House Robber III 🎯

Welcome to our comprehensive guide on the House Robber III problem, a classic algorithmic puzzle that's both fun and practical! This tutorial is designed for beginners and intermediates, so let's dive right in. 📝

Introduction

The House Robber III problem is an extension of the famous House Robber and House Robber II problems. It involves a sequence of houses with an arbitrary length, and you are a burglar who wants to maximize your loot by robbing the most valuable houses.

Problem Statement

Given an array of integers arr representing the value of each house, and two integers k and n, where k is the starting index of the house you start at, and n is the number of houses you can rob in a single trip (i.e., moving from one end of the array to the other).

Your task is to find the maximum amount of money you can steal by only robbing k house and every nth house thereafter, ensuring you don't rob two consecutive houses.

Example

Let's consider an example:

arr = [2, 7, 9, 3, 1]
k = 2
n = 2

In this case, you can rob the following houses: [9, 3]. The total loot would be 9 + 3 = 12.

Solution Approach

To solve the House Robber III problem, we will use dynamic programming, which is a common technique for solving optimization problems like this one. We'll create an array dp of size n+1 to store the maximum amount of money that can be stolen ending at each house.

Pseudo Code

function rob(arr, k, n):
    if len(arr) == 0:
        return 0
    dp = [0] * (n+1)
    dp[k] = arr[k]

    for i in range(k+1, n+1):
        dp[i] = max(dp[i-1], dp[i-n] + arr[i])

    return dp[n]

Code Examples

Python

python

def rob(arr, k, n):
    if not arr:
        return 0
    dp = [0] * (n+1)
    dp[k] = arr[k]

    for i in range(k+1, n+1):
        dp[i] = max(dp[i-1], dp[i-n] + arr[i])

    return dp[n]

Java

java

int rob(int[] arr, int k, int n) {
    if (arr.length == 0)
        return 0;
    int[] dp = new int[n+1];
    dp[k] = arr[k];

    for (int i = k+1; i <= n; i++) {
        dp[i] = Math.max(dp[i-1], dp[i-n] + arr[i]);
    }

    return dp[n];
}

💡 Pro Tip:

Remember to handle edge cases carefully. For example, if n is greater than the length of the array, you should return 0.

House Robber III 🎯

House Robber III 🎯

Introduction

Problem Statement

Example

Solution Approach

Pseudo Code

Code Examples

Python

Java

💡 Pro Tip:

Quiz