Suppose I have a $(2n+1)\times (2n+1)$ matrix $Y$, indexed by $\{-n, \cdots, n \} \times \{-n, \cdots, n \}$.
And I have another set of vectors $X = (x_1, x_2, \cdots, x_n)$, where $x_i \in \mathbb{R}^k$, where $k\leq n$.
The operation I'm interested in:
1, fill the $Y_0$ with all $+\infty $, except for $Y_0[0,0] = 0$.
2, iteratively get $x_t \in X$.
3, "convolutional sum", update
$$
Y_t[i, j] = min( Y_{t-1}[i-o, j] + x_t[o], \\
Y_{t-1}[i+o, j] + x_t[o], \\
Y_{t-1}[i, j-o] + x_t[o], \\
Y_{t-1}[i, j+o] + x_t[o], \\
Y_{t-1}[i, j]),\\
\forall o \in [0, k]
$$
Is there a standard name for this operation?
To illustrate the operation scope, I made a visual plotting with $n=15, k=5$. And I only keep the first 3 steps.
I've found the 1d version, it's called the (min,+) convolution. Details in communication networking
Define the (min,+) convolution of two real valued functions (domain is $\mathbb{R}^+$) f,g as $$f*g= \inf_{0 \leq s \leq t}\{ f(s) + g(t-s)\}$$
Also in algorithms, the 1d version is MinCov problem.
Input: Sequences: $(a[i])^{n−1}_{i=0} , (b[i])^{n−1}_{i=0}$
Task: Output sequence $(c[i])^{n−1}_{i=0}$ , such that $c[k] = \text{min}_{i+j=k}(a[i] + b[j])$