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2830. Maximize the Profit as the Salesman
Description
You are given an integer n
representing the number of houses on a number line, numbered from 0
to n  1
.
Additionally, you are given a 2D integer array offers
where offers[i] = [start_{i}, end_{i}, gold_{i}]
, indicating that i^{th}
buyer wants to buy all the houses from start_{i}
to end_{i}
for gold_{i}
amount of gold.
As a salesman, your goal is to maximize your earnings by strategically selecting and selling houses to buyers.
Return the maximum amount of gold you can earn.
Note that different buyers can't buy the same house, and some houses may remain unsold.
Example 1:
Input: n = 5, offers = [[0,0,1],[0,2,2],[1,3,2]] Output: 3 Explanation: There are 5 houses numbered from 0 to 4 and there are 3 purchase offers. We sell houses in the range [0,0] to 1^{st} buyer for 1 gold and houses in the range [1,3] to 3^{rd} buyer for 2 golds. It can be proven that 3 is the maximum amount of gold we can achieve.
Example 2:
Input: n = 5, offers = [[0,0,1],[0,2,10],[1,3,2]] Output: 10 Explanation: There are 5 houses numbered from 0 to 4 and there are 3 purchase offers. We sell houses in the range [0,2] to 2^{nd} buyer for 10 golds. It can be proven that 10 is the maximum amount of gold we can achieve.
Constraints:
1 <= n <= 10^{5}
1 <= offers.length <= 10^{5}
offers[i].length == 3
0 <= start_{i} <= end_{i} <= n  1
1 <= gold_{i} <= 10^{3}
Solutions
Solution 1: Sorting + Binary Search + Dynamic Programming
We sort all the purchase offers by $end$ in ascending order, and then use dynamic programming to solve the problem.
Define $f[i]$ to represent the maximum amount of gold we can get from the first $i$ purchase offers. The answer is $f[n]$.
For $f[i]$, we can choose not to sell the $i$th purchase offer, in which case $f[i] = f[i  1]$; or we can choose to sell the $i$th purchase offer, in which case $f[i] = f[j] + gold_i$, where $j$ is the largest index that satisfies $end_j \leq start_i$.
The time complexity is $O(n \times \log n)$, and the space complexity is $O(n)$. Here, $n$ is the number of purchase offers.

class Solution { public int maximizeTheProfit(int n, List<List<Integer>> offers) { offers.sort((a, b) > a.get(1)  b.get(1)); n = offers.size(); int[] f = new int[n + 1]; int[] g = new int[n]; for (int i = 0; i < n; ++i) { g[i] = offers.get(i).get(1); } for (int i = 1; i <= n; ++i) { var o = offers.get(i  1); int j = search(g, o.get(0)); f[i] = Math.max(f[i  1], f[j] + o.get(2)); } return f[n]; } private int search(int[] nums, int x) { int l = 0, r = nums.length; while (l < r) { int mid = (l + r) >> 1; if (nums[mid] >= x) { r = mid; } else { l = mid + 1; } } return l; } }

class Solution { public: int maximizeTheProfit(int n, vector<vector<int>>& offers) { sort(offers.begin(), offers.end(), [](const vector<int>& a, const vector<int>& b) { return a[1] < b[1]; }); n = offers.size(); vector<int> f(n + 1); vector<int> g; for (auto& o : offers) { g.push_back(o[1]); } for (int i = 1; i <= n; ++i) { auto o = offers[i  1]; int j = lower_bound(g.begin(), g.end(), o[0])  g.begin(); f[i] = max(f[i  1], f[j] + o[2]); } return f[n]; } };

class Solution: def maximizeTheProfit(self, n: int, offers: List[List[int]]) > int: offers.sort(key=lambda x: x[1]) f = [0] * (len(offers) + 1) g = [x[1] for x in offers] for i, (s, _, v) in enumerate(offers, 1): j = bisect_left(g, s) f[i] = max(f[i  1], f[j] + v) return f[1]

func maximizeTheProfit(n int, offers [][]int) int { sort.Slice(offers, func(i, j int) bool { return offers[i][1] < offers[j][1] }) n = len(offers) f := make([]int, n+1) g := []int{} for _, o := range offers { g = append(g, o[1]) } for i := 1; i <= n; i++ { j := sort.SearchInts(g, offers[i1][0]) f[i] = max(f[i1], f[j]+offers[i1][2]) } return f[n] }

function maximizeTheProfit(n: number, offers: number[][]): number { offers.sort((a, b) => a[1]  b[1]); n = offers.length; const f: number[] = Array(n + 1).fill(0); const g = offers.map(x => x[1]); const search = (x: number) => { let l = 0; let r = n; while (l < r) { const mid = (l + r) >> 1; if (g[mid] >= x) { r = mid; } else { l = mid + 1; } } return l; }; for (let i = 1; i <= n; ++i) { const j = search(offers[i  1][0]); f[i] = Math.max(f[i  1], f[j] + offers[i  1][2]); } return f[n]; }