Alex2025_qwq's blog

Yet Another Constructive Problem～
By Alex2025_qwq, history, 65 minutes ago, In English

Hi everyone! Here’s a fun problem: Construct an n*m binary matrix such that all the 1-cells form a single 4-connected component, while maximizing the number of 4-connected components formed by the 0-cells. Two cells are considered adjacent if they share an edge. Is there an O(nm) construction with a provably tight upper bound on the number of zero-components? What is the tightest upper bound you can establish, and what are the best achievable time and space complexities for the construction?

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54 minutes ago, hide # |
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note: remember to handle n=1 or m=1

(btw im editing so much lol)

my construction (0: . and 1: #):

.#.#
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.#.#
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a[i][j] = 1 if (i % 2 or j % 2) else 0

its $$$\lceil n/2 \rceil \cdot \lceil m/2 \rceil$$$ 0 components

time and space is $$$O(nm)$$$

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    . # # #
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