minSij

minSij is a mathematical notation used to denote the minimum value of a family of numbers Sij indexed by i and j. In compact form, minSij = min_{i ∈ I, j ∈ J} Sij, where I and J specify the index sets. If S is a matrix with rows indexed by i and columns by j, minSij is simply the smallest entry in the matrix. If S is a function S(i, j) defined on a finite domain, minSij is the minimum value of that function over the domain.

In many contexts Sij represents a score, cost, energy, or similarity between pairs (i, j). The minimum over i and j captures the most favorable or least costly pairing, depending on the interpretation. For example, in an assignment or transportation problem, minSij identifies the lowest cost connection among all possible i–j pairs. In clustering or graph-related tasks, it might be used to determine the weakest link or the closest pair according to S.

Computationally, calculating minSij involves scanning the relevant entries of S and keeping track of the smallest seen value. The time complexity is generally O(|I|·|J|) for a dense matrix. Handle special cases carefully: if I or J is empty, the minimum is undefined; if S contains NaNs, some environments propagate NaN as the result unless explicitly handled.

Related notions include maxSij for the maximum value, and argmin techniques that return the pair (i, j) achieving the minimum rather than the value itself.