The "Maximum White Tiles Covered by a Carpet" coding problem typically involves a line segment, often represented by an array or a list of intervals, where some segments are "white tiles" and others are empty. You are given a carpet of a fixed length and your task is to place this carpet anywhere on the line such that it covers the maximum possible number of white tiles. This problem requires careful consideration of optimal placement, often hinting at the use of techniques like sorting intervals, sliding window, or binary search to efficiently find the best position for the carpet. It tests your ability to optimize coverage within a fixed-size window over a given set of elements.
This Maximum White Tiles Covered by a Carpet interview question is frequently asked by companies like Google because it effectively evaluates a candidate's ability to apply geometric reasoning and optimize search problems. It tests your understanding of how to efficiently manage intervals and perform queries over a range. Specifically, it often involves a combination of sorting to order the white tile segments, and then either a sliding window or binary search approach to determine the optimal carpet placement. It assesses your ability to analyze the problem structure and choose the most efficient algorithmic pattern, showcasing your grasp of array manipulation, sorting, and interval management.
The "Maximum White Tiles Covered by a Carpet" problem is optimally solved using a combination of Sorting, Sliding Window, and sometimes Binary Search for specific optimizations.
carpetLen. The window represents the carpet's current placement.
left and right, defining the current window.right pointer, expand the window and calculate the number of white tiles covered within it.carpetLen, you need to "slide" the left pointer. This might involve partial tile coverage at the window's edges.left and right indices of the sorted tile array), prefix sums of tile lengths can be pre-calculated. This helps in efficiently updating the covered length as the window slides.
This array, sliding window, and prefix sum interview pattern allows for an efficient O(N log N) (due to sorting) or O(N) (after sorting) solution.Suppose tiles = [[10,11],[12,14],[15,16]] (meaning white tiles from 10-11, 12-14, 15-16) and carpetLen = 3.
Sorted tiles: [[10,11],[12,14],[15,16]] (already sorted).
Initialize maxCovered = 0, currentCovered = 0, left = 0.
Iterate right from 0 to len(tiles)-1:
right = 0 (tile [10,11]):
tiles[0][0] = 10, ends at 10 + carpetLen - 1 = 12.[10,11]. currentCovered = (11 - 10 + 1) = 2.maxCovered = 2.[10, 12].right = 1 (tile [12,14]):
tiles[1][1] = 14.14 - carpetLen + 1 = 12.[12, 14].currentCovered needs to be recalculated or adjusted.
This example highlights the complexity of partial tile coverage and the need for careful sliding window management in the Maximum White Tiles Covered by a Carpet coding problem.Let's use a simpler way to track covered length with sliding window.
tiles = [[10,11],[12,14],[15,16]], carpetLen = 3.
max_covered = 0, current_covered = 0, left = 0.
right = 0 (tile [10,11]):
start_carpet = tiles[right][0] = 10. end_carpet = start_carpet + carpetLen - 1 = 12.current_covered = min(tiles[right][1], end_carpet) - tiles[right][0] + 1 = min(11, 12) - 10 + 1 = 11 - 10 + 1 = 2.max_covered = 2.right = 1 (tile [12,14]):
start_carpet = tiles[left][0] = 10. end_carpet = start_carpet + carpetLen - 1 = 12.covered_length = 0, j = 0.
For each i (start of carpet):
carpet_start = tiles[i][0]. carpet_end = carpet_start + carpetLen - 1.
while j < len(tiles) and tiles[j][0] <= carpet_end:
current_tile_start = tiles[j][0]. current_tile_end = tiles[j][1].
covered_segment_start = max(carpet_start, current_tile_start).
covered_segment_end = min(carpet_end, current_tile_end).
if covered_segment_start <= covered_segment_end:
covered_length += (covered_segment_end - covered_segment_start + 1).
j += 1.
max_covered = max(max_covered, covered_length).This is a clearer sliding window approach.
A very common mistake in the Maximum White Tiles Covered by a Carpet coding problem is miscalculating the covered length when a carpet partially overlaps with white tiles, especially at the edges of the carpet or when a tile is split by the carpet's boundary. Forgetting to sort the white tile intervals by their start points can lead to incorrect results with the sliding window approach. Another pitfall is not efficiently updating the sum of covered tiles as the window slides, potentially re-calculating from scratch each time, which can lead to O(N^2) complexity. Edge cases, like a carpet shorter than any tile, or an empty tiles array, are also frequently missed.
To ace the Maximum White Tiles Covered by a Carpet interview question, master the sliding window technique, particularly its application to interval-based problems. Understand how to maintain and update a sum or count within a moving window efficiently. Practice sorting intervals and performing partial overlap calculations accurately. Familiarize yourself with prefix sums to quickly query sums over ranges. Always draw out a few examples and manually trace the sliding window's movement, paying attention to how left and right pointers (or their equivalents) are adjusted and how covered length is calculated. Solidifying your array, sorting, and sliding window interview pattern knowledge will be invaluable.