多边形连通和最小生成树本质上是一样的，问题在于确定权值。
下面算法由js实现，演示由svg提供。
<html>
<head>
<script>
let shown='hidden';
//核心算法
let caculatePath=function(){
/*显示和隐藏，算法无关*/
for(let i=0;i<panel.children.length;i++){
panel.children[i].style.visibility=shown;
}
lineGroup.innerHTML=null;
if(shown=='hidden'){
shown='visible';
}else{
shown='hidden';
return;
}
getEvent().target.style.visibility='visible';
let gons=getPolygons(getEvent().target);
let target=document.getElementById('target');
/*----------END----------*/
//closeList表示已经连通的图形，无需再连通
let closeList=[];
//未连通的图形
let openList=[];
//放入外框图形
closeList.push(gons[0]);
//放入其他图形
for(let i=1;i<gons.length;i++){
openList.push(gons[i]);
}
let cache={};
while(openList.length>0){
let min={dist:Number.MAX_SAFE_INTEGER};
let imin,jmin;
for(let i=0;i<openList.length;i++){
for(let j=0;j<closeList.length;j++){
//缓存之前的计算值,提高计算效率，这里应该利用最小堆，但是js没有默认实现，先不管
let cacheKey=openList[i].index+':'+closeList[j].index;
let d=cache[cacheKey];
if(!d){
d=polygon2Polygon(openList[i],closeList[j]);
}
cache[cacheKey]=d;
if(d.dist<min.dist){
min=d;
imin=i;
jmin=j;
}
}
}
//构建父子级
if(closeList[jmin].child==null){
closeList[jmin].child=[];
}
closeList[jmin].child.push(
{
node:openList[imin],
target:min.target,
source:min.source,
});
closeList.push(openList[imin]);
openList.splice(imin,1);
}
//测试
collect(gons[0],getEvent().target);
return gons[0]
};
/**
* 返回{ dist:最小距离,source:parent链接点，target：child链接点}
* @param poly1
* @param poly2
*/
let polygon2Polygon=function(poly1,poly2){
let point,min={dist:Number.MAX_SAFE_INTEGER};
for(let i=0;i<poly1.length;i++){
point=poly1[i];
let d=point2Polygon(point,poly2);
if(d.dist<min.dist){
min=d;
min.target=point;
min.source=d.anchor;
}
}
for(let i=0;i<poly2.length;i++){
point=poly2[i];
let d=point2Polygon(point,poly1);
if(d.dist<min.dist){
min=d;
min.target=d.anchor;
min.source=point;
}
}
return min;
};
//计算图形和点之间的最小距离，返回线，距离
let point2Polygon=function(point,poly){
let min=Number.MAX_SAFE_INTEGER;
let anchor,d;
for(let i=0;i<poly.length;i++){
let p1=poly[i],p2;
let i1;
if(i==poly.length-1)
i1=0;
else
i1=i+1;
p2=poly[i1];
d= distToSegment(point,p1,p2);
if(min>d.dist){
min=d.dist;
anchor=d.p;
}
}
return {
dist:min,
anchor:anchor,
}
};
/*-------------点到线段距离------------*/
function sqr(x) { return x * x }
function dist2(v, w) { return sqr(v.x - w.x) + sqr(v.y - w.y) }
//返回距离和连接点
function distToSegment(p, v, w) {
let l2 = dist2(v, w);
if (l2 == 0) return dist2(p, v);
let t = ((p.x - v.x) * (w.x - v.x) + (p.y - v.y) * (w.y - v.y)) / l2;
t = Math.max(0, Math.min(1, t));
let w1={ x: v.x + t * (w.x - v.x),
y: v.y + t * (w.y - v.y) };
return {dist:Math.sqrt(dist2(p, w1)),
p:w1};
}
/*-------------END---------------*/
/*-------无关算法分界线--------*/
//创建连线
let createLineElement=function(){
let line= document.createElementNS('http://www.w3.org/2000/svg','line');
line.style.stroke='black';
line.style['stroke-width']=2;
line.setAttributeNS('','marker-end','url(#arrow)');
return line;
};
let collect=function(p){
if(p.child){
for(let i=0;i<p.child.length;i++){
let np=createLineElement();
let s=p.child[i].source;
let t=p.child[i].target;
let x1,y1,x2,y2;
if(s instanceof Array){
x1=(s[0].x+s[1].x)/2;
y1=(s[0].y+s[1].y)/2;
}else{
x1=s.x;
y1=s.y;
}
if(t instanceof Array){
x2=(t[0].x+t[1].x)/2;
y2=(t[0].y+t[1].y)/2;
}else{
x2=t.x;
y2=t.y;
}
np.setAttribute('x1',x1);
np.setAttribute('y1',y1);
np.setAttribute('x2',x2);
np.setAttribute('y2',y2);
lineGroup.appendChild(np);
collect(p.child[i].node);
}
}
};
let getEvent = function(){
return window.event || arguments.callee.caller.arguments[0];
};
let printChild=function(p){
console.log('node:',p.index);
if(p.child){
console.log('{',p.index,'--child:');
for(let i=0;i<p.child.length;i++){
console.log('source:',p.child[i].source);
console.log('target:',p.child[i].target);
printChild(p.child[i].node);
}
console.log('--',p.index,'}');
}
}
//从path里获取所有的多边形
let getPolygons=function(p){
let d=p.getAttribute('d');
if(d){
let polygons=d.split('M');
let result=[];
for(let i=1;i<polygons.length;i++){
let seg = polygons[i];
let plist=[];
result.push(plist);
plist.index=i-1;
let status=0;
let np,pindex=0;
for(let j=0;j<seg.length;j++){
let c = seg.charAt(j);
if(status==0){
//初始化状态
if(c==' '||c=='L'||c=='Z'){
}else{
np={};
np.index=pindex++;
plist.push(np);
np.x=c;
np.y='';
status=1;
}
}else if(status == 1){
//开始写x
if(c==' '||c==','){
//开始写y
np.x=parseFloat(np.x);
status=2;
}else if(c=='L'){
}else{
np.x+=c;
}
}else if(status == 2){
if(c==' '||c=='Z'||c=='L'){
np.y=parseFloat(np.y);
status = 0;
}else{
np.y+=c;
}
}
}
}
return result;
}
}
window.onload=function(){
let panel=document.getElementById('panel');
for(let i=0,len=panel.children.length;i<len;i++){
panel.children[i].onclick=caculatePath;
}
}
</script>
</head>
<body>
<svg width='100%' height='100%' style="border:1px solid">
<defs>
<marker id="arrow" markerWidth="5" markerHeight="5" refx="0" refy="3" orient="auto" markerUnits="strokeWidth">
</marker>
</defs>
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</g>
<g id="lineGroup"></g>
</svg>
<div style="position:fixed;right:10%;height:100px;width:200px;top:50px;background-color:#ccc;text-align:center;line-height:1.95">
<div style="">点击左侧图形</div> <div>查看最小连通图</div></div>
</body>
</html>