(parentName,childName);
 * obj.death(name);
 * List<String> param_3 = obj.getInheritanceOrder();
 */

class ThroneInheritance {
public:
    ThroneInheritance(string kingName) {
        king = kingName;
    }

    void birth(string parentName, string childName) {
        g[parentName].emplace_back(childName);
    }

    void death(string name) {
        dead.insert(name);
    }

    vector<string> getInheritanceOrder() {
        ans.resize(0);
        dfs(king);
        return ans;
    }

private:
    string king;
    unordered_set<string> dead;
    unordered_map<string, vector<string>> g;
    vector<string> ans;

    void dfs(string& x) {
        if (!dead.contains(x)) {
            ans.emplace_back(x);
        }
        for (auto& y : g[x]) {
            dfs(y);
        }
    }
};

/**
 * Your ThroneInheritance object will be instantiated and called as such:
 * ThroneInheritance* obj = new ThroneInheritance(kingName);
 * obj->birth(parentName,childName);
 * obj->death(name);
 * vector<string> param_3 = obj->getInheritanceOrder();
 */

type ThroneInheritance struct {
    king string
    dead map[string]bool
    g map[string][]string
}


func Constructor(kingName string) ThroneInheritance {
    return ThroneInheritance{kingName, map[string]bool{}, map[string][]string{}}
}


func (this *ThroneInheritance) Birth(parentName string, childName string)  {
    this.g[parentName] = append(this.g[parentName], childName)
}


func (this *ThroneInheritance) Death(name string)  {
    this.dead[name] = true
}


func (this *ThroneInheritance) GetInheritanceOrder() (ans []string) {
    var dfs func(string)
    dfs = func(x string) {
        if !this.dead[x] {
            ans = append(ans, x)
        }
        for _, y := range this.g[x] {
            dfs(y)
        }
    }
    dfs(this.king)
    return
}


/**
 * Your ThroneInheritance object will be instantiated and called as such:
 * obj := Constructor(kingName);
 * obj.Birth(parentName,childName);
 * obj.Death(name);
 * param_3 := obj.GetInheritanceOrder();
 */

class ThroneInheritance {
    private king: string;
    private dead: Set<string> = new Set();
    private g: Map<string, string[]> = new Map();

    constructor(kingName: string) {
        this.king = kingName;
    }

    birth(parentName: string, childName: string): void {
        this.g.set(parentName, this.g.get(parentName) || []);
        this.g.get(parentName)!.push(childName);
    }

    death(name: string): void {
        this.dead.add(name);
    }

    getInheritanceOrder(): string[] {
        const ans: string[] = [];
        const dfs = (x: string) => {
            if (!this.dead.has(x)) {
                ans.push(x);
            }
            for (const y of this.g.get(x) || []) {
                dfs(y);
            }
        };
        dfs(this.king);
        return ans;
    }
}

/**
 * Your ThroneInheritance object will be instantiated and called as such:
 * var obj = new ThroneInheritance(kingName)
 * obj.birth(parentName,childName)
 * obj.death(name)
 * var param_3 = obj.getInheritanceOrder()
 */

public class ThroneInheritance {
    private string king;
    private HashSet<string> dead = new HashSet<string>();
    private Dictionary<string, List<string>> g = new Dictionary<string, List<string>>();
    private List<string> ans = new List<string>();

    public ThroneInheritance(string kingName) {
        king = kingName;
    }

    public void Birth(string parentName, string childName) {
        if (!g.ContainsKey(parentName)) {
            g[parentName] = new List<string>();
        }
        g[parentName].Add(childName);
    }

    public void Death(string name) {
        dead.Add(name);
    }

    public IList<string> GetInheritanceOrder() {
        ans.Clear();
        DFS(king);
        return ans;
    }

    private void DFS(string x) {
        if (!dead.Contains(x)) {
            ans.Add(x);
        }
        if (g.ContainsKey(x)) {
            foreach (string y in g[x]) {
                DFS(y);
            }
        }
    }
}

/**
 * Your ThroneInheritance object will be instantiated and called as such:
 * ThroneInheritance obj = new ThroneInheritance(kingName);
 * obj.Birth(parentName,childName);
 * obj.Death(name);
 * IList<string> param_3 = obj.GetInheritanceOrder();
 */

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