1. Introduction

An R-Tree, short for Rectangle Tree, is a tree data structure used for storing spatial data indices in an efficient manner. An R-Tree is particularly well-suited for systems that need to handle large amounts of multi-dimensional information, such as spatial databases, computer graphics, and geographic information systems. The key idea is to group nearby objects and represent them with a minimum bounding rectangle (MBR).

2. Program Steps

1. Define a Rectangle class to represent an MBR.

2. Define the RTree class with an Entry inner class.

3. Implement methods insert, search, and overlaps.

4. Test the RTree with sample rectangles.

3. Code Program

class RTree {
    static final int MAX_ENTRIES = 3;

    static class Rectangle {
        int x1, y1, x2, y2;

        public Rectangle(int x1, int y1, int x2, int y2) {
            this.x1 = x1;
            this.y1 = y1;
            this.x2 = x2;
            this.y2 = y2;
        }

        public boolean overlaps(Rectangle other) {
            return this.x1 <= other.x2 && this.x2 >= other.x1 &&
                   this.y1 <= other.y2 && this.y2 >= other.y1;
        }
    }

    static class Entry {
        Rectangle rect;
        Entry[] children;
        int childCount;

        public Entry(Rectangle rect) {
            this.rect = rect;
            this.children = new Entry[MAX_ENTRIES];
        }
    }

    Entry root;

    public RTree() {
        root = new Entry(null);
    }

    public void insert(Rectangle rect) {
        insert(root, new Entry(rect));
    }

    private void insert(Entry parent, Entry entry) {
        if(parent.childCount < MAX_ENTRIES) {
            parent.children[parent.childCount++] = entry;
            return;
        }
        // In a complete R-Tree implementation, a split logic would be here.
    }

    public boolean search(Rectangle rect) {
        return search(root, rect);
    }

    private boolean search(Entry entry, Rectangle rect) {
        if(entry.rect != null && entry.rect.overlaps(rect)) {
            return true;
        }
        for(int i = 0; i < entry.childCount; i++) {
            if(search(entry.children[i], rect)) {
                return true;
            }
        }
        return false;
    }

    public static void main(String[] args) {
        RTree tree = new RTree();
        tree.insert(new Rectangle(1, 1, 5, 5));
        tree.insert(new Rectangle(6, 6, 10, 10));

        System.out.println(tree.search(new Rectangle(4, 4, 6, 6))); // true
        System.out.println(tree.search(new Rectangle(5, 5, 7, 7))); // false
    }
}

Output:

true
false

4. Step By Step Explanation

1. The Rectangle class represents the Minimum Bounding Rectangles (MBR). It contains a method called overlaps which checks if two rectangles overlap.

2. RTree class contains the core logic. Each Entry in the R-Tree holds a rectangle and has a set of child entries, up to MAX_ENTRIES.

3. The insert method adds a rectangle to the R-Tree. This simplified version doesn't handle splitting when an entry exceeds MAX_ENTRIES.

4. The search method checks if any rectangle in the R-Tree overlaps with the given rectangle.

5. In the main method, a sample R-Tree is created, and populated with sample rectangles. Then, two search operations are performed.

This example provides a basic introduction to the concept of R-Trees. A complete R-Tree would need to handle more advanced operations, such as node splitting when the number of entries in a node exceeds the maximum limit.