How to sort lines into continous paths efficiently

Question

As part of my work I get a bunch of lines from my co-worker. (Background: I do electromagnetic designs, my co-worker does the mechanical design). These lines are in random order, but I need them in continous paths, with all line segments running in the same direction.

I've created a small example, real work consists of several thousand lines.

Create some test data:

(* create some test data *)
spiral1 = N@Table[{p Cos[p], p Sin[p]}, {p, Pi, 4 Pi, Pi/16}];
spiral2 = 
  N@Table[{p Cos[p + Pi], p Sin[p + Pi]}, {p, Pi, 5 Pi, Pi/16}];
lines = Join[Partition[spiral1, 2, 1], Partition[spiral2, 2, 1]];
(* scramble *)
randomLines = 
  MapAt[Reverse, RandomSample@lines, 
   List /@ RandomChoice[Range[Length[lines]], 
     Floor[Length[lines]/2]]];

And see what we've got:

Graphics[Table[{Hue[i/Length[randomLines]], 
   Arrow@randomLines[[i]]}, {i, Length[randomLines]}] ]

These lines are now sorted (code below):

sorted = mySort[randomLines];

The random lines were sorted into seperate paths with a continous direction of the lines. (The total direction of each path is still random, but that's OK for me).

Looks as expected:

Graphics[Table[{PointSize[0.05], Hue[i/Length[sorted]], 
   Point[sorted[[i, 1, 1]]], Arrow@sorted[[i]], GrayLevel[0], 
   Text[i, sorted[[i, 1, 1]], {0, 0}]}, {i, Length[sorted]}] ]

My question now is, how can I make my prodedural code more Mathematica-like? And hopefully more efficient? For real work, my code is too slow.

Code:

mySort[in : {{{_?NumberQ ..}, {_?NumberQ ..}} ..}] := Module[
  {sorted, pool, this, found,
   tol = 10.0*^-9},
(* start *)
  sorted = {{First[in]}};
  pool = Rest[in];
While[Length[pool] > 0,
found = False;
   (* use the last of all sorted paths that were found so far,
      the other paths are already complete *)
For[i = 1, i <= Length[pool] && Not[found], i++,
    this = pool[[i]];
    (* check if this fits onto the sorted line path )
    found = Which[
      ( this fits on end )
      Norm[sorted[[-1, -1, -1]] - this[[1]]] < tol,
        sorted[[-1]] = Append[sorted[[-1]], this];
        True,
      ( reversed this fits on end )
      Norm[sorted[[-1, -1, -1]] - this[[-1]]] < tol,
        sorted[[-1]] = Append[sorted[[-1]], Reverse@this];
        True,
      ( this fits on beginning )
      Norm[sorted[[-1, 1, 1]] - this[[-1]]] < tol,
        sorted[[-1]] = Prepend[sorted[[-1]], this];
        True,
      ( reversed this fits on beginning )
      Norm[sorted[[-1, 1, 1]] - this[[1]]] < tol,
        sorted[[-1]] = Prepend[sorted[[-1]], Reverse@this];
        True,
      ( nothing found )
      True,
        False
      ]( end of which );
    ]( end of for *);
(* remove from unsorted list *)
   pool = DeleteCases[pool, this];
If[Not[found],
    (* need to start a new path )
    sorted = Append[sorted, {this}];
    ];
   ]( end of while *);
(* return result *)
  sorted
  ]

Edit:

Real world data:

randomLines = {{{-0.1732811503690476`, 
     0.3235849266204875`}, {-0.17359189280402723`, 
     0.3225643511484479`}}, {{-0.003790604496207962`, 
     0.330308955004866`}, {-0.004423377968477279`, 
     0.331138019882757`}}, {{-0.1609083437046747`, 
     0.32692061568805486`}, {-0.16038815001632004`, 
     0.3272754550920334`}}, {{-0.15872690806364326`, 
     0.3281712493076333`}, {-0.1592959532338122`, 
     0.3279016198019645`}}, {{-0.0274412108919291`, 
     0.29769071153521204`}, {-0.02718659256118707`, 
     0.29769509804809235`}}, {{-0.005814165101646035`, 
     0.33269164676889584`}, {-0.0050985101499539795`, 
     0.3319329710614432`}}, {{-0.13803406459546397`, 
     0.3404086542534827`}, {-0.145936615236459`, 
     0.339332441377564`}}, {{-0.022642808371541714`, 
     0.32717602179538546`}, {-0.022729514890690448`, 
     0.32742192097588857`}}, {{-0.16278418842931225`, 
     0.3252461150235488`}, {-0.16319556509908598`, 
     0.3247693743061922`}}, {{-0.17411145452597812`, 
     0.31410618140280167`}, {-0.17392795919535808`, 
     0.3130552462865045`}}, {{-0.132713447337878`, 
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     0.31064530270648133`}}, {{-0.0006761538014226468`, 
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Looks random:

Graphics[Table[{Hue[i/Length[randomLines]], 
   Arrow@randomLines[[i]]}, {i, Length[randomLines]}] ]

Sort into paths:

sorted = mySort[randomLines];

This is what I expect:

Graphics[{
  Arrowheads[.02],
  Table[{PointSize[0.015], Hue[i/Length[sorted]], 
    Point[sorted[[i, 1, 1]]], Arrow@sorted[[i]], GrayLevel[0], 
    Text[i, sorted[[i, 1, 1]], {0, 0}]}, {i, Length[sorted]}]
  }]

Answer 1

I think this should do the trick and also should have decent performance:

TOL = 0.5 Sqrt[Min[Total[(randomLines[[All, 1]] - randomLines[[All, 2]])^2, {2}]]];
(*pts=DeleteDuplicates[Flatten[randomLines,1],Norm[#1-#2] <TOL&];*)
pts = Flatten[randomLines, 1];
clusters = DeleteDuplicates[Sort /@ Nearest[pts -> "Index", pts, {[Infinity], TOL}]];
pts = pts[[clusters[[All, 1]]]];
edges = Partition[Flatten[Nearest[pts -> "Index", Flatten[randomLines, 1], {[Infinity], TOL}]], 2];
A = KirchhoffMatrix[edges];
chains = Table[
  Module[{L, cycles, perm},
   L = A[[c, c]];
   cycles = SparseArrayHamiltonianCycle[L]; If[Length[cycles] == 0, perm = SparseArrayMinimumBandwidthOrdering[L][[2, 1]];
    ,
    perm = cycles[[1]];
    ];
   pts[[c[[perm]]]]
   ],
  {c, SparseArray`StronglyConnectedComponents[A]}];
Graphics[Table[{RandomColor[], Arrow[c]}, {c, chains}]]

For the synthetic example, we obtain the following:

For the real world example, we get:

This is how this works:

• First we determine the smallest edge length in order to determine a reasonable tolerance parameter TOL. We assume from here on that the distance between points that are to be identified with each other aare considerably smaller than this minimal edge length.

• We use DeleteDuplicates Nearest to perform the actual identification with respect to this tolerance parameter.

• We determine the edges edges of the underlying graph (again using the tolerance TOL).

• We build the graph Laplacian A of the underlying graph (omitting the creation of Graph data structure for performance reasons - as usual).

• Then we go through the connected compontents and consider each component's graph Laplacian L separately. If we find a Hamiltonian cycle, we use it as reordering of that component. Otherwise, the connected component c is a linear chain. In that case, we use the minimum bandwidth reordering. This works because a linear chain must have a reordering of vertices that brings the Laplacian into tridiagonal form. And the graph Laplacian can be tridiagonal only if the indices of neighboring vertices differ by $\pm 1$. So any one of the two possible minimal bandwidth reorderings for c will do.