Sum of Remoteness of All Cells

Sum of Remoteness of All Cells

What is this problem about?

In a grid (matrix), some cells are blocked and some are open. Two open cells are "connected" if there is a path between them (moving up, down, left, right). The "remoteness" of an open cell is the sum of the values of all open cells that are not in its connected component. The task is to find the total sum of remoteness for all open cells in the grid.

Why is this asked in interviews?

This problem from Media.net tests a candidate's ability to handle grid-based connectivity and use mathematical simplification to avoid O(N²) component comparisons. It evaluates skills in Graph Traversal (BFS/DFS) or Union-Find, as well as the ability to aggregate data across components efficiently.

Algorithmic pattern used

The pattern is "Connected Component Identification with Global Summing."

1. Calculate the total_sum of all values in all open cells in the grid.
2. Find all connected components of open cells using BFS or DFS.
3. For each component:
   • Calculate comp_sum (sum of values in this component).
   • Calculate comp_size (number of cells in this component).
   • The remoteness of any cell in this component is total_sum - comp_sum.
   • The contribution of this whole component to the final answer is (total_sum - comp_sum) * comp_size.
4. Sum these contributions for all components.

Example explanation

Grid values: Component A (cells 1, 2; values 10, 20), Component B (cell 3; value 50).

1. Total Sum = 10 + 20 + 50 = 80.
2. Component A:
   • Sum = 30, Size = 2.
   • Remoteness of each cell = 80 - 30 = 50.
   • Component contribution = 50 * 2 = 100.
3. Component B:
   • Sum = 50, Size = 1.
   • Remoteness = 80 - 50 = 30.
   • Component contribution = 30 * 1 = 30. Grand Total = 100 + 30 = 130.

Common mistakes candidates make

1. Manual Comparison: Trying to calculate the distance or connection between every pair of cells individually.
2. Integer Overflow: The total sum of remoteness can easily exceed 2^31 - 1, so use 64-bit integers (long).
3. Visiting Blocked Cells: Failing to correctly handle the "blocked" vs "open" cell logic during traversal.

Interview preparation tip

When a problem asks about "everything not in the same group," always calculate the "total" first. The answer for one group is usually Total - Group_Value. This is much faster than looking at all other groups one by one.