Given a quadrilateral ABCD in K_n and the distances of edges, the special frequency quadrilaterals are derived as two of the three sum distances d(A,B)+d(C,D), d(A,C)+d(B,D), and d(A,D)+d(B,C) are equal. A probability model formulated based on the special frequency quadrilaterals implies the edges in the optimal Hamiltonian cycle
are diferent from the other edges in K_n. Christodes proposed a 3/2-approximation algorithm for metric traveling salesman problem (TSP) that runs in O(n^3) time. Cornuejols and Nemhauser constructed a family of graphs where the performance ratio of Christodes algorithm is exactly 3/2 in the worst case. We apply the special frequency quadrilaterals to the family of metric TSP instances for cutting the useless edges. In the end, the complex graph is reduced to a simple graph where the optimal Hamiltonian cycle can be detected in O(n) time, where n >= 4 is the number of vertices in the graph.