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2008. Maximum Earnings From Taxi

Description

There are n points on a road you are driving your taxi on. The n points on the road are labeled from 1 to n in the direction you are going, and you want to drive from point 1 to point n to make money by picking up passengers. You cannot change the direction of the taxi.

The passengers are represented by a 0-indexed 2D integer array rides, where rides[i] = [starti, endi, tipi] denotes the ith passenger requesting a ride from point starti to point endi who is willing to give a tipi dollar tip.

For each passenger i you pick up, you earn endi - starti + tipi dollars. You may only drive at most one passenger at a time.

Given n and rides, return the maximum number of dollars you can earn by picking up the passengers optimally.

Note: You may drop off a passenger and pick up a different passenger at the same point.

 

Example 1:

Input: n = 5, rides = [[2,5,4],[1,5,1]]
Output: 7
Explanation: We can pick up passenger 0 to earn 5 - 2 + 4 = 7 dollars.

Example 2:

Input: n = 20, rides = [[1,6,1],[3,10,2],[10,12,3],[11,12,2],[12,15,2],[13,18,1]]
Output: 20
Explanation: We will pick up the following passengers:
- Drive passenger 1 from point 3 to point 10 for a profit of 10 - 3 + 2 = 9 dollars.
- Drive passenger 2 from point 10 to point 12 for a profit of 12 - 10 + 3 = 5 dollars.
- Drive passenger 5 from point 13 to point 18 for a profit of 18 - 13 + 1 = 6 dollars.
We earn 9 + 5 + 6 = 20 dollars in total.

 

Constraints:

  • 1 <= n <= 105
  • 1 <= rides.length <= 3 * 104
  • rides[i].length == 3
  • 1 <= starti < endi <= n
  • 1 <= tipi <= 105

Solutions

First, we sort $rides$ in ascending order by $start$. Then we design a function $dfs(i)$, which represents the maximum tip that can be obtained from accepting orders starting from the $i$-th passenger. The answer is $dfs(0)$.

The calculation process of the function $dfs(i)$ is as follows:

For the $i$-th passenger, we can choose to accept or not to accept the order. If we don't accept the order, the maximum tip that can be obtained is $dfs(i + 1)$. If we accept the order, we can use binary search to find the first passenger encountered after the drop-off point of the $i$-th passenger, denoted as $j$. The maximum tip that can be obtained is $dfs(j) + end_i - start_i + tip_i$. Take the larger of the two. That is:

$$ dfs(i) = \max(dfs(i + 1), dfs(j) + end_i - start_i + tip_i) $$

Where $j$ is the smallest index that satisfies $start_j \ge end_i$, which can be obtained by binary search.

In this process, we can use memoization search to save the answer of each state to avoid repeated calculations.

The time complexity is $O(m \times \log m)$, and the space complexity is $O(m)$. Here, $m$ is the length of $rides$.

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class Solution:
    def maxTaxiEarnings(self, n: int, rides: List[List[int]]) -> int:
        @cache
        def dfs(i: int) -> int:
            if i >= len(rides):
                return 0
            st, ed, tip = rides[i]
            j = bisect_left(rides, ed, lo=i + 1, key=lambda x: x[0])
            return max(dfs(i + 1), dfs(j) + ed - st + tip)

        rides.sort()
        return dfs(0)

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class Solution {
    private int m;
    private int[][] rides;
    private Long[] f;

    public long maxTaxiEarnings(int n, int[][] rides) {
        Arrays.sort(rides, (a, b) -> a[0] - b[0]);
        m = rides.length;
        f = new Long[m];
        this.rides = rides;
        return dfs(0);
    }

    private long dfs(int i) {
        if (i >= m) {
            return 0;
        }
        if (f[i] != null) {
            return f[i];
        }
        int[] r = rides[i];
        int st = r[0], ed = r[1], tip = r[2];
        int j = search(ed, i + 1);
        return f[i] = Math.max(dfs(i + 1), dfs(j) + ed - st + tip);
    }

    private int search(int x, int l) {
        int r = m;
        while (l < r) {
            int mid = (l + r) >> 1;
            if (rides[mid][0] >= x) {
                r = mid;
            } else {
                l = mid + 1;
            }
        }
        return l;
    }
}

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class Solution {
public:
    long long maxTaxiEarnings(int n, vector<vector<int>>& rides) {
        sort(rides.begin(), rides.end());
        int m = rides.size();
        long long f[m];
        memset(f, -1, sizeof(f));
        function<long long(int)> dfs = [&](int i) -> long long {
            if (i >= m) {
                return 0;
            }
            if (f[i] != -1) {
                return f[i];
            }
            auto& r = rides[i];
            int st = r[0], ed = r[1], tip = r[2];
            int j = lower_bound(rides.