What is the time complexity of Construct Quad Tree?

The optimal solution has a time complexity of O(n^2).

What is the space complexity of Construct Quad Tree?

The space complexity is O(n^2).

What data structures are used to solve Construct Quad Tree?

This problem uses Array, Divide and Conquer, Tree, Matrix. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Construct Quad Tree?

Insight 1: The problem lends itself perfectly to a divide-and-conquer (recursion) approach because the Quad-Tree structure naturally mirrors the division of the grid into quadrants. Insight 2: Determining whether a sub-grid contains all the same values is crucial for identifying leaf nodes, which halts the recursion for that branch.

Which companies ask LeetCode 427. Construct Quad Tree in interviews?

This problem is frequently asked at Palantir Technologies.

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Construct Quad Tree - LeetCode 427 Solution

Construct Quad Tree - Complete Solution Guide

Construct Quad Tree is LeetCode problem 427, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

Given a n * n matrix grid of 0's and 1's only. We want to represent grid with a Quad-Tree. Return the root of the Quad-Tree representing grid . A Quad-Tree is a tree data structure in which each internal node has exactly four children. Besides, each node has two attributes: val : True if the node represents a grid of 1's or False if the node represents a grid of 0's. Notice that you can assign the val to True or False when isLeaf is False, and both are accepted in the answer. isLeaf : True if th

Detailed Explanation

The problem requires constructing a Quad-Tree from a given n x n grid of 0s and 1s. A Quad-Tree is a tree where each internal node has exactly four children, representing four quadrants of the grid. A node is a leaf node if the corresponding grid section contains only 0s or only 1s. In that case, the 'val' attribute of the leaf node is set to the value of the grid (0 or 1), and 'isLeaf' is set to true. If the grid section contains both 0s and 1s, the node is an internal node; 'isLeaf' is set to false, the 'val' attribute can be either 0 or 1 (it doesn't matter for internal nodes), and the grid is divided into four equal sub-grids, for which four child Quad-Trees are recursively constructed (topLeft, topRight, bottomLeft, bottomRight). The input is a 2D array (matrix) of integers representing the grid. The output is the root of the constructed Quad-Tree.

Solution Approach

The solution uses a recursive Depth-First Search (DFS) approach to construct the Quad-Tree. The core idea is to repeatedly divide the grid into four quadrants until a quadrant contains only a single value (all 0s or all 1s). The `dfs` function takes the starting row, starting column, and size of the current sub-grid as input. It first checks if all the elements in the current sub-grid are the same. If they are, it creates a leaf node with the corresponding value and sets `isLeaf` to true. Otherwise, it divides the sub-grid into four equal quadrants, recursively calls itself to build the Quad-Trees for each quadrant, and then creates an internal node with these four subtrees as its children.

Step-by-Step Algorithm

Step 1: Define a recursive function `dfs(r, c, size)` that takes the starting row `r`, starting column `c`, and the size of the sub-grid `size` as input.
Step 2: Check if all elements in the current sub-grid (from `grid[r][c]` to `grid[r+size-1][c+size-1]`) are the same. This involves iterating through all elements in the sub-grid.
Step 3: If all elements are the same, create a new `Node` with `isLeaf = true` and `val` equal to the value of the elements in the sub-grid (0 or 1). Return the created node.
Step 4: If the elements are not all the same, divide the sub-grid into four equal quadrants, each with size `size/2`.
Step 5: Recursively call `dfs` for each of the four quadrants: top-left (r, c, size/2), top-right (r, c + size/2, size/2), bottom-left (r + size/2, c, size/2), and bottom-right (r + size/2, c + size/2, size/2).
Step 6: Create a new `Node` with `isLeaf = false`, and set its `topLeft`, `topRight`, `bottomLeft`, and `bottomRight` children to the nodes returned by the recursive calls. The `val` field of the internal node can be any value (typically true or false). Return this internal node.
Step 7: Call the `dfs` function initially with `r = 0`, `c = 0`, and `size = len(grid)` to start the Quad-Tree construction from the entire grid.

Key Insights

Insight 1: The problem lends itself perfectly to a divide-and-conquer (recursion) approach because the Quad-Tree structure naturally mirrors the division of the grid into quadrants.
Insight 2: Determining whether a sub-grid contains all the same values is crucial for identifying leaf nodes, which halts the recursion for that branch.
Insight 3: The choice of 'val' for internal nodes is arbitrary as long as 'isLeaf' is false. It's more important to correctly build the subtrees.
Insight 4: Understanding the base case for recursion is key - when a subgrid contains only one value.

Complexity Analysis

Time Complexity: O(n^2)

Space Complexity: O(n^2)

Topics

This problem involves: Array, Divide and Conquer, Tree, Matrix.

Companies

Asked at: Palantir Technologies.

Construct Quad Tree | LeetCode 427

Medium

Array Divide and Conquer Tree Matrix

Time: O(n^2)Space: O(n^2)

Solution Code

Available in:

/*
// Definition for a QuadTree node.
class Node {
    public boolean val;
    public boolean isLeaf;
    public Node topLeft;
    public Node topRight;
    public Node bottomLeft;
    public Node bottomRight;

    
    public Node() {
        this.val = false;
        this.isLeaf = false;
        this.topLeft = null;
        this.topRight = null;
        this.bottomLeft = null;
        this.bottomRight = null;
    }
    
    public Node(boolean val, boolean isLeaf) {
        this.val = val;
        this.isLeaf = isLeaf;
        this.topLeft = null;
        this.topRight = null;
        this.bottomLeft = null;
        this.bottomRight = null;
    }
    
    public Node(boolean val, boolean isLeaf, Node topLeft, Node topRight, Node bottomLeft, Node bottomRight) {
        this.val = val;
        this.isLeaf = isLeaf;
        this.topLeft = topLeft;
        this.topRight = topRight;
        this.bottomLeft = bottomLeft;
        this.bottomRight = bottomRight;
    }
}
*/

class Solution {
    public Node construct(int[][] grid) {
        return dfs(grid, 0, 0, grid.length);
    }

    private Node dfs(int[][] grid, int r, int c, int size) {
        // Check if all elements in the current sub-grid are the same
        int first_val = grid[r][c];
        boolean all_same = true;
        for (int i = r; i < r + size; i++) {
            for (int j = c; j < c + size; j++) {
                if (grid[i][j] != first_val) {
                    all_same = false;
                    break;
                }
            }
            if (!all_same) {
                break;
            }
        }

        // If all are the same, create a leaf node
        if (all_same) {
            return new Node(first_val == 1, true);
        }

        // If not all are the same, divide the grid and recurse
        int half = size / 2;

        Node topLeft = dfs(grid, r, c, half);
        Node topRight = dfs(grid, r, c + half, half);
        Node bottomLeft = dfs(grid, r + half, c, half);
        Node bottomRight = dfs(grid, r + half, c + half, half);

        // Create an internal node with the four children
        // The 'val' for an internal node can be arbitrary, so we use true
        return new Node(true, false, topLeft, topRight, bottomLeft, bottomRight);
    }
}

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Problem Info

Description

Problem 427: Construct Quad Tree - Detailed explanation and solution approaches.

Topics

Array Divide and Conquer Tree Matrix

Companies

Palantir Technologies

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:427

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

7 steps

1
Step 1: Define a recursive function `dfs(r, c, size)` that takes the starting row `r`, starting column `c`, and the size of the sub-grid `size` as input.
2
Step 2: Check if all elements in the current sub-grid (from `grid[r][c]` to `grid[r+size-1][c+size-1]`) are the same. This involves iterating through all elements in the sub-grid.
3
Step 3: If all elements are the same, create a new `Node` with `isLeaf = true` and `val` equal to the value of the elements in the sub-grid (0 or 1). Return the created node.
4
Step 4: If the elements are not all the same, divide the sub-grid into four equal quadrants, each with size `size/2`.
5
Step 5: Recursively call `dfs` for each of the four quadrants: top-left (r, c, size/2), top-right (r, c + size/2, size/2), bottom-left (r + size/2, c, size/2), and bottom-right (r + size/2, c + size/2, size/2).
6
Step 6: Create a new `Node` with `isLeaf = false`, and set its `topLeft`, `topRight`, `bottomLeft`, and `bottomRight` children to the nodes returned by the recursive calls. The `val` field of the internal node can be any value (typically true or false). Return this internal node.
7
Step 7: Call the `dfs` function initially with `r = 0`, `c = 0`, and `size = len(grid)` to start the Quad-Tree construction from the entire grid.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n^2)

🐢 Quadratic - Gets slow with large inputs. Consider optimization!

💾

Space Complexity

O(n^2)

📚 Quadratic space - Uses lots of memory!

📝

Why?

The time complexity comes from the fact that at each node we potentially need to iterate through the whole section of size k x k, where k reduces in half with each subsequent recursive call. In the worst case, all cells need to be visited. Space complexity is primarily driven by the tree structure, which can potentially store every single matrix entry as leaf nodes, in the worst case.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Iterative Approach with Queue

An iterative approach can be implemented using a queue to perform a level-order traversal of the tree. This could avoid recursion depth limitations but might be more complex to implement.

Time: O(n^2)Space: O(n^2)

Trade-offs: Avoids recursion, but is harder to implement. Primarily beneficial if recursion depth is a constraint.

✨

Key Insights

1Insight 1: The problem lends itself perfectly to a divide-and-conquer (recursion) approach because the Quad-Tree structure naturally mirrors the division of the grid into quadrants.
2Insight 2: Determining whether a sub-grid contains all the same values is crucial for identifying leaf nodes, which halts the recursion for that branch.
3Insight 3: The choice of 'val' for internal nodes is arbitrary as long as 'isLeaf' is false. It's more important to correctly build the subtrees.
4Insight 4: Understanding the base case for recursion is key - when a subgrid contains only one value.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Tree

Hierarchical structure with nodes. Like a family tree!

Edge Cases

Edge case 1: A 1x1 grid. The solution correctly identifies this as a leaf node and creates a Quad-Tree with a single node.
Edge case 2: A grid with all the same values (all 0s or all 1s). The solution identifies this and creates a single leaf node, avoiding unnecessary recursion.
Edge case 3: A grid with alternating 0s and 1s. This represents the worst-case scenario, where the solution needs to divide the grid down to individual cells, resulting in the maximum recursion depth and tree size.
Edge case 4: An empty grid. Although the problem constraints state n >= 2^0 = 1, it's good practice to consider cases where the grid might be empty. The code would need an initial null check.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Forgetting to check if all elements in a sub-grid are the same before dividing it. This leads to infinite recursion and incorrect Quad-Tree construction.

✗Common mistake 2: Incorrectly calculating the starting row and column indices for the sub-grids in the recursive calls. This can cause out-of-bounds access or incorrect Quad-Tree structure.

✗Common mistake 3: Not setting the 'isLeaf' attribute correctly. This is crucial for distinguishing leaf nodes from internal nodes.

✗Common mistake 4: Incorrectly setting the value of the `val` attribute. Remember that internal node `val` doesn't matter, but leaf node `val` must match the single value the leaf represents.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Image compression. Quad-Trees can be used to represent images efficiently, especially images with large areas of uniform color. Each node can represent a section of the image, and leaf nodes can store the color value. Areas with little color variation can be represented by larger leaf nodes, reducing storage space.

🏦Real-world scenario 2: Spatial indexing in geographic information systems (GIS). Quad-Trees can be used to index and search for geographic data, such as points of interest or satellite imagery. Each node can represent a geographic region, and leaf nodes can store data related to that region. This allows for efficient searching and retrieval of data within a specific area.

🎮Real-world scenario 3: Collision detection in game development. Quad-Trees can divide the game world into regions, allowing to easily detect collisions by checking if two entities are in the same region and only performing detailed collision checks when they are.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

If stuck, try working through a small example by hand. The pattern often reveals itself.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: August 7, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 13

Try solving this without using extra space first. It's a great exercise for understanding the underlying pattern.

@CodeNinjaJul 6

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJun 29

❓

Frequently Asked Questions

When should I use in-place modification vs creating a new array?

Use in-place modification when space complexity matters (O(1) space requirement) and the original array can be modified. Create a new array when you need to preserve the original data or when the problem involves significant restructuring that would complicate in-place logic.

How much time should I spend on a medium problem in interviews?

In a 45-minute interview, aim for: 5-10 minutes understanding and clarifying, 5-10 minutes discussing approach, 20-25 minutes coding, and 5-10 minutes testing. If stuck after 15 minutes of thinking, verbalize your thought process and ask for hints.

What is the importance of time and space complexity analysis?

Complexity analysis is crucial because: 1) Interviewers always ask about it, 2) It helps you choose between approaches, 3) It demonstrates CS fundamentals. Always state both time and space complexity, and be prepared to explain how you derived them.

What is the optimal approach for "Construct Quad Tree"?

For this medium-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether divide and conquer techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

🤖 Need help with this problem?