How to generate a random walk along the edges in a hexagon grid?

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Retam Paul
Retam Paul 2023-7-24
评论： Retam Paul 2023-7-31
I wrote the following function to generate a 2D hexagonal grid taking inputs a1, a2 (the direction cosines of two edges of a hexagon) and the number of hexagons on each side. I need to generate a random walk such that the path is only along the edges of the hexagon and moving forward (along the positive x-axis, the path should not trace back to the origin). Can anyone help me out? I tried biasing the paths but it didn't work. Attaching the code for generation of the hexagon grid. A simple execution of the code would be:
>> a1=[1 0];
>> a2=[-1/2 sqrt(3)/2];
>> h=10;
>> top=3;
>> btm=3;
>> hexmesh(a1,a2,h,top,btm);
function[vertices] = hexmesh(a1,a2,h,top,btm)
% a1=First vector
% a2=Second vector
% h=cells horizontal
% top=cells above
% btm=cells below
% This function plots a hexagonal grid having h*(top+btm) cells with
% a1 and a2 being the two vectors comprising the first hexagon in the
% formula hex = a0 + m*a1 + n*a2 where m,n are integers
%% Drawing the top half
numhex=0;
vertices=[];
for ii=1:h % horizontal cells
for jj=1:top % number of cells at top half
a1=[1 0];
a2=[-1/2 sqrt(3)/2];
if rem(ii-1,2) == 0
transh=(3/2)*(ii-1)*a1;
end
if rem(ii-1,2) == 1
transh=((3/2)*ii-1)*a1+ a2;
end
transv=(jj-1)*(a1+2*a2);
x1=0*a1+0*a2 + transh + transv;
x2=1*a1+0*a2 + transh + transv;
x3=0*a1+1*a2 + transh + transv;
x4=2*a1+1*a2 + transh + transv;
x5=1*a1+2*a2 + transh + transv;
x6=2*a1+2*a2 + transh + transv;
vxt=0.25.*[x1(1) x3(1) x5(1) x6(1) x4(1) x2(1) x1(1)]-0.124181; % set of x-coor points from a single polygon in top half
vyt=0.25.*[x1(2) x3(2) x5(2) x6(2) x4(2) x2(2) x1(2)];
plot(vxt,vyt,'k'); hold on;
numhex=numhex+1;
dum=[vxt(1:6);vyt(1:6)];
vertices=[vertices dum];
end
end
axis equal;
%% Drawing bottom half
a2(2)=-a2(2);
for kk=1:h
for ll=1:btm % number of cells in btm half
if rem(kk-1,2) == 0
transh=(3/2)*(kk-1)*a1;
end
if rem(kk-1,2) == 1
transh=((3/2)*kk-1)*a1+ a2;
end
transv=(ll-1)*(a1+2*a2);
x11m=0*a1+0*a2 + transh + transv;
x22m=1*a1+0*a2 + transh + transv;
x33m=0*a1+1*a2 + transh + transv;
x44m=2*a1+1*a2 + transh + transv;
x55m=1*a1+2*a2 + transh + transv;
x66m=2*a1+2*a2 + transh + transv;
vxb=0.25.*[x11m(1) x33m(1) x55m(1) x66m(1) x44m(1) x22m(1) x11m(1)]-0.124181; % set of x-coor points from a single polygon in bottom half
vyb=0.25.*[x11m(2) x33m(2) x55m(2) x66m(2) x44m(2) x22m(2) x11m(2)];
plot(vxb,vyb,'k');
numhex=numhex+1;
dum=[vxb(1:6);vyb(1:6)];
vertices=[vertices dum];
end
end
grid on
%% Drawing middle hexagons
% a3,a4=vectors comprising the middle row of hexagons
a3=[-a2(1) a2(2)];
a4=[-a2(1) -a2(2)];
for pp=1:h/2
transh=3*(pp-1)*a1;
x11m=1*a3+1*a4 + transh;
x22m=2*a3+1*a4 + transh;
x33m=3*a3+2*a4 + transh;
x44m=3*a3+3*a4 + transh;
x55m=2*a3+3*a4 + transh;
x66m=1*a3+2*a4 + transh;
vxm=0.25.*[x11m(1) x22m(1) x33m(1) x44m(1) x55m(1) x66m(1) x11m(1)]-0.124181;
vym=0.25.*[x11m(2) x22m(2) x33m(2) x44m(2) x55m(2) x66m(2) x11m(2)];
plot(vxm,vym,'k');
numhex=numhex+1;
dum=[vxm(1:6);vym(1:6)];
vertices=[vertices dum];
end

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采纳的回答

Debadipto
Debadipto 2023-7-31
Hi Retam,
You can use the following function to generate a random walk along the edges in the hexagon grid. The function takes as input the "vertices" vector generated by the "hexmesh" function.
function random_walk(vertices)
v = unique(round(vertices', 4), 'rows');
l = 1;
r = 1;
i = 1;
d = dictionary;
temp_len = 0;
% the vertices and edges of the hexagon can be seen as a graph
% so we create a directed adjacency list of vertices of the hexagon
while r < size(v, 1)
while r <= size(v, 1) && v(l, 1) == v(r, 1)
r = r + 1;
end
if temp_len == 0
temp_len = r - 1;
end
stop = r;
if mod(i, 2) == 0
while r <= size(v, 1) && l < stop
d(num2str([v(l, 1) v(l, 2)])) = num2str([v(r, 1) v(r, 2) v(r, 1) v(r, 2)]);
r = r + 1;
l = l + 1;
end
elseif i == 1 || mod(i - 3, 4) == 0
while r <= size(v, 1) && l < stop
d(num2str([v(l, 1) v(l, 2)])) = num2str([v(r, 1) v(r, 2) v(r + 1, 1) v(r + 1, 2)]);
r = r + 1;
l = l + 1;
end
else
d(num2str([v(l, 1) v(l, 2)])) = num2str([v(r, 1) v(r, 2) v(r, 1) v(r, 2)]);
r = r + 1;
l = l + 1;
while r <= size(v, 1) && l < stop - 1
d(num2str([v(l, 1) v(l, 2)])) = num2str([v(r - 1, 1) v(r - 1, 2) v(r, 1) v(r, 2)]);
r = r + 1;
l = l + 1;
end
d(num2str([v(l, 1) v(l, 2)])) = num2str([v(r - 1, 1) v(r - 1, 2) v(r - 1, 1) v(r - 1, 2)]);
l = l + 1;
end
i = i + 1;
end
x = [];
y = [];
rnd = randi(temp_len, 1, 1);
vertex = num2str([v(rnd, 1) v(rnd, 2)]);
x(end + 1) = v(rnd, 1);
y(end + 1) = v(rnd, 2);
% generate a random walk across the nodes of the graph
while isKey(d, vertex)
rnd = randi(2, 1, 1);
temp = str2num(d(vertex));
next_vertex = [];
if mod(rnd, 2) == 0
next_vertex = [temp(1, 1) temp(1, 2)];
else
next_vertex = [temp(1, 3) temp(1, 4)];
end
x(end + 1) = next_vertex(1, 1);
y(end + 1) = next_vertex(1, 2);
vertex = num2str(next_vertex);
end
plot(x, y, 'r', 'LineWidth', 2);
end
The function can be used as illustrated in the following code snippet
>> a1=[1 0];
>> a2=[-1/2 sqrt(3)/2];
>> h=10;
>> top=3;
>> btm=3;
>> vertices = hexmesh(a1,a2,h,top,btm);
>> random_walk(vertices);
Generated random walk:
Regards,
Debadipto Biswas

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