ZCO22001 - Editorial

Problem Link - Vacation

Problem Statement

Chefland can be represented as a N by M grid. The rows of the grid are numbered from 1 to N, and the columns of the grid are numbered from 1 to M. The square in row X and column Y of the grid can be uniquely identified as square (X,Y). Each grid square has a cost of either 0 or 1.

Chef is going on a vacation to Chefland for the next Q days. On the i-th of these days, Chef would like to travel from square (Ai,Bi) to square (Ci,Di). Here, it is guaranteed that both A[i] ≤C[i] and B[i]≤D[i]. Due to restrictions on movement in Chefland, it is only possible to travel downwards or to the right. In other words, if Chef is in a square (X,Y), he is only able to move to square (X,Y+1) or (X+1,Y) in one move. The cost of Chef’s trip is defined as the product of the costs of the grid squares he passes through on his journey from the starting square to the ending square. Note that this product also includes the costs of the starting square and the ending square.

For each of the Q days, find the minimum cost Chef would need to pay to travel between the starting and the ending squares.

Approach

To solve the problem efficiently, we use a prefix sum array to preprocess the grid. The prefix sum array stores the number of cells with a cost of 0 in the rectangle from the top-left corner to each cell. This allows us to quickly calculate the number of cells with a cost of 0 in any subgrid defined by (A, B) to (C, D). If the count of 0 - cost cells in this subgrid is greater than 0, it means the cost of the trip is 0, since multiplying by 0 results in 0. Otherwise, the cost is 1. By processing all queries using the prefix sum array, we can efficiently compute the result for each journey without recalculating values for overlapping regions.

Time Complexity

The preprocessing step takes O(N \times M), and each query is processed in O(1), resulting in an overall complexity of O(N \times M + Q).

Space Complexity

The space required is O(N \times M) to store the prefix sum array.