This approach involves splitting the problem into two scenarios: robbing from house 0 to house n-2, and from house 1 to house n-1. The aim is to apply the linear version of the House Robber problem to each scenario separately and then take the maximum of the two resulting summed values.
Time Complexity: O(n).
Space Complexity: O(1).
1
2def robLinear(nums):
3 prev1, prev2 = 0, 0
4 for num in nums:
5 temp = prev1
6 prev1 = max(num + prev2, prev1)
7 prev2 = temp
8 return prev1
9
10class Solution:
11 def rob(self, nums):
12 if len(nums) == 1:
13 return nums[0]
14 return max(robLinear(nums[:-1]), robLinear(nums[1:]))
15This Python solution utilizes a helper function robLinear to handle the linear robbery problem. It then applies this helper to the range excluding first and last houses respectively, returning the greatest result.
This approach implements a memoized version of the dynamic programming solution to avoid recomputing values. We explore two scenarios of the circle, taking into account the cycles explicitly by caching intermediate results.
Time Complexity: O(n).
Space Complexity: O(n) for the memoization array.
1
2public class Solution {
3 private Dictionary<int, int> memo = new Dictionary<int, int>();
4 private int RobHelper(int[] nums, int start, int end) {
5 if (start >= end) return 0;
6 if (memo.ContainsKey(start)) return memo[start];
7 int result = Math.Max(nums[start] + RobHelper(nums, start + 2, end), RobHelper(nums, start + 1, end));
8 memo[start] = result;
9 return result;
10 }
11
12 public int Rob(int[] nums) {
13 if (nums.Length == 1) return nums[0];
14 memo.Clear();
15 int case1 = RobHelper(nums, 0, nums.Length - 1);
16 memo.Clear();
17 int case2 = RobHelper(nums, 1, nums.Length);
18 return Math.Max(case1, case2);
19 }
20}
21This C# solution employs a Dictionary for managing memoization to store and retrieve calculated subproblems conveniently. The method RobHelper determines the maximum taking turns at each index, from which two scenarios for circle robbing are evaluated for the final yield optimum.