「接続行列を係数に持つ線形方程式」の線形時間DFSで解くアルゴリズムの図示です．

1. 「接続行列を係数に持つ線形方程式」
の線形時間DFS†の図示
@satanic0258
† https://twitter.com/opt_coder/status/1267774774639423488

2. 𝑐1
𝑐2 𝑐3
𝑐4 𝑐5 𝑐6
+1
−1
+1
+1
+1
−1
+1
−1
−1 −1
𝑥1
𝑥2
𝑥3 𝑥4 𝑥5
見ている頂点
訪問済み
−1
+1
0
0
0
0
+1
0
−1
0
0
0
0
+1
0
−1
0
0
0
−1
0
0
+1
0
0
0
+1
0
0
−1
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1
𝑐2
𝑐3
𝑐4
𝑐5
𝑐6
=

3. 𝑐1
𝑐2 𝑐3
𝑐4 𝑐5 𝑐6
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1
𝑥3 𝑥4 𝑥5
見ている頂点
訪問済み
−1
+1
0
0
0
0
+1
0
−1
0
0
0
0
+1
0
−1
0
0
0
−1
0
0
+1
0
0
0
+1
0
0
−1
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1
𝑐2
𝑐3
𝑐4
𝑐5
𝑐6
=
−1
+1
𝑥2

4. 𝑐1
𝑐2 𝑐3
𝑐4 𝑐5 𝑐6
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1
𝑥3 𝑥4 𝑥5
見ている頂点
訪問済み
−1
+1
0
0
0
0
+1
0
−1
0
0
0
0
+1
0
−1
0
0
0
−1
0
0
+1
0
0
0
+1
0
0
−1
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1
𝑐2
𝑐3
𝑐4
𝑐5
𝑐6
=
−1
+1
𝑥2

5. 𝑐1
𝑐2 𝑐3
𝑐4 𝑐5 𝑐6
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1
𝑥3 𝑥4 𝑥5
見ている頂点
訪問済み
−1
+1
0
0
0
0
+1
0
−1
0
0
0
0
+1
0
−1
0
0
0
−1
0
0
+1
0
0
0
+1
0
0
−1
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1
𝑐2
𝑐3
𝑐4
𝑐5
𝑐6
=
−1
+1
𝑥2

6. 𝑐1
𝑐2 𝑐3
𝑐4 𝑐5 𝑐6
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1
𝑥3 = −𝑐4 𝑥4 𝑥5
見ている頂点
訪問済み
𝑥1
𝑥2
𝑥3 = −𝑐4
𝑥4
𝑥5
−1
+1
0
0
0
0
+1
0
−1
0
0
0
0
+1
0
−1
0
0
0
−1
0
0
+1
0
0
0
+1
0
0
−1
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1
𝑐2
𝑐3
𝑐4
𝑐5
𝑐6
=
−1
+1
𝑥2

7. 𝑐1
𝑐2 + 𝑐4 𝑐3
0 𝑐5 𝑐6
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1
𝑥3 = −𝑐4 𝑥4 𝑥5
見ている頂点
訪問済み
−1
+1
0
0
0
0
+1
0
−1
0
0
0
0
0
0
0
0
0
0
−1
0
0
+1
0
0
0
+1
0
0
−1
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1
𝑐2 + 𝑐4
𝑐3
0
𝑐5
𝑐6
=
𝑥1
𝑥2
𝑥3 = −𝑐4
𝑥4
𝑥5
−1
+1
𝑥2

8. 𝑐1
𝑐2 + 𝑐4 𝑐3
0 𝑐5 𝑐6
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1
𝑥3 = −𝑐4 𝑥4 𝑥5
見ている頂点
訪問済み
−1
+1
0
0
0
0
+1
0
−1
0
0
0
0
0
0
0
0
0
0
−1
0
0
+1
0
0
0
+1
0
0
−1
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1
𝑐2 + 𝑐4
𝑐3
0
𝑐5
𝑐6
=
𝑥1
𝑥2
𝑥3 = −𝑐4
𝑥4
𝑥5
−1
+1
𝑥2

9. 𝑐1
𝑐2 + 𝑐4 𝑐3
0 𝑐5 𝑐6
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1
𝑥3 = −𝑐4 𝑥4 = 𝑐5 𝑥5
見ている頂点
訪問済み
−1
+1
0
0
0
0
+1
0
−1
0
0
0
0
0
0
0
0
0
0
−1
0
0
+1
0
0
0
+1
0
0
−1
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1
𝑐2 + 𝑐4
𝑐3
0
𝑐5
𝑐6
=
𝑥1
𝑥2
𝑥3 = −𝑐4
𝑥4 = 𝑐5
𝑥5
−1
+1
𝑥2

10. 𝑐1
𝑐2 + 𝑐4
+𝑐5
𝑐3
0 0 𝑐6
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1
𝑥3 = −𝑐4 𝑥4 = 𝑐5 𝑥5
見ている頂点
訪問済み
−1
+1
0
0
0
0
+1
0
−1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
+1
0
0
−1
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1
𝑐2 + 𝑐4 + 𝑐5
𝑐3
0
0
𝑐6
=
𝑥1
𝑥2
𝑥3 = −𝑐4
𝑥4 = 𝑐5
𝑥5
−1
+1
𝑥2

11. 𝑐1
𝑐2 + 𝑐4
+𝑐5
𝑐3
0 0 𝑐6
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥3 = −𝑐4 𝑥4 = 𝑐5 𝑥5
見ている頂点
訪問済み
−1
+1
0
0
0
0
+1
0
−1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
+1
0
0
−1
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1
𝑐2 + 𝑐4 + 𝑐5
𝑐3
0
0
𝑐6
=
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥2
𝑥3 = −𝑐4
𝑥4 = 𝑐5
𝑥5
−1
+1
𝑥2

12. 𝑐1 + 𝑐2
+𝑐4 + 𝑐5
0 𝑐3
0 0 𝑐6
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥3 = −𝑐4 𝑥4 = 𝑐5 𝑥5
見ている頂点
訪問済み
0
0
0
0
0
0
+1
0
−1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
+1
0
0
−1
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1 + 𝑐2 + 𝑐4 + 𝑐5
0
𝑐3
0
0
𝑐6
=
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥2
𝑥3 = −𝑐4
𝑥4 = 𝑐5
𝑥5
−1
+1
𝑥2

13. 𝑐1 + 𝑐2
+𝑐4 + 𝑐5
0 𝑐3
0 0 𝑐6
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥3 = −𝑐4 𝑥4 = 𝑐5 𝑥5
見ている頂点
訪問済み
0
0
0
0
0
0
+1
0
−1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
+1
0
0
−1
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1 + 𝑐2 + 𝑐4 + 𝑐5
0
𝑐3
0
0
𝑐6
=
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥2
𝑥3 = −𝑐4
𝑥4 = 𝑐5
𝑥5
−1
+1
𝑥2

14. 𝑐1 + 𝑐2
+𝑐4 + 𝑐5
0 𝑐3
0 0 𝑐6
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥3 = −𝑐4 𝑥4 = 𝑐5 𝑥5
見ている頂点
訪問済み
0
0
0
0
0
0
+1
0
−1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
+1
0
0
−1
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1 + 𝑐2 + 𝑐4 + 𝑐5
0
𝑐3
0
0
𝑐6
=
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥2
𝑥3 = −𝑐4
𝑥4 = 𝑐5
𝑥5
−1
+1
𝑥2

15. 𝑐1 + 𝑐2
+𝑐4 + 𝑐5
0 𝑐3
0 0 𝑐6
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥3 = −𝑐4 𝑥4 = 𝑐5 𝑥5 = −𝑐6
見ている頂点
訪問済み
0
0
0
0
0
0
+1
0
−1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
+1
0
0
−1
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1 + 𝑐2 + 𝑐4 + 𝑐5
0
𝑐3
0
0
𝑐6
=
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥2
𝑥3 = −𝑐4
𝑥4 = 𝑐5
𝑥5 = −𝑐6
−1
+1
𝑥2

16. 𝑐1 + 𝑐2
+𝑐4 + 𝑐5
0 𝑐3 + 𝑐6
0 0 0
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥3 = −𝑐4 𝑥4 = 𝑐5 𝑥5 = −𝑐6
見ている頂点
訪問済み
0
0
0
0
0
0
+1
0
−1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1 + 𝑐2 + 𝑐4 + 𝑐5
0
𝑐3 + 𝑐6
0
0
0
=
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥2
𝑥3 = −𝑐4
𝑥4 = 𝑐5
𝑥5 = −𝑐6
−1
+1
𝑥2

17. 𝑐1 + 𝑐2
+𝑐4 + 𝑐5
0 𝑐3 + 𝑐6
0 0 0
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥3 = −𝑐4 𝑥4 = 𝑐5 𝑥5 = −𝑐6
見ている頂点
訪問済み
0
0
0
0
0
0
+1
0
−1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1 + 𝑐2 + 𝑐4 + 𝑐5
0
𝑐3 + 𝑐6
0
0
0
=
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥2 = −𝑐3 − 𝑐6
𝑥3 = −𝑐4
𝑥4 = 𝑐5
𝑥5 = −𝑐6
−1
+1
𝑥2 = −𝑐3 − 𝑐6

18. 𝑐1 + 𝑐2 + 𝑐3
+𝑐4 + 𝑐5 + 𝑐6
0 0
0 0 0
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥3 = −𝑐4 𝑥4 = 𝑐5 𝑥5 = −𝑐6
見ている頂点
訪問済み
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1 + 𝑐2 + 𝑐3 + 𝑐4 + 𝑐5 + 𝑐6
0
0
0
0
0
=
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥2 = −𝑐3 − 𝑐6
𝑥3 = −𝑐4
𝑥4 = 𝑐5
𝑥5 = −𝑐6
−1
+1
𝑥2 = −𝑐3 − 𝑐6

19. 𝑐1 + 𝑐2 + 𝑐3
+𝑐4 + 𝑐5 + 𝑐6
0 0
0 0 0
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥3 = −𝑐4 𝑥4 = 𝑐5 𝑥5 = −𝑐6
見ている頂点
訪問済み
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1 + 𝑐2 + 𝑐3 + 𝑐4 + 𝑐5 + 𝑐6
0
0
0
0
0
=
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥2 = −𝑐3 − 𝑐6
𝑥3 = −𝑐4
𝑥4 = 𝑐5
𝑥5 = −𝑐6
−1
+1
𝑥2 = −𝑐3 − 𝑐6

20. 𝑐1 + 𝑐2 + 𝑐3
+𝑐4 + 𝑐5 + 𝑐6
0 0
0 0 0
+1
−1
+1
+1
+1
−1
−1 −1
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥3 = −𝑐4 𝑥4 = 𝑐5 𝑥5 = −𝑐6
見ている頂点
訪問済み
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
𝑥1
𝑥2
𝑥3
𝑥4
𝑥5
𝑐1 + 𝑐2 + 𝑐3 + 𝑐4 + 𝑐5 + 𝑐6
0
0
0
0
0
=
𝑥1 = 𝑐2 + 𝑐4 + 𝑐5
𝑥2 = −𝑐3 − 𝑐6
𝑥3 = −𝑐4
𝑥4 = 𝑐5
𝑥5 = −𝑐6
−1
+1
𝑥2 = −𝑐3 − 𝑐6
𝑐1 + 𝑐2 + 𝑐3 + 𝑐4 + 𝑐5 + 𝑐6 = 0 のときのみ以下の解を持つ