Distance between distance between [ x_0,y_00] to [x_10,y_100] coordinates in Cell array

I have a cell array having [x,y] coordinates as shown below.

{{[ x_0,y_00]},
{[ x_1,y_10],[ x_1,y_11],[ x_1,y_12],[ x_1,y_13],[ x_1,y_14],[ x_1,y_15]},
{[ x_2,y_20],[ x_2,y_21],[ x_2,y_22],[ x_2,y_23],[ x_2,y_24],[ x_2,y_25]},
{[ x_3,y_30],[ x_3,y_31],[ x_3,y_32],[ x_3,y_33],[ x_3,y_34],[ x_3,y_35]},
{[ x_4,y_40],[ x_4,y_41],[ x_4,y_42],[ x_4,y_43],[ x_4,y_44],[ x_4,y_45]},
{[ x_5,y_50],[ x_5,y_51],[ x_5,y_52],[ x_5,y_53],[ x_5,y_54],[ x_5,y_55]},
{[ x_6,y_60],[ x_6,y_61],[ x_6,y_62],[ x_6,y_63],[ x_6,y_64],[ x_6,y_65]},
{[ x_7,y_70],[ x_7,y_71],[ x_7,y_72],[ x_7,y_73],[ x_7,y_74],[ x_7,y_75]},
{[ x_8,y_80],[ x_8,y_81],[ x_8,y_82],[ x_8,y_83],[ x_8,y_84],[ x_8,y_85]},
{[ x_9,y_90],[ x_9,y_91],[ x_9,y_92],[ x_9,y_93],[ x_9,y_94],[ x_9,y_95]},
{[x_10,y_100]}}

The values of y coordinates are random. How can I find the minimum distance between [ x_0,y_00] to [x_10,y_100]. distance? Can someone please guide me.

[Guillaume]

The question is really badly explained. If I understand correctly, your cell array represent a directly acyclic graph where the points on each layer of your upper level cell array is connected to each point on the next layer, and you want to find the shortest path going from the 1st cell to the last cell using only one point from each intermediate cell array.

Is that correct?

[rest12]

Yes exactly

Answer 1

[Guillaume]

I would build a digraph from your cell array, where each node is a point and the edges are weighted by the distance between the points. The difficult bit will be to build the graph but once that is done, the shortest path is trivially found using shortestpath.

%demo cell array, x goes from 0 to 10, y is random between -10 and 10
c = [{{[0, 0]}}; arrayfun(@(x) num2cell([repmat(x, 5, 1), sort(randperm(21, 5))'-11], 2)', (1:9)', 'UniformOutput', false); {{[10, 0]}}]

%build graph:
g = digraph;   %empty graph to start with
vartnodenames = {'ID', 'X', 'Y'};  %variable names for node table
%add 1st layer as a node
g = g.addnode(table(1, c{1}{1}(1), c{1}{1}(2), 'VariableNames', vartnodenames));
lastlayerids = 1; %will be used in a loop to build edges
lastnode = 1;  %node index last inserted
lastcoordinates = c{1}{1};  %coordinates of points of previous layer in the loop
%add other layers
for layer = 2:numel(c)
   newlayerids = lastnode + (1:numel(c{layer}));  %ids of nodes in new layer
   newcoordinates = vertcat(c{layer}{:});         %coordinate of nodes in new layer
   g = g.addnode(table(newlayerids', newcoordinates(:, 1), newcoordinates(:, 2), 'VariableNames', vartnodenames));
   [s, e] = ndgrid(lastlayerids, newlayerids);   %cartesian product of nodes from previous layer and nodes from new layer => edges between the two layers
   distances = hypot(lastcoordinates(:, 1) - newcoordinates(:, 1)', lastcoordinates(:, 2) - newcoordinates(:, 2)');   %weigth for edges = distance between layer nodes
   g = g.addedge(s, e, distances);
   lastnode = lastnode + numel(c{layer});         %update for next layer
   lastlayerids = newlayerids;
   lastcoordinates = newcoordinates;
end

%find shortest path between 1st and last node
[nodes, distance, edges] = shortestpath(g, 1, lastnode);

%plot
h = plot(g, 'XData', g.Nodes.X, 'YData', g.Nodes.Y);
highlight(h, nodes);
highlight(h, 'Edges', edges);

Answer 2

[KSSV]

A = {{rand(2,1)},{rand(2,1)},{rand(2,1)},{rand(2,1)},{rand(2,1)}} ;
A = cell2mat([A{:}]) ;
A = A';
d = pdist2(A(1,:),A) ;

[val,idx] = max(d) ;
iwant = A(idx,:)

[rest12]

thanks for your reply. But I think maybe my question is not very clear. Let me try to explain to you here. I need to find the minimum total distance between [ x_0,y_00] to [x_10,y_100]. Since there are different coordinates between [ x_0,y_00] to [x_10,y_100] the distance value could be different. So I need to find the minimum distance path using these points.

For example. the minimum distance path could be:
[ x_0,y_00] --> [ x_1,y_10] --> [ x_2,y_22] --> [ x_3,y_33] --> [ x_4,y_41] --> [ x_5,y_55] ........... [x_10,y_100]
Or it could be :
[ x_0,y_00] --> [ x_1,y_18] --> [ x_2,y_23] --> [ x_3,y_38] --> [ x_4,y_47] --> [ x_5,y_50] ........... [x_10,y_100]