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Elementary Graph Algorithms

Elementary Graph Algorithms. Representations of graphs: undirected graph. 一個 Undirected graph ( 無向圖 ) ，有 5 個點 7 個邊： 無向圖 G=(V,E) ， V 代表點集合， E 代表邊集合。 E 中的元素形式為集合 {u,v} ，代表邊的兩端。. 1. 2. 3. 5. 4. 2. 5. 1. 1. 3. 4. 5. 2. 2. 4. 3. 2. 3. 5. 4. 1. 2. 4. 5. Adjacency-list.

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Elementary Graph Algorithms

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  1. Elementary Graph Algorithms

  2. Representations of graphs:undirected graph • 一個Undirected graph (無向圖)，有5個點7個邊： • 無向圖G=(V,E)，V代表點集合，E代表邊集合。E中的元素形式為集合{u,v}，代表邊的兩端。 1 2 3 5 4 Elementary Graph Algorithms

  3. 2 5 1 1 3 4 5 2 2 4 3 2 3 5 4 1 2 4 5 Adjacency-list • 可藉由Adjacency-list表示，將每個點相鄰的點集合用一個List存起來。 Elementary Graph Algorithms

  4. Adjacency-array • 可藉由Adjacency-array表示，利用一個二維陣列將每對點之間是否有邊連起來，有存1，沒有存0。 Elementary Graph Algorithms

  5. Representations of graphs: Directed graph • 一個Directed graph (有向圖)，有5個點8個邊： • 有向圖G=(V,E)，V代表點集合，E代表邊集合。E中的元素形式為(u,v)，u代表起點，v代表中點。 1 2 3 5 4 Elementary Graph Algorithms

  6. 2 5 1 3 4 2 3 4 3 1 4 4 5 Adjacency-list • 可藉由Adjacency-list表示，將每個點所指向的點集合存在List中。 Elementary Graph Algorithms

  7. Adjacency-array • 可藉由Adjacency-array表示，利用一個二維陣列A存，若(u,v)是一個邊則A[u][v]=1反之A[u][v]=0。 Elementary Graph Algorithms

  8. Breadth-first search • Breadth-first search(簡稱BFS，先廣搜尋)是最簡單的圖形搜尋演算法之一。 • 用於搜尋圖形G中，所有自點s開始出發，有路徑可以到達的點。 • BFS利用Queue作為儲存將要探索的點的資料結構。 Elementary Graph Algorithms

  9. Breadth-first search • BFS是許多重要的圖形演算法的原型，如 Prim’s minimum spanning tree演算法以及Dijkstra’s single-source shortest-path演算法。 Elementary Graph Algorithms

  10. r s t u Q ∞ 0 ∞ ∞ (a) s 0 ∞ ∞ ∞ ∞ v w x y r s t u Q 1 0 ∞ ∞ (b) w r 1 1 ∞ 1 ∞ ∞ v w x y r s t u Q 1 0 2 ∞ (c) r t x 1 2 2 ∞ 1 2 ∞ v w x y BFS的運作範例 Elementary Graph Algorithms

  11. BFS的運作範例 r s t u Q 1 0 2 ∞ (d) t x v 2 2 2 2 1 2 ∞ v w x y r s t u Q 1 0 2 3 (e) v u x 2 2 3 2 1 2 ∞ v w x y r s t u Q 1 0 2 3 (f) v u y 2 3 3 2 1 2 3 v w x y Elementary Graph Algorithms

  12. BFS的運作範例 r s t u Q 1 0 2 3 (g) u y 3 3 2 1 2 3 v w x y r s t u Q 1 0 2 3 (h) y 3 2 1 2 3 v w x y r s t u Q 1 0 2 3 (i) empty 2 1 2 3 v w x y Elementary Graph Algorithms

  13. BFS演算法 • 起始點是s。 • 初始化時，將所有的點塗成白色。 • 已經發現的點，塗成綠色。 • 已經展開所有相鄰節點的點，塗成紅色。 • u.d儲存s到u的距離。 • u.π儲存自s到u的最短路徑中，u之前的一個點。 Elementary Graph Algorithms

  14. Time Complexity: O(|E|+|V|) BFS(G,s) • for each vertex u∈ G.V-{s} • u.color = WHITE • u.d = ∞ • u.π = NIL • s.color = GREEN • s.d = 0 • s.π = NIL • Q = empty • Enqueue(Q,s) • while Q≠empty • u = Dequeue(Q) • for each v∈ G.Adj[u] • if v.color == WHITE • v.color = GREEN • v.d = u.d + 1 • v.π = u • Enqueue(Q,v) • u.color = RED Elementary Graph Algorithms

  15. BFS演算法的性質 • 執行過BFS之後，自s可達的點都被塗成紅色。 • 如v是s可達的點，則v.d代表s到v的最短路徑長。 • 如v是s可達的點，則(v.π, v)是某條自s到v最短路徑的一個邊。 Elementary Graph Algorithms

  16. r s t u 1 0 2 3 2 1 2 3 v w x y BFS演算法的性質 • 邊集合T={(v.π,v): v自s可達}形成一個breadth-first tree。 藍色的邊形成一個Breadth-first tree Elementary Graph Algorithms

  17. Print path演算法 • 可在執行過BFS演算法的圖上印出s到v的最短路徑。如無路徑也會印出沒有路徑的訊息。 Print-Path(G,s,v) • if v == s • print s • elseif v.π== NIL • print “no path from” s “to” v • else Print-Path(G,s,v.π) • print v Elementary Graph Algorithms

  18. Depth-first search • Depth-first search(簡稱DFS，先深搜尋)是最簡單的圖形搜尋演算法之一。同樣用於搜尋圖形G中，所有自點s開始出發，有路徑可以到達的點。 • DFS利用Stack作為儲存已經開始探索但尚未結束的點的資料結構。 • 相較於BFS，DFS可以利用程式語言的遞迴來避免自行實做資料結構。 Elementary Graph Algorithms

  19. DFS演算法 • 起始點是s。 • 初始化時，將所有的點塗成白色。 • 點初次被發現的時候，塗成灰色。 • 點做完DFS-Visit的時候，塗成黑色。 • u.d儲存u被發現的時間。 • u.f儲存u做完DFS-Visit的時間。 • u.d < u.f Elementary Graph Algorithms

  20. DFS運作範例 發現時間 (a) (b) v w v w u u 1/ 1/ 2/ y z y z x x (c) (d) v w v w u u 1/ 2/ 1/ 2/ 3/ 4/ 3/ y z y z x x Elementary Graph Algorithms

  21. DFS運作範例 (e) (f) v w v w u u 1/ 2/ 1/ 2/ B B 4/ 3/ 4/5 3/ y z y z x x (g) v w u (h) v w u 1/ 2/ 1/ 2/7 B B 4/5 3/6 4/5 3/6 y z x y z x Elementary Graph Algorithms

  22. DFS運作範例 (i) v w u (j) v w u 1/ 2/7 1/8 2/7 B B F F 4/5 3/6 4/5 3/6 y z x y z x (k) v w u (l) v w u 1/8 2/7 9/ 1/8 2/7 9/ B F B C F 4/5 3/6 4/5 3/6 y z x y z x Elementary Graph Algorithms

  23. DFS運作範例 (m) v w u (n) v w u 1/8 2/7 9/ 1/8 2/7 9/ B C B F C F B 4/5 3/6 10/ 4/5 3/6 10/ y z x y z x (o) (o) v w v w u u 1/8 2/7 9/ 1/8 2/7 9/12 B B C C F F B B 4/5 3/6 10/11 4/5 3/6 10/11 y z y z x x Elementary Graph Algorithms

  24. DFS演算法 DFS(G) • for each vertex u∈ G.V • do u.color = WHITE • u.π = NIL • time = 0 • for each vertex u ∈ G.V • if u.color == WHITE • DFS-Visit(G,u) Elementary Graph Algorithms

  25. DFS-Visit演算法 DFS-Visit(G, u) • time = time+1 //u has just been discovered • u.d = time • u.color = GRAY • for each v ∈ G.Adj[u] • if v.color == WHITE • v.π = u • DFS-Visit(G, v) • u.color = BLACK //DFS-Visit(G, u) is done • time = time + 1 • u.f = time Elementary Graph Algorithms

  26. DFS的性質 • DFS演算法的時間複雜度為O(|V|+|E|)。 • 執行過DFS演算法之後，所有的點都是黑色的。 • {(v.π,v):v.π≠NIL}將形成一個Depth-first forest，也就是一個元素為Depth-first tree的集合。 • 如u到v之間在Depth-first forest中有一路徑，則稱u是v的ancestor，稱v是u的descendant。 Elementary Graph Algorithms

  27. 邊的分類 • 假定邊為(u,v) • Tree edges: 若初次發現v時是藉由(u,v)，即v.π=u，則此邊稱作tree edge。 • Back edges: 若v是u的ancestor，則此邊稱作back edge。 • Forward edges: 若v是u的descendant且(u,v)不是tree edge，則此邊稱作forward edge。 • Cross edges: 不屬於前三類的邊均稱為Cross edges。 Elementary Graph Algorithms

  28. (a) t z s y 11/16 3/6 2/9 1/10 B F C B 14/15 4/5 7/8 12/13 C C C u w v x (b) s t z v u w y x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ( s (z (y (x x) y) (w w) z) s) (t (v v) (u u) t) Elementary Graph Algorithms

  29. (c) s t C B F z C u v B C y w C x Elementary Graph Algorithms

  30. Topological sort • 對一DAG (Directed acyclic graph，有向無迴圈圖) G=(V,E)，Topological sort(拓樸排序)是一對V的排序，其中若(u,v) ∈ E則在排序中u必須排在v之前。 • 常用於決定排程。 Elementary Graph Algorithms

  31. 內褲 襪子 手錶 長褲 鞋子 襯衫 皮帶 領帶 夾克 襪子 內褲 長褲 鞋子 手錶 襯衫 皮帶 領帶 夾克 一個表示穿衣服先後順序的圖 經拓樸排序後，可依此順序穿上所有衣物。 Elementary Graph Algorithms

  32. Topological sort演算法 • 利用DFS演算法，計算出每一個點u的Finish time (u.f)。 • 當一個點執行完畢DFS-Visit時，將該點放入一公用的Linked List前端。 • 最後此Linked List存放的順序即為一Topological sort。 Elementary Graph Algorithms

  33. 內褲 襪子 17/18 11/16 手錶 9/10 12/15 長褲 鞋子 13/14 1/8 襯衫 皮帶 領帶 2/5 6/7 夾克 3/4 襪子 內褲 長褲 鞋子 手錶 襯衫 皮帶 領帶 夾克 17/18 11/16 12/15 13/14 9/10 1/8 6/7 2/5 3/4 Topological sort操作範例 每個點旁的數字代表Discovery time/Finish Time 以上為公用的Linked List在執行完DFS之後的順序。

  34. Lemma 22.11: A directed graph G is acyclic iff DFS(G) yields no back edges. Pf:  Suppose there is a back edge (u,v), v is an ancestor of u. Thus there is a path from v to u and a cycle exists. Suppose G has a cycle c. We show DFS(G) yields a back edge. Let v be the first vertex to be discovered in c, and (u,v) be the preceding edge in c. At time v.d, there is a path of white vertices from v to u. By the white-path thm., u becomes a descendant of v in the DF forest. Thus (u,v)is a back edge. Elementary Graph Algorithms

  35. Thm 22.12 TOPOLOGICAL-SORT(G) produces a topological sort of G pf: • Suppose DFS is run to determinate finishing times for vertices. • It suffices to show that for any pair of u,v , if there is an edge from u to v, then v.f < u.f. • When (u,v) is explored by DFS(G), v cannot be gray. • Therefore v must be either white or black. 1. If v is white, it becomes a descendant of u, so v.f<u.f 2. If v is black, then v.f<u.f Elementary Graph Algorithms

  36. Strongly connected components • 對一個有向圖G=(V,E)而言，一個Strongly Connected Component C是一個滿足下列條件的點集合： • C⊆V • 對任C中相異兩點u及v，存在一條路徑由u到v且有另一路徑由v到u。 Elementary Graph Algorithms

  37. Elementary Graph Algorithms

  38. Strongly connected components演算法 • 呼叫DFS(G)對所有點u，計算出u.f，即finishing time。 • 計算出GT，即點集合與G相同，而邊連接方向相反的圖。 • 呼叫DFS(GT)，但在DFS主迴圈中，選擇點的順序是先挑取u.f值較大的點u。(即以u.f遞減順序挑取。) • 在DFS(GT)的Depth-first forest中，每一個樹均是一個Strongly connected component。 Elementary Graph Algorithms

  39. a b c d 13/14 11/16 1/10 8/9 G: 12/15 3/4 2/7 5/6 cd e f g h abe h a b c d 13/14 11/16 1/10 8/9 fg GT: 12/15 3/4 2/7 5/6 e f g h

  40. Lemma 22.13 : Let C and C’ be distinct strongly connected components in directed graph G=(V, E), let u, v in C and u’, v’ in C’, and suppose there is a path from u to u’ in G. Then there cannot also be a path from v’ to v in G. Def: Let U  V, define d(U) = min u U { u.d } f (U) = max u U { u.f } Elementary Graph Algorithms

  41. Lemma 22.14 Let C and C’ be distinct strongly connected components in directed graph G=(V, E). Suppose that there is an edge (u, v) in E, where u in C and v in C’. Then f (C) > f ( C’). pf: Case (1): • If d(C) < d(C’): let x be the 1st discovered vertex in C. • At time x.d all vertices in C and C’ are white. • There is a path from x to all (white) vertices in C and to all vertices in C’: x~↝ u  v ~↝ w. • All vertices in C and C’ are descendants of x. Thus x.f = f (C) > f(C’). Elementary Graph Algorithms

  42. Lemma 22.14 Let C and C’ be distinct strongly connected components in directed graph G=(V, E). Suppose that there is an edge (u, v) in E, where u in C and v in C’. Then f (C) > f ( C’). pf: Case (2): • If d(C) > d(C’): let y be the 1st discovered vertex in C’. • At time y.d all vertices in C’ are white and there is a path from y to all vertices in C’. • I.e. all vertices in C’ are descendants of y. • So y.f = f(C’). There cannot be a path from C’ to C. Why? • Thus any w in C has w.f > y.f and so f(C) > f(C’). Elementary Graph Algorithms

  43. Corollary 22.15: Let C and C’ be distinct strongly connected components in directed graph G=(V, E). Suppose that there is an edge (u, v) in ET, where u in C and v in C’. Then f ( C ) < f ( C’ ). Pf: It is clear that (v, u) is in E. By the previous lemma we have f( C’ ) > f ( C ). Elementary Graph Algorithms

  44. Theorem 22.16 Strongly-Connected-Component(G) correctly computes the strongly connected components of a directed graph. pf: • By induction on the number (k) of depth-first trees found in the DFS of GT, where each tree forms a strongly connected component. • Basis: trivial for k=0. • Inductive step: assume each of the first k DFS trees produced in line 3 is a SCC. • Consider the (k+1)-st tree produced. Let u be the root of this tree and let u be in SCC C. • Thus u.f = f ( C ) > f ( C’ ) for any other SCC C’ yet to be visited. Note we visit in decreasing finishing time. Elementary Graph Algorithms

  45. All other vertices in C are descendant of u in its DFS. • By inductive hypothesis and the Corollary 15 any edges in GT that leave C must be to SCC’s already visited. • Thus, no vertex in any SCC other than C will be a descendant of u during the DFS of GT. • Thus, the vertices of the DFS tree in GT that is rooted at u form exactly one SCC.  Elementary Graph Algorithms

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