Maximum Number of Points From Grid Queries

Maximum Number of Points From Grid Queries

What is this problem about?

The Maximum Number of Points From Grid Queries coding problem presents a grid of integers and a list of queries. Each query consists of an integer val. For each query, you need to find the maximum number of cells you can visit in the grid starting from (0,0), such that all visited cells have values strictly less than val. You can move to adjacent cells (up, down, left, right).

Why is this asked in interviews?

J.P. Morgan, Meta, Google, and Bloomberg use this hard problem to test a combination of graph traversal (BFS/DFS), sorting, and a Disjoint Set Union (DSU) or a modified BFS/DFS that processes queries offline. The key is that queries can be processed more efficiently if sorted, allowing incremental updates to the reachable cells.

Algorithmic pattern used

Offline Processing with BFS/DFS and Sorting: This problem is a classic example where offline processing of queries (sorting them) combined with a graph traversal or DSU can yield an efficient solution.

1. Prepare Data: Create a list of all cells (value, row, col) and sort them by value in ascending order. Create a list of queries (val, original_index) and sort them by val in ascending order.
2. Modified BFS/DFS or DSU:
   • Initialize a visited grid and a reachable_count = 0.
   • Iterate through the sorted queries. For each query (query_val, original_idx):
     • While there are cells in the sorted cells list whose value < query_val:
       • Add these cells to a queue for BFS (or process them with DSU).
       • If a cell is added, perform a BFS/DFS from it. All its neighbors with value < query_val that haven't been visited become part of the current connected component. Update reachable_count.
       • With DSU: union newly connected cells, and keep track of component sizes.
     • Store the current reachable_count for answers[original_idx].

With BFS: Iterate through sorted queries. Maintain a max_heap of reachable cells (value, r, c) encountered so far. When a query Q comes, add all cells (val, r, c) where val < Q to the max_heap (or a min_heap and extract if value is too low). Actually, a simpler BFS variant:

1. Combine cells (value, r, c) and queries (val, query_idx). Sort both collections by value/val.
2. Iterate through sorted queries. For each query (Q_val, Q_idx):
   • Expand the reachable area: while there are grid cells (G_val, G_r, G_c) with G_val < Q_val that have not been "processed" yet, add (G_r, G_c) to a component and perform BFS/DFS from them to explore all adjacent cells with values less than Q_val. Keep a running count of all reachable cells.
   • The current reachable_count is the answer for Q_idx.

This needs careful merging of cell processing and query answering. A more direct way is a single BFS:

• Sort queries by value.
• Store (value, row, col) for all grid cells and sort them.
• Initialize ans[query_count].
• Use a BFS/DFS from (0,0) to find reachable cells.
• When processing queries, if query_val > current_max_reachable_cell_value, expand the reachable set.

This problem is very similar to "Swim in Rising Water" or "Path With Minimum Effort", where sorting the edges/values and using a MST-like approach (DSU) or a priority queue BFS is key. Specifically, sort all cells by their values. Process queries and cells together in increasing order of value. When a query with val arrives, all cells with grid[r][c] < val are potentially traversable. Use a DSU structure where cells are nodes. As you iterate through sorted grid cells:

• If grid[r][c] < query.val: Add (r,c) to DSU. Union it with valid neighbors.
• The answer for query is the size of the component containing (0,0).

Common mistakes candidates make

• Not sorting queries: Processing queries independently leads to repeated work.
• Repeated BFS/DFS for each query: Too slow if N queries and N*M grid cells.
• Incorrectly handling connectivity: DSU is well-suited for dynamically building components.

Interview preparation tip

For the Array Matrix Union Find Breadth-First Search Sorting Two Pointers Heap (Priority Queue) interview pattern, problems requiring "maximum reachable cells under a threshold" are often solved by sorting both values and queries, then incrementally expanding reachable regions using DSU or a priority queue based BFS. Understand the core idea of offline query processing.