---
id: wiki-2026-0508-directed-acyclic-graph-dependenc
title: Directed Acyclic Graph Dependency Management
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [DAG, Build Graph, Task Dependency, Topological Sort]
duplicate_of: none
source_trust_level: A
confidence_score: 0.94
verification_status: applied
tags: [DAG, dependency, build-system, scheduler, topological-sort]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: networkx/airflow
---

# Directed Acyclic Graph Dependency Management

## 매 한 줄
> **"매 DAG = nodes (tasks) + directed edges (must-run-before) + 매 cycle 금지"**. 매 1960s Make 의 build graph 부터 매 2026 Airflow/Dagster pipelines, Bazel/Turborepo monorepo, Spark physical plan, Git commit history, React fiber tree 까지 — 매 dependency resolution 의 universal data structure.

## 매 핵심

### 매 핵심 연산
- **Topological Sort**: 매 valid execution order. Kahn's O(V+E) or DFS.
- **Cycle Detection**: 매 DAG validity check.
- **Transitive Reduction**: 매 minimal edge set with same reachability.
- **Critical Path**: 매 longest path = makespan lower bound.
- **Incremental Recompute**: 매 dirty subgraph 만 재실행.

### 매 응용
1. **Build systems**: Make, Bazel, Buck, Turborepo, Nx.
2. **Workflow orchestration**: Airflow, Dagster, Prefect, Argo Workflows.
3. **ML training pipelines**: Kubeflow, MLflow, ZenML.
4. **Spreadsheet recalc**: Excel, Google Sheets formula engine.
5. **VCS**: Git commit DAG, Mercurial.
6. **React/Solid reactivity**: 매 signal dependency graph.

### 매 schedule strategies
- **List scheduling**: 매 ready tasks → workers (greedy).
- **HEFT**: 매 heterogeneous earliest finish time (cloud).
- **Critical Path Method (CPM)**: 매 longest path 기반 prioritization.
- **Work-stealing**: 매 dynamic load balancing (Tokio, Rayon).

## 💻 패턴

### Topological Sort (Kahn's Algorithm)
```python
from collections import deque, defaultdict

def topo_sort(nodes, edges):
    indegree = defaultdict(int)
    graph = defaultdict(list)
    for u, v in edges:
        graph[u].append(v); indegree[v] += 1
    queue = deque([n for n in nodes if indegree[n] == 0])
    order = []
    while queue:
        u = queue.popleft(); order.append(u)
        for v in graph[u]:
            indegree[v] -= 1
            if indegree[v] == 0: queue.append(v)
    if len(order) != len(nodes): raise ValueError("Cycle detected")
    return order
```

### Parallel DAG Executor (asyncio)
```python
import asyncio
async def run_dag(tasks, deps):
    """tasks: {name: async_fn}, deps: {name: [prereqs]}."""
    completed = {}; pending = dict(deps)
    async def run(name):
        await asyncio.gather(*(completed[d] for d in deps.get(name, [])))
        return await tasks[name]()
    completed = {n: asyncio.create_task(run(n)) for n in tasks}
    return await asyncio.gather(*completed.values())
```

### Incremental Build (Content-Hash)
```python
def needs_rebuild(node, hashes, prev_hashes):
    own_hash = hash_inputs(node.sources, [hashes[d] for d in node.deps])
    if prev_hashes.get(node.name) != own_hash:
        hashes[node.name] = own_hash
        return True
    hashes[node.name] = own_hash
    return False
```

### Critical Path
```python
def critical_path(graph, durations):
    order = topo_sort(graph.nodes, graph.edges)
    earliest = {n: durations[n] for n in graph.nodes}
    for u in order:
        for v in graph.successors(u):
            earliest[v] = max(earliest[v], earliest[u] + durations[v])
    return max(earliest.values()), earliest
```

### Cycle Detection (DFS)
```python
WHITE, GRAY, BLACK = 0, 1, 2
def has_cycle(graph):
    color = {n: WHITE for n in graph}
    def dfs(u):
        color[u] = GRAY
        for v in graph[u]:
            if color[v] == GRAY: return True
            if color[v] == WHITE and dfs(v): return True
        color[u] = BLACK
        return False
    return any(dfs(n) for n in graph if color[n] == WHITE)
```

### Airflow DAG (Practical)
```python
from airflow.decorators import dag, task
from datetime import datetime

@dag(start_date=datetime(2026,1,1), schedule="@daily", catchup=False)
def etl():
    @task
    def extract(): return fetch()
    @task
    def transform(data): return clean(data)
    @task
    def load(clean): warehouse.write(clean)
    load(transform(extract()))
etl()
```

## 매 결정 기준
| 상황 | Approach |
|---|---|
| Build deterministic, hermetic | Bazel / Buck (content-hash) |
| Data pipeline, scheduled | Airflow / Dagster |
| Monorepo JS/TS | Turborepo / Nx |
| ML experiment tracking | Kubeflow / MLflow / ZenML |
| In-process reactive UI | Signals (Solid/Vue/Svelte) |
| Real-time stream graph | Flink / Spark Structured Streaming |

**기본값**: 매 explicit DAG (declarative) > 매 implicit ordering — 매 visualization + audit + parallel scheduling 가능.

## 🔗 Graph
- 부모: [[Graph Theory]] · [[Topological Sort]]
- 변형: [[Build Graph]]
- 응용: [[Bazel]] · [[Airflow]] · [[Turborepo 환경 구성]] · [[Spark]]
- Adjacent: [[Incremental-Computation]]

## 🤖 LLM 활용
**언제**: 매 multi-step agent plan 의 dependency 표현, 매 RAG indexing pipeline orchestration.
**언제 X**: 매 cyclic feedback loop 가 본질적 (RL, gradient descent) — 매 DAG 외 unrolled iteration.

## ❌ 안티패턴
- **Hidden side effects**: 매 task 가 state 직접 mutate → 매 incremental build 깨짐.
- **Ignoring transitive dependencies**: 매 missing edge → race condition.
- **Single-task megasinks**: 매 fan-in bottleneck — 매 break into shards.
- **Cycle by feature flag**: 매 conditional dependencies 가 implicit cycle 만들 수 있음.
- **Over-fine granularity**: 매 nano-tasks → scheduler overhead > work.

## 🧪 검증 / 중복
- Verified (Kahn 1962; Bazel docs 2026; Airflow 3.x docs; CLRS Ch.22).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — full content with topo, parallel exec, incremental, Airflow |
