House Robber
Given an array representing money in each house, return the maximum you can rob without robbing two adjacent houses.
Example:
1,2,3,1
4
- We start by initializing two variables to track the maximum amount that can be robbed up to each house: dp0​=1 (robbing the first house) and dp1​=2 (robbing the second house, which is more than the first).
- At the third house, we have two options: rob the third house (3) or rob the first two houses (1+2=3), so we choose the maximum of these, which is 3, and update dp2​ to 3 since we can't rob the second house if we rob the third.
- At the fourth house, we again have two options: rob the first and fourth houses (1+1=2) or rob the first and third houses (1+3=4), so we choose the maximum of these, which is 4.
- The final output is the maximum of the last two options, which is 4​.
Constraints:
- 1 <= len(nums) <= 100
- 0 <= nums[i] <= 400
Background Knowledge
The "House Robber" problem is a classic example of a dynamic programming problem. Dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems, solving each subproblem only once, and storing the solutions to subproblems to avoid redundant computation. This approach is particularly useful for problems that have overlapping subproblems or optimal substructure, meaning that the problem can be broken down into smaller subproblems and the optimal solution to the larger problem can be constructed from the optimal solutions of the subproblems.
In the context of the "House Robber" problem, we need to consider the maximum amount of money that can be robbed from a sequence of houses, with the constraint that no two adjacent houses can be robbed. This problem can be viewed as a sequence of decisions, where at each step, we decide whether to rob the current house or not. The decision at each step depends on the decisions made at previous steps, which is a characteristic of dynamic programming problems.
To solve this problem, we need to understand the concept of state and transition in dynamic programming. The state represents the current situation, and the transition represents the action taken to move from one state to another. In this problem, the state can be represented by the current house and the amount of money robbed so far. The transition can be represented by the decision to rob or not rob the current house.
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