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 while taking into consideration data preservation characteristics of the plurality of operators and taking into consideration structural-order preservation characteristics of the plurality of operators by: obtaining a set of operators on which the operator depends; obtaining a set of vector nodes of the hierarchical data in an input schema of the operator, wherein a vector node represents a vector including scalars that are represented by a set of scalar nodes; and for each of the vector nodes, analyzing the set of scalar nodes to determine the dependencies, wherein a scalar node represents a scalar; 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 by inserting at least one context split operator and at least one context merge operator into the new graph.