Given nonnegative real numbers a distance labeling with constraint (or simply an -labeling) of a graph G is a function such that whenever the distance between u and v is i for The span of an labeling f is the difference between the maximum value and the minimum value of f. We seek the infimum span over all possible -labelings. It is called the -span of G and is denoted by By the property of span, to study the -labelings, it suffices to consider those labelings with This article studies -labelings for paths.