Interval Tree using GNU Tree-based container

Consider a situation where we have a set of intervals and we need the following operations to be implemented efficiently: 

1. Add an interval
2. Remove an interval
3. Given an interval x, find if x overlaps with any of the existing intervals.

An Interval Tree can be implemented as an augmented binary-search tree (preferably self-balancing), and thus enable us to perform the required operations in O(logN) time complexity. 

Each node of the tree will store the following information: 

1. An interval i: represented as a pair [low, high].
2. Metadata max of right-endpoints: The maximum of the right-endpoints of all the intervals stored in the subtree rooted at this node. Storing this metadata is how we are augmenting the tree.

Example Interval Tree used in Interval Tree | Set 1: 

In Interval Tree | Set 1, we saw how to implement an interval tree using a simple BST (which is not self-balancing). In this article, we will use the built-in GNU tree-based container to implement an Interval Tree. The benefits of doing so are: 

• We do not have to code our own tree data structure.
• We get the default operations like insert and delete out-of-the-box.
• We get to use the in-built Red-Black tree implementation, which means our tree will be self-balancing.

We will be using GNU policy-based implementation of tree data structure. 

The article Policy-based data structures in g++ introduce GNU policy-based data structure along with the required header files. 

We will define our own Node_update policy such that we can maintain the maximum of right-endpoints of intervals in the subtree as metadata in the nodes of our tree. 

The syntax for defining a custom Node_policy is: 

C++

#include <iostream> // Include the iostream header for cout and endl
#include <functional> // For std::function

// Define a custom node update policy struct
template <
    typename Const_Node_Iterator,  // Constant node iterator type
    typename Node_Iterator,        // Node iterator type
    typename Cmp_Fn_,              // Comparator function type
    typename Allocator_>           // Allocator type
struct custom_node_update_policy {
    // Define a nested struct to represent metadata type
    struct metadata_type {
        // Define members of metadata type here
    };

    // Define a method similar to the apply() method in Java
    void apply(Node_Iterator it,
               Const_Node_Iterator endIt)
    {
        // Implementation of the method
        // Note: C++ doesn't have function objects like Java
    }

    // Other methods can be added here as needed
};

using std::cout; // Import cout into the global namespace
using std::endl; // Import endl into the global namespace

int main() {
    // Your program logic goes here
    // You can create instances of
    // custom_node_update_policy and call its methods
    // For example:
    custom_node_update_policy<int, int,
                              std::function<int(int)>,
                              void*> policy;
    policy.apply(10, 20); // Example call to apply method

    // Output to demonstrate that the program is running
    cout << "Program executed successfully." << endl;

    return 0;
}

Java

import java.util.function.Function;

public class Program {
    // Define a generic class with type parameters
    public static class CustomNodeUpdatePolicy<
        ConstNodeIterator, NodeIterator, CmpFn, Allocator> {
        // Define a nested class to represent metadata type
        public class MetadataType {
            // Define members of metadata type here
        }

        // Define a method similar to the operator() in C++
        public void apply(NodeIterator it,
                          ConstNodeIterator endIt)
        {
            // Implementation of the method
            // Note: Java doesn't have operator overloading
            // like C++
        }

        // Other methods can be added here as needed
    }

    public static void main(String[] args)
    {
        // Your program logic goes here
        // You can create instances of
        // CustomNodeUpdatePolicy and call its methods For
        // example:
        CustomNodeUpdatePolicy<Integer, Integer,
                               Function<Integer, Integer>,
                               Object> policy
            = new CustomNodeUpdatePolicy<>();
        policy.apply(10,
                     20); // Example call to apply method

        // Output to demonstrate that the program is running
        System.out.println(
            "Program executed successfully.");
    }
}

C#

using System;

public class Program
{
    // Define a generic class with type parameters
    public class CustomNodeUpdatePolicy<ConstNodeIterator, NodeIterator, CmpFn, Allocator>
    {
        // Define a nested class to represent metadata type
        public class MetadataType
        {
            // Define members of metadata type here
        }

        // Define a method similar to the operator() in C++
        public void Apply(NodeIterator it, ConstNodeIterator endIt)
        {
            // Implementation of the method
            // Note: C# doesn't have operator overloading like C++
        }

        // Other methods can be added here as needed
    }

    public static void Main(string[] args)
    {
        // Your program logic goes here
        // You can create instances of CustomNodeUpdatePolicy and call its methods
        // For example:
        CustomNodeUpdatePolicy<int, int, Func<int, int, int>, object> policy = new CustomNodeUpdatePolicy<int, int, Func<int, int, int>, object>();
        policy.Apply(10, 20); // Example call to Apply method

        // Output to demonstrate that the program is running
        Console.WriteLine("Program executed successfully.");
    }
}

Javascript

// Define a generic class with type parameters
class CustomNodeUpdatePolicy {
    constructor(ConstNodeIterator, NodeIterator, CmpFn, Allocator) {
        this.ConstNodeIterator = ConstNodeIterator;
        this.NodeIterator = NodeIterator;
        this.CmpFn = CmpFn;
        this.Allocator = Allocator;
    }

    // Define a nested class to represent metadata type
    MetadataType() {
        // Define members of metadata type here
    }

    // Define a method similar to the operator() in Python
    apply(it, endIt) {
        // Implementation of the method
        // Note: JavaScript doesn't have operator overloading like Python
    }

    // Other methods can be added here as needed
}

// Your program logic goes here
// You can create instances of CustomNodeUpdatePolicy and call its methods
// For example:
const policy = new CustomNodeUpdatePolicy(Number, Number, (x, y) => x + y, Object);
policy.apply(10, 20);  // Example call to Apply method

// Output to demonstrate that the program is running
console.log("Program executed successfully.");

Python3

# Define a generic class with type parameters
class CustomNodeUpdatePolicy:
    def __init__(self, ConstNodeIterator, NodeIterator, CmpFn, Allocator):
        self.ConstNodeIterator = ConstNodeIterator
        self.NodeIterator = NodeIterator
        self.CmpFn = CmpFn
        self.Allocator = Allocator

    # Define a nested class to represent metadata type
    class MetadataType:
        pass
        # Define members of metadata type here

    # Define a method similar to the operator() in C++
    def apply(self, it, endIt):
        pass
        # Implementation of the method
        # Note: Python doesn't have operator overloading like C++

    # Other methods can be added here as needed

# Your program logic goes here
# You can create instances of CustomNodeUpdatePolicy and call its methods
# For example:
policy = CustomNodeUpdatePolicy(int, int, lambda x, y: x + y, object)
policy.apply(10, 20)  # Example call to Apply method

# Output to demonstrate that the program is running
print("Program executed successfully.")

Output

Program executed successfully.

• type_of_our_metadata: int in our case since we want to store the metadata “max of right-endpoints of intervals in the subtree”.
• void operator()(node_iterator it, const_node_iterator end_it): method which is called internally to restore node-invariance, i.e. maintaining correct metadata, after invariance has been invalidated.
• it: node_iterator to the node whose invariance we need to restore.
• end_it: const_node_iterator to a just-after-leaf node.

See GNU tree-based containers for more details.

We will also define a method overlapSearch which searches for any interval in the tree overlapping with a given interval i.

// pseudocode for overlapSearch
Interval overlapSearch(Interval i) {
    // start from root
    it = root_node
    while (it not null) {
        if (doOVerlap(i, it->interval)) {
              // overlap found
               return it->interval
        }
        if (left_child exists
                     AND
             left_child->max_right_endpoint
                   >= it->left_endpoint) {
            // go to left child
            it = it->left_child
        }
        else {
            // go to right child
            it = it->right_child
        }
    }
    // no overlapping interval found
    return NO_INTERVAL_FOUND
}

Below is the implementation of the Interval Tree:

C++

// CPP program for above approach
#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>

using namespace std;
using namespace __gnu_pbds;

typedef pair<int, int> Interval;

// An invalid interval, used as
// return value to denote that no
// matching interval was found
const Interval NO_INTERVAL_FOUND = { 1, 0 };

// interval update policy struct
template <class Node_CItr,
          class Node_Itr,
          class Cmp_Fn, class _Alloc>
struct interval_node_update_policy {

    // Our metadata is maximum of
    // right-endpoints of intervals in the
    // sub-tree, which is of type int
    typedef int metadata_type;

    // An utility function to check
    // if given two intervals overlap
    bool doOverlap(Interval i1,
                   Node_CItr i2)
    {
        return (i1.first <= (*i2)->second
                && (*i2)->first <= i1.second);
    }

    // Search for any interval that
    // overlaps with Interval i
    Interval overlapSearch(Interval i)
    {
        for (Node_CItr it = node_begin();
             it != node_end();) {
            if (doOverlap(i, it)) {
                return { (*it)->first,
                         (*it)->second };
            }

            if (it.get_l_child() != node_end()
                && it.get_l_child()
                           .get_metadata()
                       >= i.first) {
                it = it.get_l_child();
            }
            else {
                it = it.get_r_child();
            }
        }
        return NO_INTERVAL_FOUND;
    }

    // To restore the node-invariance
    // of the node pointed to by
    // (it). We need to derive the
    // metadata for node (it) from
    // its left-child and right-child.
    void operator()(Node_Itr it,
                    Node_CItr end_it)
    {
        int max_high = (*it)->second;

        if (it.get_l_child() != end_it) {
            max_high = max(
                max_high,
                it.get_l_child()
                    .get_metadata());
        }

        if (it.get_r_child() != end_it) {
            max_high = max(
                max_high,
                it.get_r_child()
                    .get_metadata());
        }

        // The max of right-endpoint
        // of this node and the max
        // right-endpoints of children.
        const_cast<int&>(
            it.get_metadata())
            = max_high;
    }

    virtual Node_CItr node_begin() const = 0;
    virtual Node_CItr node_end() const = 0;
    virtual ~interval_node_update_policy() {}
};

// IntervalTree data structure
// rb_tree_tag: uses red-black search tree
// interval_node_update_policy:
// our custom Node_update policy
typedef tree<Interval,
             null_type,
             less<Interval>,
             rb_tree_tag,
             interval_node_update_policy>
    IntervalTree;

// Driver Code
int main()
{
    IntervalTree IT;
    Interval intvs[] = { { 15, 20 },
                         { 10, 30 },
                         { 17, 19 },
                         { 5, 20 },
                         { 12, 15 },
                         { 30, 40 } };

    for (Interval intv : intvs) {
        IT.insert(intv);
    }

    Interval toSearch = { 25, 29 };
    cout << "\nSearching for interval ["
         << toSearch.first << ", "
         << toSearch.second << "]";
    Interval res = IT.overlapSearch(toSearch);
    if (res == NO_INTERVAL_FOUND)
        cout << "\nNo Overlapping Interval\n";
    else
        cout << "\nOverlaps with ["
             << res.first << ", "
             << res.second << "]\n";

    Interval toErase = { 10, 30 };
    IT.erase(toErase);
    cout << "\nDeleting interval ["
         << toErase.first << ", "
         << toErase.second
         << "]\n";

    cout << "\nSearching for interval ["
         << toSearch.first << ", "
         << toSearch.second << "]";
    res = IT.overlapSearch(toSearch);
    if (res == NO_INTERVAL_FOUND)
        cout << "\nNo Overlapping Interval\n";
    else
        cout << "\nOverlaps with ["
             << res.first << ", "
             << res.second << "]\n";
    return 0;
}

Java

import java.util.Comparator;
import java.util.TreeSet;

// Class to represent intervals
class Interval {
    int first, second;

    public Interval(int first, int second) {
        this.first = first;
        this.second = second;
    }
}

// Class representing the Interval Tree
class IntervalTree {
    TreeSet<Interval> treeSet;

    // Constructor for IntervalTree class
    public IntervalTree() {
        // Use a TreeSet with a custom comparator based on the second endpoint of intervals
        treeSet = new TreeSet<>(Comparator.comparingInt(interval -> interval.second));
    }

    // Method to insert an interval into the tree
    public void insert(Interval interval) {
        treeSet.add(interval);
    }

    // Method to search for any interval that overlaps with the given interval
    public Interval overlapSearch(Interval interval) {
        for (Interval i : treeSet) {
            if (doOverlap(interval, i)) {
                return new Interval(i.first, i.second);
            }
        }
        return NO_INTERVAL_FOUND;
    }

    // Method to remove an interval from the tree
    public void erase(Interval interval) {
        treeSet.remove(interval);
    }

    // Helper method to check if two intervals overlap
    private boolean doOverlap(Interval i1, Interval i2) {
        return (i1.first <= i2.second && i2.first <= i1.second);
    }

    // Declare NO_INTERVAL_FOUND as static or public
    public static final Interval NO_INTERVAL_FOUND = new Interval(1, 0);
}

// Main class to test the IntervalTree
public class Main {
    public static void main(String[] args) {
        IntervalTree IT = new IntervalTree();
        Interval[] intvs = {
                new Interval(15, 20),
                new Interval(10, 30),
                new Interval(17, 19),
                new Interval(5, 20),
                new Interval(12, 15),
                new Interval(30, 40)
        };

        // Insert intervals into the tree
        for (Interval intv : intvs) {
            IT.insert(intv);
        }

        // Search for overlapping interval
        Interval toSearch = new Interval(25, 29);
        System.out.println("Searching for interval [" + toSearch.first + ", " + toSearch.second + "]");
        Interval res = IT.overlapSearch(toSearch);
        if (res == IntervalTree.NO_INTERVAL_FOUND)
            System.out.println("No Overlapping Interval");
        else
            System.out.println("Overlaps with [" + res.first + ", " + res.second + "]");

        // Remove an interval from the tree
        Interval toErase = new Interval(10, 30);
        IT.erase(toErase);
        System.out.println("Deleting interval [" + toErase.first + ", " + toErase.second + "]");

        // Search for overlapping interval after deletion
        System.out.println("Searching for interval [" + toSearch.first + ", " + toSearch.second + "]");
        res = IT.overlapSearch(toSearch);
        if (res == IntervalTree.NO_INTERVAL_FOUND)
            System.out.println("No Overlapping Interval");
        else
            System.out.println("Overlaps with [" + res.first + ", " + res.second + "]");
    }
}

C#

using System;
using System.Collections.Generic;

// Class representing an interval
class Interval
{
    public int Start { get; set; }
    public int End { get; set; }

    public Interval(int start, int end)
    {
        Start = start;
        End = end;
    }
}

// Class representing an Interval Tree
class IntervalTree
{
    private readonly SortedSet<Interval> intervals;

    public IntervalTree()
    {
        intervals = new SortedSet<Interval>(Comparer<Interval>.Create((a, b) => a.Start.CompareTo(b.Start)));
    }

    // Insert an interval into the Interval Tree
    public void Insert(Interval interval)
    {
        intervals.Add(interval);
    }

    // Search for any interval that overlaps with the given query interval
    public Interval OverlapSearch(Interval queryInterval)
    {
        foreach (var interval in intervals)
        {
            if (DoOverlap(queryInterval, interval))
            {
                return interval;
            }
        }
        return null;
    }

    // Remove an interval from the Interval Tree
    public void Erase(Interval interval)
    {
        intervals.Remove(interval);
    }

    // Check if two intervals overlap
    private bool DoOverlap(Interval i1, Interval i2)
    {
        return (i1.Start <= i2.End && i2.Start <= i1.End);
    }
}

class Program
{
    static void Main()
    {
        IntervalTree intervalTree = new IntervalTree();
        Interval[] intervals = {
            new Interval(15, 20),
            new Interval(10, 30),
            new Interval(17, 19),
            new Interval(5, 20),
            new Interval(12, 15),
            new Interval(30, 40)
        };

        // Insert intervals into the Interval Tree
        foreach (var interval in intervals)
        {
            intervalTree.Insert(interval);
        }

        Interval toSearch = new Interval(25, 29);
        Console.Write($"\nSearching for interval [{toSearch.Start}, {toSearch.End}]: ");
        Interval result = intervalTree.OverlapSearch(toSearch);
        if (result == null)
            Console.WriteLine("No Overlapping Interval");
        else
            Console.WriteLine($"Overlaps with [{result.Start}, {result.End}]");

        Interval toErase = new Interval(10, 30);
        intervalTree.Erase(toErase);
        Console.Write($"\nDeleting interval [{toErase.Start}, {toErase.End}]: ");

        Console.Write($"\nSearching for interval [{toSearch.Start}, {toSearch.End}]: ");
        result = intervalTree.OverlapSearch(toSearch);
        if (result == null)
            Console.WriteLine("No Overlapping Interval");
        else
            Console.WriteLine($"Overlaps with [{result.Start}, {result.End}]");
    }
}

Javascript

// JavaScript Program for the above approach
// Define a Node class representing a node in the interval tree
class Node {
    constructor(interval) {
        this.interval = interval;
        this.max = interval[1];
        this.left = null;
        this.right = null;
    }
}

// Define an IntervalTree class to manage the interval tree operations
class IntervalTree {
    constructor() {
        this.root = null;
    }

    // insert a new interval into the interval tree
    insert(root, interval) {
        if (!root) {
            return new Node(interval);
        }
        if (interval[0] < root.interval[0]) {
            root.left = this.insert(root.left, interval);
        } else {
            root.right = this.insert(root.right, interval);
        }
        root.max = Math.max(root.max, interval[1]);
        return root;
    }

    // check for interval operlap in the interval tree
    overlap(root, interval) {
        if (!root) {
            return null;
        }
        if (root.interval[0] <= interval[1] && root.interval[1] >= interval[0]) {
            return root.interval;
        }
        if (root.left && root.left.max >= interval[0]) {
            return this.overlap(root.left, interval);
        }
        return this.overlap(root.right, interval);
    }

    // delete an interval from the interval tree
    delete(root, interval) {
        if (!root) {
            return root;
        }
        if (interval[0] < root.interval[0]) {
            root.left = this.delete(root.left, interval);
        } else if (interval[0] > root.interval[0]) {
            root.right = this.delete(root.right, interval);
        } else {
            if (!root.left) {
                let temp = root.right;
                root = null;
                return temp;
            } else if (!root.right) {
                let temp = root.left;
                root = null;
                return temp;
            }
            let temp = this.minValueNode(root.right);
            root.interval = temp.interval;
            root.right = this.delete(root.right, temp.interval);
        }
        return root;
    }

    // helper function to find the node with the minimum value in subtree
    minValueNode(root) {
        let current = root;
        while (current.left !== null) {
            current = current.left;
        }
        return current;
    }
}

// Example usage of the IntervalTree
const intervals = [[15, 20], [10, 30], [17, 19], [5, 20], [12, 15], [30, 40]];
const IT = new IntervalTree();

for (const interval of intervals) {
    IT.root = IT.insert(IT.root, interval);
}

// Search for an interval and print the result
const to_search = [25, 29];
console.log(`Searching for interval ${JSON.stringify(to_search)}`);
let res = IT.overlap(IT.root, to_search);
if (res) {
    console.log(`Overlaps with ${JSON.stringify(res)}`);
} else {
    console.log("No Overlapping Interval");
}

// Delete an interval and print the result
const to_erase = [10, 30];
console.log(`Deleting interval ${JSON.stringify(to_erase)}`);
IT.root = IT.delete(IT.root, to_erase);

// Search for an interval again after deletion and print the result
console.log(`Searching for interval ${JSON.stringify(to_search)}`);
res = IT.overlap(IT.root, to_search);
if (res) {
    console.log(`Overlaps with ${JSON.stringify(resultAfterDeletion)}`);
} else {
    console.log("No Overlapping Interval");
}
// This code is contributed by Yash Agarwal(yashagarwal2852002)

Python3

# Define a Node class representing a node in the interval tree
class Node:
    def __init__(self, interval):
        self.interval = interval
        self.max = interval[1]
        self.left = None
        self.right = None

# Define an IntervalTree class to manage the interval tree operations
class IntervalTree:
    def __init__(self):
        self.root = None

    # Insert a new interval into the interval tree
    def insert(self, root, interval):
        if not root:
            return Node(interval)
        if interval[0] < root.interval[0]:
            root.left = self.insert(root.left, interval)
        else:
            root.right = self.insert(root.right, interval)
        root.max = max(root.max, interval[1])
        return root

    # Check for interval overlap in the interval tree
    def overlap(self, root, interval):
        if not root:
            return None
        if (root.interval[0] <= interval[1] and root.interval[1] >= interval[0]):
            return root.interval
        if root.left and root.left.max >= interval[0]:
            return self.overlap(root.left, interval)
        return self.overlap(root.right, interval)

    # Delete an interval from the interval tree
    def delete(self, root, interval):
        if not root:
            return root
        if interval[0] < root.interval[0]:
            root.left = self.delete(root.left, interval)
        elif interval[0] > root.interval[0]:
            root.right = self.delete(root.right, interval)
        else:
            if root.left is None:
                temp = root.right
                root = None
                return temp
            elif root.right is None:
                temp = root.left
                root = None
                return temp
            temp = self.minValueNode(root.right)
            root.interval = temp.interval
            root.right = self.delete(root.right, temp.interval)
        return root

    # Helper function to find the node with the minimum value in a subtree
    def minValueNode(self, root):
        current = root
        while(current.left is not None):
            current = current.left
        return current

# Example usage of the IntervalTree
intervals = [(15, 20), (10, 30), (17, 19), (5, 20), (12, 15), (30, 40)]
IT = IntervalTree()
for interval in intervals:
    IT.root = IT.insert(IT.root, interval)

# Search for an interval and print the result
to_search = (25, 29)
print(f"Searching for interval {to_search}")
res = IT.overlap(IT.root, to_search)
if res:
    print(f"Overlaps with {res}")
else:
    print("No Overlapping Interval")

# Delete an interval and print the result
to_erase = (10, 30)
print(f"Deleting interval {to_erase}")
IT.root = IT.delete(IT.root, to_erase)

# Search for an interval again after deletion and print the result
print(f"Searching for interval {to_search}")
res = IT.overlap(IT.root, to_search)
if res:
    print(f"Overlaps with {res}")
else:
    print("No Overlapping Interval")

Output

Searching for interval [25, 29]
Overlaps with [10, 30]

Deleting interval [10, 30]

Searching for interval [25, 29]
No Overlapping Interval

Time Complexity : O(log N), All operations are logarithmic in size, i.e. O(logN), where N is the number of intervals stored in the tree. 
Auxiliary Space: O(N)

We are able to achieve logarithmic worst-case complexity because a red-black tree is used internally which is self-balancing.