#2503

Hard

auto_awesomeTwo-pointer scanning with invariant tracking

LeetCode Problem Workspace

Maximum Number of Points From Grid Queries

Solve the Maximum Number of Points From Grid Queries problem using two-pointer scanning and invariant tracking.

array two pointers breadth first search union find sorting

Problem Statement

You are given an m x n integer matrix called grid and an array queries of size k. For each query, you need to start at the top-left cell of the grid and calculate the maximum number of points you can accumulate by traversing the grid.

For each query, you can traverse the grid and revisit cells, but you need to ensure that the points gained during the traversal are maximized. After completing the traversal for each query, return the maximum points as an array answer with the same size as queries.

Examples

Example 1

Input: grid = [[1,2,3],[2,5,7],[3,5,1]], queries = [5,6,2]

Output: [5,8,1]

The diagrams above show which cells we visit to get points for each query.

Example 2

Input: grid = [[5,2,1],[1,1,2]], queries = [3]

Output: [0]

We can not get any points because the value of the top left cell is already greater than or equal to 3.

Constraints

m == grid.length
n == grid[i].length
2 <= m, n <= 1000
4 <= m * n <= 105
k == queries.length
1 <= k <= 104
1 <= grid[i][j], queries[i] <= 106

Solution Approach

Two-Pointer Scanning

Since the queries are provided beforehand, we can answer them in any order. Two-pointer scanning is applied to efficiently navigate the grid, tracking which cells are visited and ensuring the traversal follows the rules to maximize points. Preprocessing the grid helps minimize redundant work across queries.

Invariant Tracking

By tracking invariants during the traversal, we can avoid recalculating unnecessary parts of the grid. This includes managing the state of the traversal to ensure we can calculate the maximum points effectively while respecting query constraints.

Optimization with Sorting

Sorting the queries beforehand and using the two-pointer approach in combination with efficient data structures allows for an optimized solution. This prevents recalculating points for each query and ensures the solution scales efficiently with larger grids and query arrays.

Complexity Analysis

Metric	Value
Time	O(k \log k + (n \cdot m) \log (n \cdot m) + k \cdot \alpha(n \cdot m))
Space	O((n \cdot m) + k)

The time complexity is O(k \log k + (n \cdot m) \log (n \cdot m) + k \cdot \alpha(n \cdot m)) due to the sorting of queries and the optimized traversal with invariant tracking. The space complexity is O((n \cdot m) + k) to handle the grid and queries.

What Interviewers Usually Probe

Tests understanding of two-pointer strategies in grid-based problems.
Assesses the ability to manage grid traversal with invariant tracking and efficient data structures.
Evaluates optimization skills when processing multiple queries in a grid setting.

Common Pitfalls or Variants

Common pitfalls

Not leveraging the precomputation of queries, which leads to redundant computations.
Incorrectly handling the grid traversal state, leading to errors in point calculation.
Inefficient query handling that doesn't scale well with large inputs or multiple queries.

Follow-up variants

Consider handling grids with larger dimensions to test the efficiency of the solution.
Test with queries that are all smaller than the values in the grid to ensure the algorithm handles edge cases.
Consider implementing a dynamic programming approach instead of sorting for different performance trade-offs.

FAQ

What is the two-pointer scanning approach for solving Maximum Number of Points From Grid Queries?

The two-pointer scanning approach helps you efficiently traverse the grid by utilizing a pointer system that tracks visited cells and ensures points are maximized during traversal.

How does invariant tracking help in the Maximum Number of Points From Grid Queries problem?

Invariant tracking ensures that once a part of the grid has been processed for a query, it is not recalculated unnecessarily for subsequent queries, improving overall efficiency.

What is the time complexity of the solution for Maximum Number of Points From Grid Queries?

The time complexity of the solution is O(k \log k + (n \cdot m) \log (n \cdot m) + k \cdot \alpha(n \cdot m)), driven by sorting and efficient grid traversal.

What are the common pitfalls when solving the Maximum Number of Points From Grid Queries problem?

Common pitfalls include not precomputing queries, mishandling grid traversal states, and failing to optimize query processing, leading to inefficiencies.

How can GhostInterview help solve the Maximum Number of Points From Grid Queries problem?

GhostInterview’s solver aids in applying two-pointer strategies, tracks traversal invariants, and guides you in optimizing query handling for grid problems.

terminal

Solution

Solution 1: Offline Query + BFS + Priority Queue (Min Heap)

According to the problem description, each query is independent, the order of the queries does not affect the result, and we are required to start from the top left corner each time, counting the number of cells that can be accessed and whose value is less than the current query value.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

class Solution:
    def maxPoints(self, grid: List[List[int]], queries: List[int]) -> List[int]:
        m, n = len(grid), len(grid[0])
        qs = sorted((v, i) for i, v in enumerate(queries))
        ans = [0] * len(qs)
        q = [(grid[0][0], 0, 0)]
        cnt = 0
        vis = [[False] * n for _ in range(m)]
        vis[0][0] = True
        for v, k in qs:
            while q and q[0][0] < v:
                _, i, j = heappop(q)
                cnt += 1
                for a, b in pairwise((-1, 0, 1, 0, -1)):
                    x, y = i + a, j + b
                    if 0 <= x < m and 0 <= y < n and not vis[x][y]:
                        heappush(q, (grid[x][y], x, y))
                        vis[x][y] = True
            ans[k] = cnt
        return ans

Continue Practicing

#2617 minimum number of visited cells in a grid #2812 find the safest path in a grid #1631 path with minimum effort