documentclass{article}
usepackage{tikz}

newcommand{NEpath}[4]{
    fill[white!25]  (#1) rectangle +(#2,#3);
    fill[fill=white]
    (#1)
    foreach dir in {#4}{
        ifnumdir=0
        -- ++(1,0)
        else
        -- ++(0,1)
        fi
    } |- (#1);
    draw[help lines] (#1) grid +(#2,#3);
    draw[dashed] (#1) -- +(#3,#3);
    coordinate (prev) at (#1);
    foreach dir in {#4}{
        ifnumdir=0
        coordinate (dep) at (1,0);
        else
        coordinate (dep) at (0,1);
        fi
        draw[line width=2pt,-stealth] (prev) -- ++(dep) coordinate (prev);
    };
}

begin{document}
    begin{tikzpicture}
    NEpath{0,0}{6}{6}{1,1,0,1,1,0,0,0,1,0,1,0};
    end{tikzpicture}
end{document}