Let $S$ be a set of $n^2$ symbols. Let $A$ be an $n\times n$ square grid with each cell labeled by a distinct symbol in $S$. Let $B$ be another $n\times n$ square grid, also with each cell labeled by a distinct symbol in $S$. Then each symbol in $S$ labels two cells, one in $A$ and one in $B$. Define the \emph{combined distance} between two symbols in $S$ as the distance between the two cells in $A$ plus the distance between the two cells in $B$ that are labeled by the two symbols. Bel\'en Palop asked the following question at the open problems session of CCCG 2009: How to arrange the symbols in the two grids such that the minimum combined distance between any two symbols is maximized? In this paper, we give a partial answer to Bel\'en Palop's question. Define $c_p(n) = \max_{A,B}\min_{s,t \in S} \{\dist_p(A,s,t) + \dist_p(B,s,t) \}$, where $A$ and $B$ range over all pairs of $n\times n$ square grids labeled by the same set $S$ of $n^2$ distinct symbols, and where $\dist_p(A,s,t)$ and $\dist_p(B,s,t)$ are the $L_p$ distances between the cells in $A$ and in $B$, respectively, that are labeled by the two symbols $s$ and $t$. We present asymptotically optimal bounds $c_p(n) = \Theta(\sqrt{n})$ for all $p=1,2,...,\infty$. The bounds also hold for generalizations to $d$-dimensional grids for any constant $d \ge 2$. Our proof yields a simple linear-time constant-factor approximation algorithm for maximizing the minimum combined distance between any two symbols in two grids.
Deep Dive into Spreading grid cells.
Let $S$ be a set of $n^2$ symbols. Let $A$ be an $n\times n$ square grid with each cell labeled by a distinct symbol in $S$. Let $B$ be another $n\times n$ square grid, also with each cell labeled by a distinct symbol in $S$. Then each symbol in $S$ labels two cells, one in $A$ and one in $B$. Define the \emph{combined distance} between two symbols in $S$ as the distance between the two cells in $A$ plus the distance between the two cells in $B$ that are labeled by the two symbols. Bel'en Palop asked the following question at the open problems session of CCCG 2009: How to arrange the symbols in the two grids such that the minimum combined distance between any two symbols is maximized? In this paper, we give a partial answer to Bel'en Palop’s question. Define $c_p(n) = \max_{A,B}\min_{s,t \in S} \{\dist_p(A,s,t) + \dist_p(B,s,t) \}$, where $A$ and $B$ range over all pairs of $n\times n$ square grids labeled by the same set $S$ of $n^2$ distinct symbols, and where $\dist_p(A,s,t)$ an
Let S be a set of n 2 symbols. Let A be an n × n square grid with each cell labeled by a distinct symbol in S. Let B be another n × n square grid, also with each cell labeled by a distinct symbol in S. Then each symbol in S labels two cells, one in A and one in B. Define the combined distance between two symbols in S as the distance between the two cells in A plus the distance between the two cells in B that are labeled by the two symbols. Belén Palop asked the following question at the open problems session of CCCG 2009 [1]: How to arrange the symbols in the two grids such that the minimum combined distance between any two symbols is maximized?
In the original setting of this question as posed by Belén Palop, the two grids A and B are axis-parallel, each grid cell is a unit square, and the distance between two cells is the L 1 distance between the cell centers. Thus the distance between two cells sharing an edge is 1, and the distance between two cells sharing only a vertex is 2. We refer to Figure 1 for an example. Note that the question is also interesting for the other norms L p , p = 2, . . . , ∞, in particular, L ∞ . In this paper, we give a partial answer to Belén Palop’s question. To be precise, let n ≥ 2, and define
where A and B range over all pairs of n × n square grids labeled by the same set S of n 2 distinct symbols, and where dist p (A, s, t) and dist p (B, s, t) are the L p distances between (the centers of) the cells in A and in B, respectively, that are labeled by the two symbols s and t. Our main result is the following theorem:
Our bounds on the minimum combined distance can be generalized to d-dimensional grids for any integer d ≥ 2. Define c d p (n) analogous to (1) except that A and B range over all pairs of n × • • • × n hypercubic grids labeled by the same set S of n d distinct symbols. Then c 2 p (n) = c p (n) for all p = 1, 2, . . . , ∞. Theorem 2 in the following is a straightforward extension to Theorem 1:
Our proof for the lower bound is constructive and, in conjunction with the upper bound, yields a simple linear-time constant-factor approximation algorithm for the optimization problem of maximizing the minimum combined distance between any two symbols in two grids.
In this section, we prove the lower bound c ∞ (n) ≥ 2 n/3 in Theorem 1. For convenience, let
be the set of center coordinates of the grid cells of A, and label each cell of A by its center coordinates. Let
) is a positive integer to be specified. To prove the lower bound, we will construct B from A by moving cells in the same grid such that the combined distance between any two symbols in A and B is Ω(k).
We first consider the special case that n = k 2 for some integer k ≥ 2. Assign a color (i, j) to each cell (x, y) such that i = x mod k and j = y mod k. To transform A into B, we move each cell (x, y) of color (i, j) to a cell (x ′ , y ′ ) of the same color (i, j) such that
(2)
Then each cell in A is moved to a distinct cell in B. The cells of color (0, 0) remain at the same positions in the grid. We refer to Figure 2 for an example.
6,1 4,5 5,5 6,5 7,5 8,5 3,5 0,4 1,4 2,4 4,4 5,4 6,4 7,4 8,4 3,4 0,3 1,3 2,3 4,3 5,3 6,3 7,3 8,3 3,3 0,2 1,2 2,2 4,2 5,2 6,2 7,2 8,2 3,2 0,1 1,1 2,1 4,1 5,1 7,1 8,1 3,1 0,0 1,0 2,0 4,0 5,0 6,0 7,0 8,0 3,0 Consider any two cells (x 1 , y 1 ) and (x 2 , y 2 ) in A that are moved to two cells
The combined L ∞ distance between the corresponding two symbols (x 1 , y 1 ) and (x 2 , y 2 ) is
We will show that this combined distance is at least k. Let
If (i 1 , j 1 ) = (i 2 , j 2 ), then the combined distance is at least 2k because the L ∞ distance between any two cells of the same color is at least k. It remains to show that the combined distance is at least k even if (i 1 , j 1 ) = (i 2 , j 2 ). Assume without loss of generality that
This implies that the two values |i 2i 1 -k| and |i 2i 1 + k| are both at least 1. In summary, we have
Let k be the largest integer such that 3k ≤ ⌈n/k⌉; we will show later that n/3 ≤ k ≤ n/3 . Again assign a color (i, j) to each cell (x, y) such that i = x mod k and j = y mod k. To transform A into B, we move each cell (x, y) of color (i, j) to a cell (x ′ , y ′ ) of the same color (i, j) such that
Then each cell in A is moved to a distinct cell in B. The cells of color (0, 0) remain at the same positions in the grid. We refer to Figure 3 for an example.
In this section, we prove the upper bound c ∞ (n) ≤ √ n -1 + √ n -1 in Theorem 1. Let A and B be two arbitrary n × n square grids labeled by the same set S of n 2 symbols. We will show that there are two symbols in S such that the combined L ∞ distance between them in the two grids A and B is at most Let U be the set of cells in an arbitrary (u + 1) × (u + 1) sub-grid of the n × n grid A, where u is an integer to be specified, 1 ≤ u ≤ n -1. Then the L ∞ distance between any two cells in U is at most u. Let V be the set of cells in B that are labeled by the same symbols that label the cells in
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