Self-Aligned Quadruple Patterning (SAQP) is an important manufacturing technique for sub 14nm technology node. Although various routing algorithms for SAQP have been proposed, it is not easy to find a dense SAQP compliant routing pattern efficiently. Even though a grid for SAQP compliant routing pattern was proposed, it is not easy to find a valid routing pattern on the grid. The routing pattern of SAQP on the grid consists of three types of routing. Among them, third type has turn prohibition constraint on the grid. Typical routing algorithms often fail to find a valid routing for third type. In this paper, a simple directed grid-graph for third type is proposed. Valid SAQP compliant two dimensional routing patterns are found effectively by utilizing the proposed directed grid-graph. Experiments show that SAQP compliant routing patterns are found efficiently by our proposed method.