2008. Maximum Earnings From Taxi
Approach
DP on city positions; store rides by start; at each point take best ending ride or skip.
Key Techniques
Array problems involve manipulating elements stored in a contiguous block of memory. Key techniques include two-pointer traversal, prefix sums, sliding windows, and in-place partitioning. In C#, arrays are zero-indexed and fixed in size — use List<T> when you need dynamic resizing.
Binary search reduces search space by half each step, giving O(log n) time. Beyond sorted arrays, apply it on the answer space ("binary search on result") when you can define a monotonic predicate — e.g., "can we achieve X with k resources?"
Dynamic programming solves problems by breaking them into overlapping sub-problems and storing results to avoid redundant work. The key steps are: define state, write a recurrence relation, set base cases, and choose top-down (memoization) or bottom-up (tabulation). DP often yields O(n²) → O(n) time improvements over brute force.
// Approach: DP on city positions; store rides by start; at each point take best ending ride or skip.
// Time: O(n + r log r) Space: O(n + r)
public class Solution
{
public long MaxTaxiEarnings(int n, int[][] rides)
{
List<(int end, int earn)>[] startToEndAndEarns = new List<(int, int)>[n];
// dp[i] := the maximum dollars you can earn starting at i
long[] dp = new long[n + 1];
for (int i = 1; i < n; ++i)
startToEndAndEarns[i] = new List<(int, int)>();
foreach (var ride in rides)
{
int start = ride[0];
int end = ride[1];
int tip = ride[2];
int earn = end - start + tip;
startToEndAndEarns[start].Add((end, earn));
}
for (int i = n - 1; i >= 1; --i)
{
dp[i] = dp[i + 1];
foreach (var (end, earn) in startToEndAndEarns[i])
dp[i] = Math.Max(dp[i], dp[end] + earn);
}
return dp[1];
}
}