DirReduc.java

import java.util.Stack;
class DirReduc {
    static String inverse(String direction) {
        String inv = "";

        switch (direction) {
            case "NORTH": inv = "SOUTH";
                          break;
            case "SOUTH": inv = "NORTH";
                          break;
            case "EAST" : inv = "WEST";
                          break;
            case "WEST" : inv = "EAST";
                          break;

        }

        return inv;
    }

    static String[] dirReduc(String[] arr) {
        int size = arr.length;
        int idx = 0;
        Stack<String> stack = new Stack<String>();

        while (idx < size) {
            if (!stack.empty() && stack.peek() == inverse(arr[idx])) {
                stack.pop();
            } else {
                stack.push(arr[idx]);
            }
            idx++;
        }

        return stack.toArray(new String[stack.size()]);
    }

    static void p(String [] arr) {
        System.out.print("[");
        for (int i = 0; i < arr.length; i++) {
            if (i == arr.length - 1) {
                System.out.print(arr[i]);
            } else System.out.print(arr[i] + ", ");
        }
        System.out.print("]\n");
    }

    public static void main(String [] argc) {
        String[] arr = {"NORTH", "SOUTH", "EAST", "WEST"};
        p(dirReduc(arr));
        String[] arr1 = {"NORTH", "EAST", "WEST", "SOUTH", "WEST", "WEST"};
        p(dirReduc(arr1));
        String[] arr2 = {"NORTH", "WEST", "SOUTH", "EAST"};
        p(dirReduc(arr2));
        String[] arr3 = {"NORTH", "SOUTH", "SOUTH", "EAST", "WEST", "NORTH", "WEST"};
        p(dirReduc(arr3));
        String[] arr4 = {"NORTH", "SOUTH", "SOUTH", "EAST", "WEST", "NORTH"};
        p(dirReduc(arr4));
    }
}

shell.sh

$ javac DirReduc.java 
$ java DirReduc
[]
[WEST, WEST]
[NORTH, WEST, SOUTH, EAST]
[WEST]
[]