Graph Algorithms L.F.O.A. Lecture Full Of Acronyms Graph Search - PDF document

15-251: Great Theoretical Ideas in Computer Science Lecture 11 Graph Algorithms L.F.O.A. Lecture Full Of Acronyms Graph Search Algorithms The most basic graph algorithms: Given a graph G = (V,E) … • Check if vertex s can reach vertex t. BFS: Breadth-first search • Decide if G is connected. DFS: Depth-first search • Identify connected components of G. All reduce to: AFS: Arbitrary-first search “Given s∈V , identify all nodes reachable from s .” (We’ll call this set C ONN C OMP (s).) What problems do these algorithms solve? Algorithm AFS(G,s) does exactly this. Bonus of AFS(G,s): Bonus of AFS(G,s): Finds a spanning tree of C ONN C OMP (s) rooted at s. Finds a spanning tree of C ONN C OMP (s) rooted at s. Given G = (V,E), a spanning tree is Given G = (V,E), a spanning tree is a tree T = ( V,Eʹ) such that Eʹ ⊆ E. a tree T = ( V,Eʹ) such that Eʹ ⊆ E. More informally, a minimal set of edges p q r connecting up all vertices of G. w x s y z t u v 1

Bonus of AFS(G,s): Bonus of AFS(G,s): Finds a spanning tree of C ONN C OMP (s) rooted at s. Finds a spanning tree of C ONN C OMP (s) rooted at s. Given G = (V,E), a spanning tree is Given G = (V,E), a spanning tree is a tree T = ( V,Eʹ) such that Eʹ ⊆ E. a tree T = ( V,Eʹ) such that Eʹ ⊆ E. p q r p q r w x s y z w x s y z t u v t u v AFS(G,s): Finding all nodes reachable from s AFS(G,s): Finding all nodes reachable from s G p q r V = { a,b,c,p,q,r,s,t,u,v,w,x,y,z } a w x s y z E = { {a,b},{a,c},{b,c},{p,q},{p,x},{q,r}, b c {q,s},{r,y},{s,u},{s,x},{s,y},{t,u}, t u v {t,x},{u,v},{v,y},{w,x},{y,z} } “Duh, it’s these ones.” But it’s not so obvious when the input looks like… AFS(G,s): Finding all nodes reachable from s G: 1 2 3 4 // Has a “bag” data structure holding tiles // Each tile has a vertex name written on it s = 1 Put s into bag 5 6 7 8 While bag is not empty: Pick an Arbitrary tile v from bag If v is “unmarked”: AFS(G,s): “Mark” v For each neighbor w of v: Put s into bag Put w into bag While bag is not empty: Pick arbitrary tile v from bag Intent: If v is “unmarked”: 1 “Marked” vertices should be those reachable from s. “Mark” v For each neighbor w of v: w in bag means we want to keep exploring from w. Put w into bag 2

G: G: 1 2 3 4 1 2 3 4 s = 1 s = 1 5 6 7 8 5 6 7 8 AFS(G,s): AFS(G,s): Put s into bag Put s into bag While bag is not empty: While bag is not empty: Pick arbitrary tile v from bag Pick arbitrary tile v from bag If v is “unmarked”: If v is “unmarked”: 1 1 “Mark” v “Mark” v For each neighbor w of v: For each neighbor w of v: Put w into bag Put w into bag ✓ G: G: 1 2 3 4 1 2 3 4 s = 1 s = 1 5 6 7 8 5 6 7 8 1 1 AFS(G,s): AFS(G,s): Put s into bag Put s into bag While bag is not empty: While bag is not empty: Pick arbitrary tile v from bag Pick arbitrary tile v from bag If v is “unmarked”: If v is “unmarked”: “Mark” v “Mark” v For each neighbor w of v: For each neighbor w of v: Put w into bag Put w into bag ✓ ✓ G: G: 1 2 3 4 1 2 3 4 s = 1 s = 1 5 6 7 8 5 6 7 8 1 1 AFS(G,s): AFS(G,s): Put s into bag Put s into bag While bag is not empty: While bag is not empty: Pick arbitrary tile v from bag Pick arbitrary tile v from bag If v is “unmarked”: If v is “unmarked”: 2 5 2 5 “Mark” v “Mark” v For each neighbor w of v: For each neighbor w of v: 6 6 Put w into bag Put w into bag 3

✓ ✓ G: G: 1 2 3 4 1 2 3 4 s = 1 s = 1 5 6 7 8 5 6 7 8 6 AFS(G,s): AFS(G,s): Put s into bag Put s into bag While bag is not empty: While bag is not empty: Pick arbitrary tile v from bag Pick arbitrary tile v from bag If v is “unmarked”: If v is “unmarked”: 2 5 2 5 “Mark” v “Mark” v For each neighbor w of v: For each neighbor w of v: 6 Put w into bag Put w into bag ✓ ✓ G: G: 1 2 3 4 1 2 3 4 s = 1 s = 1 ✓ ✓ 5 6 7 8 5 6 7 8 6 6 AFS(G,s): AFS(G,s): Put s into bag Put s into bag While bag is not empty: While bag is not empty: Pick arbitrary tile v from bag Pick arbitrary tile v from bag If v is “unmarked”: If v is “unmarked”: 2 5 2 5 “Mark” v “Mark” v 1 2 For each neighbor w of v: For each neighbor w of v: 7 5 Put w into bag Put w into bag ✓ ✓ G: G: 1 2 3 4 1 2 3 4 s = 1 s = 1 ✓ ✓ 5 6 7 8 5 6 7 8 6 AFS(G,s): AFS(G,s): Put s into bag Put s into bag While bag is not empty: While bag is not empty: Pick arbitrary tile v from bag Pick arbitrary tile v from bag If v is “unmarked”: If v is “unmarked”: 2 5 2 5 “Mark” v “Mark” v 1 2 1 2 For each neighbor w of v: For each neighbor w of v: 7 7 5 5 Put w into bag Put w into bag 4

✓ ✓ G: G: 1 2 3 4 1 2 3 4 s = 1 s = 1 ✓ ✓ ✓ 5 6 7 8 5 6 7 8 7 7 AFS(G,s): AFS(G,s): Put s into bag Put s into bag While bag is not empty: While bag is not empty: Pick arbitrary tile v from bag Pick arbitrary tile v from bag If v is “unmarked”: If v is “unmarked”: 2 5 2 5 “Mark” v “Mark” v 1 2 1 2 For each neighbor w of v: For each neighbor w of v: 5 5 Put w into bag Put w into bag ✓ ✓ G: G: 1 2 3 4 1 2 3 4 s = 1 s = 1 ✓ ✓ ✓ ✓ 5 6 7 8 5 6 7 8 7 7 AFS(G,s): AFS(G,s): Put s into bag Put s into bag While bag is not empty: While bag is not empty: Pick arbitrary tile v from bag Pick arbitrary tile v from bag If v is “unmarked”: If v is “unmarked”: 2 5 2 5 “Mark” v “Mark” v 1 2 2 1 2 2 For each neighbor w of v: For each neighbor w of v: 6 6 5 3 5 3 Put w into bag Put w into bag ✓ ✓ G: G: 1 2 3 4 1 2 3 4 s = 1 s = 1 ✓ ✓ ✓ ✓ 5 6 7 8 5 6 7 8 1 AFS(G,s): AFS(G,s): Put s into bag Put s into bag While bag is not empty: While bag is not empty: Pick arbitrary tile v from bag Pick arbitrary tile v from bag If v is “unmarked”: If v is “unmarked”: 2 5 2 5 “Mark” v “Mark” v 1 2 2 2 2 For each neighbor w of v: For each neighbor w of v: 6 6 5 3 5 3 Put w into bag Put w into bag 5

✓ ✓ G: G: 1 2 3 4 1 2 3 4 s = 1 s = 1 ✓ ✓ ✓ ✓ 5 6 7 8 5 6 7 8 1 et cetera AFS(G,s): AFS(G,s): Put s into bag Put s into bag While bag is not empty: While bag is not empty: Pick arbitrary tile v from bag Pick arbitrary tile v from bag If v is “unmarked”: If v is “unmarked”: 2 5 2 5 “Mark” v “Mark” v 2 2 2 2 For each neighbor w of v: For each neighbor w of v: 6 6 5 3 5 3 Put w into bag Put w into bag Analysis of AFS Why does AFS halt? Every time a bunch of tiles is added to bag, Want to show: When this algorithm halts, it’s because some vertex v just got marked. { marked vertices } ∴ we add at most |V| bunches of tiles to the bag = (since each vertex is marked ≤ 1 time). .{ vertices reachable from s }. ∴ at most finitely many AFS(G,s): tiles ever go into the bag. Put s into bag { marked } ⊆ { reachable } : This is clear. While bag is not empty: Each iteration through Pick arbitrary tile v from bag loop removes 1 tile. { reachable } ⊆ { marked } : If v is “unmarked”: “Mark” v ∴ AFS halts after finitely Wait, why does the algorithm even halt?! For each neighbor w of v: many iterations. Put w into bag A more careful analysis A more careful analysis Every time a bunch of tiles is added to bag, Every time a bunch of tiles is added to bag, it’s because some vertex v just got marked. it’s because some vertex v just got marked. In this case, we add deg(v) tiles to the bag. In this case, we add deg(v) tiles to the bag. ∴ total number of tiles that ever enter the bag is ∴ total number of tiles that ever enter the bag is = 2|E| = 2|E| ≤ ≤ AFS(G,s): AFS(G,s): Put s into bag Put s into bag While bag is not empty: While bag is not empty: Each iteration through Each iteration through Pick arbitrary tile v from bag Pick arbitrary tile v from bag loop removes 1 tile. loop removes 1 tile. If v is “unmarked”: If v is “unmarked”: “Mark” v “Mark” v ∴ AFS halts after finitely ∴ AFS halts after ≤ 2|E| For each neighbor w of v: For each neighbor w of v: many iterations. many iterations. Put w into bag Put w into bag 6