Note about Dynamic Programming

Dynamic programming is a method used to solve complex problems by breaking them down into simpler subproblems. By solving each subproblem only once and storing the results, it avoids redundant computations, leading to more efficient solutions.

To implement dynamic programming:

  • Identify Subproblems: Divide the main problem into smaller, independent subproblems.
  • Store Solutions: Solve each subproblem and store the solution.
  • Use Solutions: Use the stored solutions to build up the solution to the main problem.

There are two approaches to dynamic programming:

  1. Top-Down (Memoization): Start from the top (main problem) and break it down into subproblems. Store the results of subproblems in a data structure (e.g., hash map) to avoid redundant computations.
  2. Bottom-Up (Tabulation): Start from the bottom (subproblems) and build up the solution to the main problem. This approach is often more efficient in terms of space complexity.

1) Professional Robber

You are a professional robber planning to rob houses along a street. Each house has a certain amount of money stashed, the only constraint stopping you from robbing each of them is that adjacent houses have security systems connected and it will automatically contact the police if two adjacent houses were broken into on the same night.

Given an integer array nums representing the amount of money of each house, return the maximum amount of money you can rob tonight without alerting the police.

Example 1:

Input: nums = [1,2,3,1]
Output: 4
Explanation: Rob house 1 (money = 1) and then rob house 3 (money = 3).
Total amount you can rob = 1 + 3 = 4.
Example 2:

Input: nums = [2,7,9,3,1]
Output: 12
Explanation: Rob house 1 (money = 2), rob house 3 (money = 9) and rob house 5 (money = 1).
Total amount you can rob = 2 + 9 + 1 = 12.

Time Complexity: O(n)
Space Complexity: O(1)

2) Count Odd Numbers in an Interval Range

Given two non-negative integers low and high. Return the count of odd numbers between low and high (inclusive).

Input: low = 3, high = 7
Output: 3
Explanation: The odd numbers between 3 and 7 are [3,5,7].

Input: low = 8, high = 10
Output: 1
Explanation: The odd numbers between 8 and 10 are [9].

Time Complexity: O(1)
Space Complexity: O(1)

3) First bad version

You are a product manager and currently leading a team to develop a new product. Unfortunately, the latest version of your product fails the quality check. Since each version is developed based on the previous version, all the versions after a bad version are also bad.

Suppose you have n versions [1, 2, …, n] and you want to find out the first bad one, which causes all the following ones to be bad.

You are given an API bool isBadVersion(version) which returns whether version is bad. Implement a function to find the first bad version. You should minimize the number of calls to the API.

Input: n = 5
Output: 4
Explanation:
call isBadVersion(3) -> false
call isBadVersion(5) -> true
call isBadVersion(4) -> true
Then 4 is the first bad version.

Input: n = 1
Output: 1

Time Complexity: O(lgn)
Space Complexity: O(1)