최소 신장 트리

Minimum spanning tree

모든 정점을 이어 주되 순환 없이 간선 비용 합계를 최소화한다.

다른 이름: MST
···
html
<canvas id="stage"></canvas><div class="controls"><button id="restart" type="button">처음부터</button><button id="step" type="button">한 단계</button><label>속도 <input id="speed" type="range" min="1" max="5" value="3"></label></div>
css
body{display:block}#stage{position:absolute;inset:0;width:100%;height:100%}.controls{position:absolute;bottom:9px;left:50%;transform:translateX(-50%);display:flex;align-items:center;gap:7px;white-space:nowrap}.controls button,.controls label{font:12px sans-serif;color:var(--fg);background:var(--surface);border:1px solid var(--line);border-radius:6px;padding:5px 8px}.controls button{cursor:pointer}.controls input{width:75px;vertical-align:middle;accent-color:var(--accent)}@media(max-width:500px){.controls{display:none}}
js
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const title = "최소 신장 트리 · 간선 선택";
const canvas = document.getElementById('stage');
const ctx = canvas.getContext('2d');
const restart = document.getElementById('restart');
const step = document.getElementById('step');
const speed = document.getElementById('speed');
let index = 0, last = performance.now(), width = 300, height = 200;
function colors(){ const s=getComputedStyle(document.documentElement); const v=(key)=>s.getPropertyValue(key).trim(); return {bg:v('--bg'),fg:v('--fg'),muted:v('--muted'),line:v('--line'),surface:v('--surface'),accent:v('--accent'),accent2:v('--accent-2'),accent3:v('--accent-3')}; }
function resize(){ const dpr=Math.min(devicePixelRatio||1,2); width=innerWidth;height=innerHeight;canvas.width=Math.round(width*dpr);canvas.height=Math.round(height*dpr);ctx.setTransform(dpr,0,0,dpr,0,0);draw(); }
function text(value,x,y,color,size=12,align='left'){ ctx.fillStyle=color;ctx.font='600 '+size+'px sans-serif';ctx.textAlign=align;ctx.textBaseline='middle';ctx.fillText(String(value),x,y); }
function line(x1,y1,x2,y2,color,thick=2){ctx.strokeStyle=color;ctx.lineWidth=thick;ctx.beginPath();ctx.moveTo(x1,y1);ctx.lineTo(x2,y2);ctx.stroke();}
function dot(x,y,r,color){ctx.fillStyle=color;ctx.beginPath();ctx.arc(x,y,r,0,Math.PI*2);ctx.fill();}
function draw(){
  if(!frames.length)return;
  const p=colors(), frame=frames[index], d=frame.data, detail=width>500;
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  text(title,12,18,p.fg,detail?16:13);text(frame.caption,12,detail?43:39,p.muted,detail?13:10);
  text((index+1)+' / '+frames.length,width-12,18,p.muted,10,'right');
  const top=detail?62:53, bottom=detail?height-47:height-9, span=Math.max(10,bottom-top), areaW=Math.min(width-22,detail?710:width-22), left=(width-areaW)/2;
  if(frame.kind==='graph'){
    const nodes=d.nodes, edges=d.edges, positions=Object.fromEntries(nodes.map(n=>[n.id,{x:left+areaW*(.06+.88*n.x),y:top+span*(.11+.78*n.y)}]));
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      const chosen=d.selected?.includes(key), active=d.activeEdge===key;ctx.globalAlpha=active||chosen?1:.58;line(pa.x,pa.y,pb.x,pb.y,active?p.accent2:chosen?p.accent:p.muted,active?4:chosen?3:2);ctx.globalAlpha=1;
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      if(d.groups){const group=d.groups[nodes.findIndex(n=>n.id===node.id)];color=[p.accent,p.accent2,p.accent3][nodes.findIndex(n=>n.id===group)%3];}
      dot(pt.x,pt.y,detail?18:12,color);ctx.strokeStyle=p.line;ctx.lineWidth=1;ctx.stroke();text(node.id,pt.x,pt.y,selected||active||front||d.groups?'#fff':p.fg,detail?13:10,'center');
      if(d.distances)text(Number.isFinite(d.distances[node.id])?d.distances[node.id]:'∞',pt.x,pt.y+(detail?29:22),p.muted,10,'center');
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    const table=frame.kind==='table',bits=frame.kind==='bits';const rows=bits?1:table?d.values.length:d.rows,cols=bits?d.bits.length:table?d.values[0].length:d.cols;
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    for(let r=0;r<rows;r++)for(let c=0;c<cols;c++){const n=r*cols+c,x=ox+c*cell,y=oy+r*cell,active=bits?d.active?.includes(c):table?d.active?.[0]===r&&d.active?.[1]===c:d.active===n,source=table?d.sources?.some(v=>v[0]===r&&v[1]===c):false,visited=bits?d.bits[c]:table?false:d.visited?.includes(n),front=d.frontier?.includes(n),blocked=d.blocked?.includes(n),path=d.path?.includes(n),queen=d.queens?.[r]===c;
      ctx.fillStyle=blocked?p.line:active?p.accent2:path||queen?p.accent3:front?p.accent2:visited?p.accent:source?p.accent3:bits&&d.bits[c]?p.accent:p.surface;ctx.fillRect(x+2,y+2,cell-4,cell-4);ctx.strokeStyle=p.muted;ctx.globalAlpha=.42;ctx.strokeRect(x+1,y+1,cell-2,cell-2);ctx.globalAlpha=1;
      const value=bits?d.bits[c]:table?d.values[r][c]:queen?'Q':d.start===n?'S':d.goal===n?'G':blocked?'':visited?'·':'';
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    const rowH=Math.min(31,span/d.intervals.length),unit=(areaW-28)/10,ox=left+10,oy=top+(span-rowH*d.intervals.length)/2;
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    const cx=width/2,cy=top+span/2,r=Math.min(areaW,span)*.35,loc=(v,rr)=>({x:cx+Math.sin(v*2*Math.PI)*rr,y:cy-Math.cos(v*2*Math.PI)*rr});ctx.strokeStyle=p.muted;ctx.globalAlpha=.6;ctx.lineWidth=2;ctx.beginPath();ctx.arc(cx,cy,r,0,Math.PI*2);ctx.stroke();ctx.globalAlpha=1;
    d.keys.forEach((key,i)=>{const point=loc(key,r),server=d.servers.find(s=>s.id===d.owners[i]),dest=loc(server.pos,r);line(point.x,point.y,dest.x,dest.y,d.active===i?p.accent2:p.line,1);dot(point.x,point.y,d.active===i?6:4,d.active===i?p.accent2:p.fg);});
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    const side=Math.min(areaW,span)-8,ox=(width-side)/2,oy=top+(span-side)/2;ctx.strokeStyle=p.muted;ctx.globalAlpha=.5;ctx.strokeRect(ox,oy,side,side);ctx.globalAlpha=1;ctx.beginPath();ctx.arc(ox,oy+side,side,Math.PI*1.5,0);ctx.strokeStyle=p.muted;ctx.stroke();
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}
function advance(){index=(index+1)%frames.length;draw();}
restart.addEventListener('click',()=>{index=0;last=performance.now();draw();});
step.addEventListener('click',()=>{advance();last=performance.now();});
addEventListener('resize',resize);resize();
function loop(now){const delay=900-Number(speed.value)*140;if(now-last>delay){advance();last=now;}requestAnimationFrame(loop);}requestAnimationFrame(loop);

모든 마을에 케이블을 놓되 총 길이를 최소화하는 문제입니다. 한 마을에서 다른 마을까지의 최단 경로와는 목표가 다릅니다. 데모의 진한 선은 현재까지 채택한 간선입니다.

크루스칼 알고리즘은 간선을 가중치 오름차순으로 정렬하고, 양 끝이 아직 다른 연결 집합일 때만 간선을 채택합니다. 같은 집합이면 순환이 생기므로 건너뜁니다. 유니온 파인드로 집합을 관리하면 정렬이 지배하는 O(E log E) 시간에 구할 수 있습니다.

연결된 무향 가중 그래프의 네트워크 설계에 씁니다. 그래프가 끊어져 있으면 하나의 신장 트리가 아니라 각 성분의 최소 신장 숲이 됩니다.

언제 쓰나

전체 연결 비용을 줄일 때. 두 점 사이 한 경로의 최소 비용이라면 최단 경로 알고리즘을 쓰세요.

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