Divide and conquer

분할 정복

Split a problem, solve its parts, and combine their results.

···
css
body{display:block} .stage{position:absolute;inset:0;display:flex;flex-direction:column;padding:clamp(9px,3vmin,20px);gap:5px}
.top{display:flex;align-items:center;justify-content:space-between;gap:8px;font-size:clamp(11px,2.7vmin,14px);font-weight:700;white-space:nowrap}
.top span:last-child{color:var(--muted);font-weight:500}canvas{width:100%;height:0;flex:1 1 0;min-height:0}
.controls{display:none;align-items:center;gap:9px;font-size:12px;color:var(--muted)}
button{border:1px solid var(--line);background:var(--surface);color:var(--fg);border-radius:7px;padding:5px 10px;cursor:pointer}
input{width:100px;accent-color:var(--accent)}
@media(min-width:600px){.controls{display:flex}}
js
const ALGORITHM_KIND = "divide-and-conquer";

const kind = ALGORITHM_KIND;
const root = document.createElement('div'); root.className = 'stage';
root.innerHTML = '<div class="top"><span id="state"></span><span id="count"></span></div><canvas></canvas><div class="controls"><button id="reset">다시 시작</button><button id="step">한 단계</button><label>속도 <input id="speed" type="range" min="1" max="5" value="3"></label></div>';
document.body.append(root);
const canvas = root.querySelector('canvas'), ctx = canvas.getContext('2d');
const title = root.querySelector('#state'), count = root.querySelector('#count');
const style = getComputedStyle(document.documentElement);
const colors = Object.fromEntries(['bg','fg','muted','line','accent','accent-2','accent-3','surface'].map(k => [k, style.getPropertyValue('--' + k).trim()]));
const palette = {base:colors.accent, active:colors['accent-2'], done:colors['accent-3'], text:colors.fg, faint:colors.muted, line:colors.line, surface:colors.surface};
let generation = 0, states = [], index = 0, elapsed = 0;
function rng(seed){return function(){seed |= 0;seed = seed + 0x6D2B79F5 | 0;let t = Math.imul(seed ^ seed >>> 15, 1 | seed);t = t + Math.imul(t ^ t >>> 7, 61 | t) ^ t;return ((t ^ t >>> 14) >>> 0) / 4294967296}}
function shuffled(){const a=[3,8,2,7,4,9,1,6,5];const random=rng(42+generation*23);for(let i=a.length-1;i>0;i--){const j=Math.floor(random()*(i+1));[a[i],a[j]]=[a[j],a[i]]}return a}
function push(values, active=[], done=[], note='', extra={}){states.push({values:[...values],active:[...active],done:[...done],note,extra})}
function makeStates(){
  states=[];
  const a=shuffled(), n=a.length;
  push(a,[],[],'시작');
  if(kind==='big-o') {states=[];for(let size=1;size<=12;size++)push([],[],[],'입력 크기 n = '+size,{size});return}
  if(kind==='bubble-sort'){for(let end=n-1;end>0;end--){for(let j=0;j<end;j++){push(a,[j,j+1],Array.from({length:n-end-1},(_,k)=>n-1-k),'이웃한 두 값 비교');if(a[j]>a[j+1]){[a[j],a[j+1]]=[a[j+1],a[j]];push(a,[j,j+1],[],'순서가 반대라 교환')}}push(a,[],Array.from({length:n-end},(_,k)=>n-1-k),'가장 큰 값 확정')}push(a,[],a.map((_,i)=>i),'정렬 완료')}
  if(kind==='selection-sort'){for(let i=0;i<n;i++){let min=i;for(let j=i+1;j<n;j++){push(a,[min,j],Array.from({length:i},(_,k)=>k),'최솟값 탐색');if(a[j]<a[min])min=j}[a[i],a[min]]=[a[min],a[i]];push(a,[i,min],Array.from({length:i+1},(_,k)=>k),'최솟값을 앞에 배치')}}
  if(kind==='insertion-sort'){for(let i=1;i<n;i++){let j=i;while(j>0&&a[j-1]>a[j]){push(a,[j-1,j],[],'앞의 값과 비교');[a[j-1],a[j]]=[a[j],a[j-1]];push(a,[j-1,j],Array.from({length:i+1},(_,k)=>k),'왼쪽으로 삽입');j--}push(a,[],Array.from({length:i+1},(_,k)=>k),'왼쪽 부분 정렬됨')}}
  if(kind==='merge-sort'){function merge(lo,hi){if(hi-lo<=1)return;const mid=(lo+hi)>>1;push(a,Array.from({length:hi-lo},(_,i)=>lo+i),[],'구간 분할: '+lo+'–'+(hi-1));merge(lo,mid);merge(mid,hi);const sorted=[...a.slice(lo,mid)], right=[...a.slice(mid,hi)];let l=0,r=0;for(let k=lo;k<hi;k++){a[k]=r>=right.length||(l<sorted.length&&sorted[l]<=right[r])?sorted[l++]:right[r++];push(a,[k],Array.from({length:k-lo+1},(_,i)=>lo+i),'작은 값부터 병합')}}merge(0,n);push(a,[],a.map((_,i)=>i),'정렬 완료')}
  if(kind==='quick-sort'){function quick(lo,hi){if(lo>=hi)return;const pivot=a[hi];let i=lo;push(a,[hi],[],'피벗 '+pivot+' 선택');for(let j=lo;j<hi;j++){push(a,[j,hi],[],'피벗과 비교');if(a[j]<pivot){[a[i],a[j]]=[a[j],a[i]];push(a,[i,j],[],'작은 값을 왼쪽으로');i++}}[a[i],a[hi]]=[a[hi],a[i]];push(a,[i],[i],'피벗 자리 확정');quick(lo,i-1);quick(i+1,hi)}quick(0,n-1);push(a,[],a.map((_,i)=>i),'정렬 완료')}
  if(kind==='heap-sort'){function sift(size,i){while(2*i+1<size){let child=2*i+1;if(child+1<size&&a[child+1]>a[child])child++;push(a,[i,child],[],'부모와 큰 자식 비교');if(a[i]>=a[child])break;[a[i],a[child]]=[a[child],a[i]];push(a,[i,child],[],'큰 값을 위로');i=child}}for(let i=(n>>1)-1;i>=0;i--)sift(n,i);for(let end=n-1;end>0;end--){[a[0],a[end]]=[a[end],a[0]];push(a,[0,end],Array.from({length:n-end},(_,k)=>end+k),'최댓값을 끝으로');sift(end,0)}push(a,[],a.map((_,i)=>i),'정렬 완료')}
  if(kind==='counting-sort'){const freq=Array(10).fill(0);for(let i=0;i<n;i++){freq[a[i]]++;push(freq,[a[i]],[],'값 '+a[i]+'의 개수 세기',{histogram:true})}let k=0;for(let value=1;value<=9;value++)while(freq[value]-- > 0){a[k++]=value;push(a,[k-1],Array.from({length:k},(_,i)=>i),'개수대로 출력')}push(a,[],a.map((_,i)=>i),'정렬 완료')}
  if(kind==='binary-search'){states=[];const b=[1,2,3,4,5,6,7,8,9];let lo=0,hi=8,target=7;push(b,[],[],'정렬된 배열에서 7 찾기');while(lo<=hi){let mid=(lo+hi)>>1;push(b,[mid],[],'가운데 '+b[mid]+'와 비교',{range:[lo,hi]});if(b[mid]===target){push(b,[mid],[mid],'7 찾음');break}if(b[mid]<target)lo=mid+1;else hi=mid-1;push(b,[],[],'절반 제외',{range:[lo,hi]})}}
  if(kind==='two-pointers'){states=[];const b=[1,2,3,4,5,6,7,8,9];let l=0,r=8;push(b,[l,r],[],'합계 10 찾기');while(l<r){push(b,[l,r],[],'합 '+(b[l]+b[r])+' 비교');if(b[l]+b[r]===10){push(b,[l,r],[l,r],'합계 10인 쌍');l++;r--}else if(b[l]+b[r]<10)l++;else r--}}
  if(kind==='sliding-window'){states=[];const b=[2,1,4,3,5,2,6,1,4];let sum=b.slice(0,3).reduce((x,y)=>x+y,0);push(b,[0,1,2],[],'크기 3, 합 '+sum);for(let start=1;start<=b.length-3;start++){sum+=b[start+2]-b[start-1];push(b,[start,start+1,start+2],[],'왼쪽 제거 · 오른쪽 추가 → 합 '+sum)}}
  if(kind==='stack'||kind==='queue'){states=[];const b=[];const acts=kind==='stack'?[['push',3],['push',8],['push',2],['pop'],['push',7],['pop'],['pop']]:[['enqueue',3],['enqueue',8],['enqueue',2],['dequeue'],['enqueue',7],['dequeue'],['dequeue']];push(b,[],[],'빈 '+(kind==='stack'?'스택':'큐'));for(const [op,value] of acts){let changed;if(op==='push'||op==='enqueue'){b.push(value);changed=b.length-1}else{changed=kind==='stack'?b.length-1:0;b.splice(changed,1)}push(b,[Math.max(0,Math.min(changed,b.length-1))],[],op+(value?' '+value:''),{operation:op})}}
  if(kind==='linked-list'){states=[];const nodes=[3,8,2,7].map(value=>({value,next:null}));for(let i=0;i<nodes.length-1;i++)nodes[i].next=nodes[i+1];const head=nodes[0];function snapshot(){const result=[];for(let node=head;node;node=node.next)result.push(node.value);return result}push(snapshot(),[],[],'노드와 다음 포인터');let node=head,i=0;while(node){push(snapshot(),[i++],[],'처음부터 '+node.value+' 방문');node=node.next}nodes[1].next={value:5,next:nodes[1].next};push(snapshot(),[2],[],'포인터 연결을 바꿔 5 삽입');node=head;i=0;while(node){push(snapshot(),[i++],[],'다시 순회: '+node.value);node=node.next}}
  if(kind==='hash-table'){states=[];const names=['A','F','B','G','C','H'];const buckets=Array.from({length:5},()=>[]);push([],[],[],'5개 버킷',{buckets:buckets.map(x=>[...x])});for(const name of names){const bucket=(name.charCodeAt(0)-65)%5;buckets[bucket].push(name);push([], [bucket],[],'키 '+name+' → 버킷 '+bucket,{buckets:buckets.map(x=>[...x])})}}
  if(kind==='binary-search-tree'){states=[];const order=[5,3,8,1,4,7,9],tree=[];for(const value of order){let i=0;while(tree[i]!==undefined)i=value<tree[i]?2*i+1:2*i+2;tree[i]=value;push(tree,[i],[],'값 '+value+' 삽입',{tree:true})}let i=0;while(tree[i]!==4){push(tree,[i],[],'4와 '+tree[i]+' 비교',{tree:true});i=4<tree[i]?2*i+1:2*i+2}push(tree,[i],[i],'값 4 찾음',{tree:true})}
  if(kind==='heap-priority-queue'){states=[];const h=[];function insert(x){h.push(x);let i=h.length-1;push(h,[i],[],'우선순위 '+x+' 추가');while(i>0){let p=(i-1)>>1;if(h[p]>=h[i])break;[h[i],h[p]]=[h[p],h[i]];i=p;push(h,[i],[],'큰 값을 부모로')}}for(const x of [3,8,2,7,5])insert(x);for(let t=0;t<2;t++){const max=h[0];h[0]=h.pop();push(h,[0],[],'최댓값 '+max+' 꺼냄');let i=0;while(2*i+1<h.length){let c=2*i+1;if(c+1<h.length&&h[c+1]>h[c])c++;if(h[i]>=h[c])break;[h[i],h[c]]=[h[c],h[i]];i=c;push(h,[i],[],'힙 복구')}}}
  if(kind==='recursion'){states=[];for(let d=1;d<=5;d++)push(Array.from({length:d},(_,i)=>5-i),[d-1],[],'factorial('+ (6-d) +') 호출',{call:true});let result=1;for(let d=5;d>=1;d--){result*=6-d;push(Array.from({length:d-1},(_,i)=>5-i),[],[],'반환값 '+result,{call:true})}}
  if(kind==='divide-and-conquer'){states=[];const b=[8,3,7,2,6,1,5,4];function visit(lo,hi){push(b,Array.from({length:hi-lo},(_,i)=>lo+i),[],'범위 '+lo+'–'+(hi-1)+' 분할',{range:[lo,hi-1]});if(hi-lo>1){const mid=(lo+hi)>>1;visit(lo,mid);visit(mid,hi);const left=b.slice(lo,mid),right=b.slice(mid,hi);let l=0,r=0;for(let k=lo;k<hi;k++)b[k]=r>=right.length||(l<left.length&&left[l]<=right[r])?left[l++]:right[r++];push(b,Array.from({length:hi-lo},(_,i)=>lo+i),[],'작은 결과를 병합')}}visit(0,b.length);push(b,[],b.map((_,i)=>i),'전체 결과')}
  if(kind==='prefix-sum'){states=[];const b=[2,1,4,3,5,2,6,1,4], prefix=[0];push(b,[],[],'원본 배열',{prefix:[0]});for(let i=0;i<b.length;i++){prefix.push(prefix[i]+b[i]);push(b,[i],Array.from({length:i+1},(_,j)=>j),'누적합 P['+(i+1)+'] = '+prefix[i+1],{prefix:[...prefix]})}push(b,[2,3,4,5],[],'구간 2–5 합 = P[6] − P[2] = '+(prefix[6]-prefix[2]),{prefix:[...prefix]})}
}
function resize(){const dpr=Math.min(devicePixelRatio||1,2);const w=canvas.clientWidth,h=canvas.clientHeight;canvas.width=Math.max(1,Math.round(w*dpr));canvas.height=Math.max(1,Math.round(h*dpr));ctx.setTransform(dpr,0,0,dpr,0,0);draw()}
function text(value,x,y,size=12,color=palette.text,align='center'){ctx.fillStyle=color;ctx.font='600 '+size+'px system-ui,sans-serif';ctx.textAlign=align;ctx.fillText(String(value),x,y)}
function round(x,y,w,h,r=7){ctx.beginPath();ctx.roundRect(x,y,w,h,r)}
function draw(){if(!states.length)return;const s=states[index], w=canvas.clientWidth,h=canvas.clientHeight;ctx.clearRect(0,0,canvas.width,canvas.height);title.textContent=s.note;count.textContent=(index+1)+' / '+states.length;const values=s.values;
 if(kind==='big-o'){const pad=26,gw=w-pad*2,gh=h-48;ctx.strokeStyle=palette.line;ctx.beginPath();ctx.moveTo(pad,12);ctx.lineTo(pad,h-27);ctx.lineTo(w-pad,h-27);ctx.stroke();const curves=[['O(1)',x=>1,palette.done],['O(log n)',x=>Math.log2(x+1)*5,palette.base],['O(n)',x=>x*3,palette.active],['O(n²)',x=>x*x*.65,palette.faint]];for(const [label,fn,color] of curves){ctx.strokeStyle=color;ctx.lineWidth=2;ctx.beginPath();for(let x=1;x<=s.extra.size;x++){const px=pad+(x/12)*gw,py=h-27-Math.min(gh,fn(x)/100*gh);if(x===1)ctx.moveTo(px,py);else ctx.lineTo(px,py)}ctx.stroke()}curves.forEach(([label,_,color],i)=>text(label,pad+38+i*(gw/4),h-6,10,color));return}
 if(kind==='hash-table'){const buckets=s.extra.buckets;const gap=w/5;for(let i=0;i<5;i++){const x=i*gap+gap*.08;round(x,h*.18,gap*.84,h*.64);ctx.fillStyle=s.active.includes(i)?palette.active:palette.surface;ctx.fill();ctx.strokeStyle=palette.line;ctx.stroke();text(i,x+gap*.42,h*.14,12,palette.faint);buckets[i].forEach((v,j)=>text(v,x+gap*.42,h*.3+j*20,12,s.active.includes(i)?colors.bg:palette.text))}return}
 if(kind==='binary-search-tree'||kind==='heap-priority-queue'){const nodes=values;const positions=[];for(let i=0;i<nodes.length;i++){const depth=Math.floor(Math.log2(i+1)),first=2**depth-1,slot=i-first;positions.push([(slot+1)*w/(2**depth+1),(depth+1)*h/4])}ctx.strokeStyle=palette.line;for(let i=1;i<nodes.length;i++){const p=(i-1)>>1;ctx.beginPath();ctx.moveTo(...positions[p]);ctx.lineTo(...positions[i]);ctx.stroke()}nodes.forEach((v,i)=>{const [x,y]=positions[i];ctx.beginPath();ctx.arc(x,y,Math.min(17,w/22),0,Math.PI*2);ctx.fillStyle=s.active.includes(i)?palette.active:palette.base;ctx.fill();text(v,x,y+4,12,colors.bg)});return}
 if(kind==='stack'||kind==='queue'||kind==='linked-list'||kind==='recursion'){const gap=w/Math.max(6,values.length+1);values.forEach((v,i)=>{const x=(i+1)*gap-gap*.34,y=h*.3;round(x,y,gap*.68,h*.33);ctx.fillStyle=s.active.includes(i)?palette.active:palette.base;ctx.fill();text(v,x+gap*.34,y+h*.19,14,colors.bg);if(kind==='linked-list'&&i<values.length-1)text('→',x+gap*.84,y+h*.2,16,palette.faint)});if(kind==='stack')text('위쪽에서 넣고 꺼냄',w/2,h*.83,11,palette.faint);if(kind==='queue')text('앞에서 꺼내고 뒤에 넣음',w/2,h*.83,11,palette.faint);if(kind==='recursion')text('호출 스택의 깊이',w/2,h*.83,11,palette.faint);return}
 const n=values.length,gap=w/(n+1),barWidth=Math.min(gap*.74,44),max=Math.max(9,...values);for(let i=0;i<n;i++){const x=(i+1)*gap-barWidth/2,barH=Math.max(7,values[i]/max*(h-42));const active=s.active.includes(i),done=s.done.includes(i);const outOfRange=s.extra.range&&(i<s.extra.range[0]||i>s.extra.range[1]);round(x,h-23-barH,barWidth,barH,Math.min(6,barWidth/4));ctx.fillStyle=outOfRange?palette.line:active?palette.active:done?palette.done:palette.base;ctx.fill();text(values[i],x+barWidth/2,h-7,Math.min(12,Math.max(9,gap*.3)),palette.text)}
 if(s.extra.prefix){const p=s.extra.prefix;text('P: '+p.join('  '),w/2,14,Math.min(12,w/30),palette.faint)}
}
function advance(){index=(index+1)%states.length;draw()}
function startIndex(){return kind==='big-o'?5:kind==='heap-priority-queue'?2:0}
function reset(){generation++;makeStates();index=startIndex();draw()}
root.querySelector('#reset').addEventListener('click',reset);root.querySelector('#step').addEventListener('click',advance);
addEventListener('resize',resize);makeStates();index=startIndex();resize();const speed=root.querySelector('#speed');setInterval(()=>{if(document.hidden)return;elapsed+=100;const baseDelay=kind==='divide-and-conquer'?1500:1050;if(elapsed>=baseDelay-Number(speed.value)*160){elapsed=0;advance()}},100);

Two people can review separate halves of a large book and combine their findings, reducing how much each must handle at once.

Divide into smaller parts, recursively conquer them, then combine results. Merge sort sorts and merges both halves; binary search discards one half, so combining is trivial. Complexity depends on the number of subproblems and the cost of combining.

Use it when parts shrink, can be solved independently, and have a clear combine step. Splitting in half is not automatically faster; overlapping subproblems may call for memoization.

When to use

Use when smaller independent results can be combined.

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