One embodiment of the present invention provides a system that routes a set of pairs of points during the design of an integrated circuit (IC) chip. The system comprises a routing engine which is configured to search for a path to connect a current pair of points in the set of pairs of points, wherein the path comprises a set of rectangles and vertices. The routing engine uses a routing database, which keeps track of previously routed nets that can obstruct the routing of the current pair of points. The system further comprises a satisfiability (SAT) solver which is capable of solving a set of constraints, wherein the set of constraints are associated with the routability of the set of pairs of points. The SAT solver additionally comprises a SAT database which maintains the set of constraints and a current partial solution to the set of constraints. The SAT database is used to update the routing database if the current partial solution changes.