CF 1742C - Stripes

CF 1742C - Stripes

Rating: 900
Tags: implementation
Solve time: 5m 41s
Verified: yes

Solution

Problem Understanding

The task presents an 8×8 grid where some rows have been painted red and some columns have been painted blue. The painting happens sequentially: when a stripe is painted, it overwrites all the cells along that row or column, even if they already had a color. The grid we receive as input shows the final colors, including unpainted cells represented by dots. The goal is to determine which color, red or blue, was painted last. That is equivalent to asking which type of stripe, a fully red row or a fully blue column, dominates in the final grid such that it must have been painted last.

Each test case provides exactly 8 lines of 8 characters, and there can be up to 4000 test cases. Since the grid is very small, any operation linear or quadratic in the number of cells is acceptable. The main consideration is correctly interpreting the final coloring in terms of the sequence of painting, not the painting process itself. Edge cases include situations where multiple stripes of each color exist, but only one color could have been last. For example, if the bottom row is entirely red and all columns have partial blue, the last stripe must be red, because the final red row could only have been painted last to appear intact. Conversely, if a column is entirely blue and all rows have partial red, the last stripe must be blue.

Approaches

A naive approach would try to reconstruct the painting sequence by checking every possible order of rows and columns. One could enumerate all possible sequences of stripe applications and see which results in the observed final grid. This works in principle but is unnecessarily complicated. The number of possible stripe orderings grows exponentially with the number of painted stripes, making such brute-force impractical even for this small grid if attempted carelessly.

The key insight comes from observing that the last stripe determines a fully filled line of its color in the final grid. A red row painted last leaves the entire row red regardless of previous blue columns. Similarly, a blue column painted last leaves the entire column blue. Therefore, to determine which color was painted last, it suffices to check for any row that is entirely red or any column that is entirely blue. If there exists a fully red row, the last stripe must be red; if there exists a fully blue column, the last stripe must be blue. The problem guarantees that the grid is generated by such stripes, so there is always at least one stripe painted last that can be detected this way.

This observation reduces the problem from trying to reconstruct sequences to simple scanning of rows and columns. We iterate through all rows to see if any are entirely red, and through all columns to see if any are entirely blue. Because the grid is fixed at 8×8, this scanning takes at most 64 operations per test case, which is negligible even for 4000 test cases. The insight leverages the structure of the problem: stripes fully fill their row or column and overwrite previous colors, making the last stripe determinable from completeness in the final grid.

Algorithm Walkthrough

1. Read the number of test cases. For each test case, read the 8×8 grid as a list of strings.
2. Iterate through each of the 8 rows. For a row, check if all 8 cells are 'R'. If such a row is found, output 'R' for this test case and continue to the next test case. This step works because a completely red row can only occur if a red stripe was painted last over that row, ensuring all cells are red.
3. If no fully red row is found, iterate through the 8 columns. For a column, check if all 8 cells are 'B'. If such a column is found, output 'B'. The existence of a fully blue column implies that a blue stripe was painted last over that column, making the last stripe blue.
4. Continue this process for all test cases. Since the problem guarantees at least one stripe exists, one of these two conditions will always be met, so no additional handling for empty grids is required.

This algorithm works because the last stripe of a given color is the only one that can leave an entire row or column completely in that color. Any previously painted stripes would have been overwritten in part by the last stripe. Therefore, scanning for a fully red row or fully blue column is sufficient to identify the last stripe unambiguously.

Python Solution

PythonRun

The solution first consumes the number of test cases and handles the empty line before each grid. The fully red row check is done before the column check because the problem guarantees that at least one stripe exists and only one color could be last. Iterating through the rows and columns uses Python's all function to verify completeness, which is concise and avoids off-by-one errors. The order of these checks matches the guarantee that there is exactly one last stripe. Stripping the input ensures no trailing newline affects the row check.

Worked Examples

For the first sample input, the grid has a fully red row at index 3 (0-based). Iterating through rows finds that row and immediately returns 'R'. No column check is needed. For the second sample input, there is no fully red row, but column 0 and column 7 are fully blue. The row check fails, then the column check finds the blue column and outputs 'B'.

These traces show that the algorithm correctly identifies the last stripe based on complete rows or columns, even when multiple stripes of each color exist in the grid.

Complexity Analysis

The time complexity is comfortably below the 1-second limit. Space usage is minimal since only the current grid needs to be held in memory.

Test Cases

PythonRun