1. A computer-implemented method, comprising: receiving a linear processing graph, wherein the linear processing graph includes a plurality of operators, wherein each operator in the plurality is connected to at least one other operator by an arc, wherein hierarchical data flows on arcs, wherein the operators read and replace identified subregions within the hierarchical data flowing into the operators on the arcs, and wherein the operators do not modify the hierarchical data outside of these identified subregions; for each operator in the linear processing graph, finding a minimal set of dependent upstream operators on which that operator depends by examining how the identified subregions are created in the linear processing graph through obtaining a set of operators on which that operator depends, by analyzing dependencies carried by a set of vector nodes of the hierarchical data in an input schema of the operator, and, for each of the vector nodes, by analyzing an associated set of scalar nodes, wherein finding the minimum set of operators includes taking into consideration data preservation characteristics of the plurality of operators and taking into consideration structural-order preservation characteristics of the plurality of operators; and rewriting the linear processing graph to create a new graph that expresses dependencies based on the minimal set of dependent upstream operators for each operator.