我在这方面做了一些工作，因为我也需要类似的东西，但我推迟了算法的开发。你帮我得到了一些冲动：D
我还需要源代码，所以在这里。我在 Mathematica 中解决了它，但由于我没有大量使用函数特性，我想它很容易翻译成任何程序语言。
历史观点
首先我决定开发圆的算法，因为交点更容易计算。它仅取决于中心和半径。
我能够使用 Mathematica 方程求解器，它的性能很好。
看看：
这很容易。我刚刚加载了具有以下问题的求解器：
For each circle
Solve[
Find new coördinates for the circle
Minimizing the distance to the geometric center of the image
Taking in account that
Distance between centers > R1+R2 *for all other circles
Move the circle in a line between its center and the
geometric center of the drawing
]
就这么简单，Mathematica 完成了所有工作。
我说“哈！这很简单，现在让我们来看看矩形吧！”。但我错了……
矩形蓝色
矩形的主要问题是查询交叉点是一个讨厌的函数。比如：
因此，当我试图用很多这些条件来为 Mathematica 提供方程时，它的表现非常糟糕，以至于我决定做一些程序性的事情。
我的算法最终如下：
Expand each rectangle size by a few points to get gaps in final configuration
While There are intersections
sort list of rectangles by number of intersections
push most intersected rectangle on stack, and remove it from list
// Now all remaining rectangles doesn't intersect each other
While stack not empty
pop rectangle from stack and re-insert it into list
find the geometric center G of the chart (each time!)
find the movement vector M (from G to rectangle center)
move the rectangle incrementally in the direction of M (both sides)
until no intersections
Shrink the rectangles to its original size
您可能会注意到“最小移动”条件并未完全满足（仅在一个方向上）。但是我发现向任何方向移动矩形来满足它，有时最终会导致用户感到困惑的地图发生变化。
在设计用户界面时，我选择将矩形移动得更远一点，但以更可预测的方式。您可以更改算法以检查其当前位置周围的所有角度和所有半径，直到找到一个空白位置，尽管它会要求更高。
无论如何，这些是结果示例（之前/之后）：
编辑>更多示例here
如您所见，“最小运动”并不满足，但结果已经足够好。
我将在此处发布代码，因为我的 SVN 存储库遇到了一些问题。问题解决后我会删除它。
编辑：
您也可以使用R-Trees 来查找矩形交点，但对于处理少量矩形来说似乎有点过头了。而且我还没有实现算法。也许其他人可以将您指向您选择的平台上的现有实现。
警告！代码是第一种方法.. 质量还不是很好，而且肯定有一些错误。
这是数学。
(*Define some functions first*)
Clear["Global`*"];
rn[x_] := RandomReal[{0, x}];
rnR[x_] := RandomReal[{1, x}];
rndCol[] := RGBColor[rn[1], rn[1], rn[1]];
minX[l_, i_] := l[[i]][[1]][[1]]; (*just for easy reading*)
maxX[l_, i_] := l[[i]][[1]][[2]];
minY[l_, i_] := l[[i]][[2]][[1]];
maxY[l_, i_] := l[[i]][[2]][[2]];
color[l_, i_]:= l[[i]][[3]];
intersectsQ[l_, i_, j_] := (* l list, (i,j) indexes,
list={{x1,x2},{y1,y2}} *)
(*A rect does intesect with itself*)
If[Max[minX[l, i], minX[l, j]] < Min[maxX[l, i], maxX[l, j]] &&
Max[minY[l, i], minY[l, j]] < Min[maxY[l, i], maxY[l, j]],
True,False];
(* Number of Intersects for a Rectangle *)
(* With i as index*)
countIntersects[l_, i_] :=
Count[Table[intersectsQ[l, i, j], {j, 1, Length[l]}], True]-1;
(*And With r as rectangle *)
countIntersectsR[l_, r_] := (
Return[Count[Table[intersectsQ[Append[l, r], Length[l] + 1, j],
{j, 1, Length[l] + 1}], True] - 2];)
(* Get the maximum intersections for all rectangles*)
findMaxIntesections[l_] := Max[Table[countIntersects[l, i],
{i, 1, Length[l]}]];
(* Get the rectangle center *)
rectCenter[l_, i_] := {1/2 (maxX[l, i] + minX[l, i] ),
1/2 (maxY[l, i] + minY[l, i] )};
(* Get the Geom center of the whole figure (list), to move aesthetically*)
geometryCenter[l_] := (* returs {x,y} *)
Mean[Table[rectCenter[l, i], {i, Length[l]}]];
(* Increment or decr. size of all rects by a bit (put/remove borders)*)
changeSize[l_, incr_] :=
Table[{{minX[l, i] - incr, maxX[l, i] + incr},
{minY[l, i] - incr, maxY[l, i] + incr},
color[l, i]},
{i, Length[l]}];
sortListByIntersections[l_] := (* Order list by most intersecting Rects*)
Module[{a, b},
a = MapIndexed[{countIntersectsR[l, #1], #2} &, l];
b = SortBy[a, -#[[1]] &];
Return[Table[l[[b[[i]][[2]][[1]]]], {i, Length[b]}]];
];
(* Utility Functions*)
deb[x_] := (Print["--------"]; Print[x]; Print["---------"];)(* for debug *)
tableForPlot[l_] := (*for plotting*)
Table[{color[l, i], Rectangle[{minX[l, i], minY[l, i]},
{maxX[l, i], maxY[l, i]}]}, {i, Length[l]}];
genList[nonOverlap_, Overlap_] := (* Generate initial lists of rects*)
Module[{alist, blist, a, b},
(alist = (* Generate non overlapping - Tabuloid *)
Table[{{Mod[i, 3], Mod[i, 3] + .8},
{Mod[i, 4], Mod[i, 4] + .8},
rndCol[]}, {i, nonOverlap}];
blist = (* Random overlapping *)
Table[{{a = rnR[3], a + rnR[2]}, {b = rnR[3], b + rnR[2]},
rndCol[]}, {Overlap}];
Return[Join[alist, blist] (* Join both *)];)
];
主要
clist = genList[6, 4]; (* Generate a mix fixed & random set *)
incr = 0.05; (* may be some heuristics needed to determine best increment*)
clist = changeSize[clist,incr]; (* expand rects so that borders does not
touch each other*)
(* Now remove all intercepting rectangles until no more intersections *)
workList = {}; (* the stack*)
While[findMaxIntesections[clist] > 0,
(*Iterate until no intersections *)
clist = sortListByIntersections[clist];
(*Put the most intersected first*)
PrependTo[workList, First[clist]];
(* Push workList with intersected *)
clist = Delete[clist, 1]; (* and Drop it from clist *)
];
(* There are no intersections now, lets pop the stack*)
While [workList != {},
PrependTo[clist, First[workList]];
(*Push first element in front of clist*)
workList = Delete[workList, 1];
(* and Drop it from worklist *)
toMoveIndex = 1;
(*Will move the most intersected Rect*)
g = geometryCenter[clist];
(*so the geom. perception is preserved*)
vectorToMove = rectCenter[clist, toMoveIndex] - g;
If [Norm[vectorToMove] < 0.01, vectorToMove = {1,1}]; (*just in case*)
vectorToMove = vectorToMove/Norm[vectorToMove];
(*to manage step size wisely*)
(*Now iterate finding minimum move first one way, then the other*)
i = 1; (*movement quantity*)
While[countIntersects[clist, toMoveIndex] != 0,
(*If the Rect still intersects*)
(*move it alternating ways (-1)^n *)
clist[[toMoveIndex]][[1]] += (-1)^i i incr vectorToMove[[1]];(*X coords*)
clist[[toMoveIndex]][[2]] += (-1)^i i incr vectorToMove[[2]];(*Y coords*)
i++;
];
];
clist = changeSize[clist, -incr](* restore original sizes*);
HTH！
编辑：多角度搜索
我对算法进行了更改，允许在所有方向上进行搜索，但优先考虑几何对称性强加的轴。
以更多周期为代价，这导致了更紧凑的最终配置，如下所示：
更多示例here。
主循环的伪代码改为：
Expand each rectangle size by a few points to get gaps in final configuration
While There are intersections
sort list of rectangles by number of intersections
push most intersected rectangle on stack, and remove it from list
// Now all remaining rectangles doesn't intersect each other
While stack not empty
find the geometric center G of the chart (each time!)
find the PREFERRED movement vector M (from G to rectangle center)
pop rectangle from stack
With the rectangle
While there are intersections (list+rectangle)
For increasing movement modulus
For increasing angle (0, Pi/4)
rotate vector M expanding the angle alongside M
(* angle, -angle, Pi + angle, Pi-angle*)
re-position the rectangle accorging to M
Re-insert modified vector into list
Shrink the rectangles to its original size
为了简洁起见，我不包括源代码，但如果您认为可以使用它，请询问它。我认为，如果你这样做，最好切换到 R-trees（这里需要大量的间隔测试）