How to Solve Maximum Number of Points From Grid Queries

This problem requires calculating the maximum points from grid queries, leveraging a two-pointer approach with invariant tracking. The queries are given beforehand, allowing for precomputation. Using this method optimizes answering queries efficiently in O(k \log k + (n \cdot m) \log (n \cdot m)) time complexity.

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

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.

How GhostInterview Helps

• GhostInterview’s solver helps you optimize solutions for grid traversal problems by guiding you to use two-pointer strategies effectively.
• The solver highlights common pitfalls like redundant calculations and helps you avoid inefficient query handling.
• GhostInterview also helps you master problem-specific patterns like invariant tracking, which is crucial for problems involving grid-based traversal.

Topic Pages

• Array
• Two Pointers
• Breadth-First Search
• Union Find
• Sorting
• Heap (Priority Queue)
• Matrix

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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.