ECE250: Algorithms and Data Structures

Elementary Graph Algorithms– Part B

Materials from CLRS: Chapter 22.3

Ladan Tahvildari, PEng, SMIEEE Professor

Software Technologies Applied Research (STAR) Group

Dept. of Elect. & Comp. Eng.

University of Waterloo

Acknowledgements

v The following resources have been used to prepare materials for this course: Ø  MIT OpenCourseWare Ø  Introduction To Algorithms (CLRS Book) Ø  Data Structures and Algorithm Analysis in C++ (M. Wiess) Ø  Data Structures and Algorithms in C++ (M. Goodrich)

v Thanks to many people for pointing out mistakes, providing suggestions, or helping to improve the quality of this course over the last ten years: Ø  http://www.stargroup.uwaterloo.ca/~ece250/

acknowledgment/ Lecture 25 ECE250 2

Lecture 25 ECE250 3

Depth-First Search v A depth-first search (DFS) in a graph G is like

wandering in a labyrinth with a string and a can of paint Ø  We start at vertex s, tying the end of our string to the point

and painting s “visited (discovered)”. Next we label s as our current vertex called u

Ø  Now, we travel along an arbitrary edge (u,v)

Ø  If edge (u,v) leads us to an already visited vertex v we return to u

Ø  If vertex v is unvisited, we unroll our string, move to v, paint v “visited”, set v as our current vertex, and repeat the previous steps

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Depth-First Search (cont’)

v Eventually, we will get to a point where all incident edges on u lead to visited vertices

v We then backtrack by unrolling our string to a previously visited vertex v. Then v becomes our current vertex and we repeat the previous steps

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DFS Algorithm

Visit all children recursively

DFS(G) 01 for each vertex u ∈ G.V() 02 u.setcolor(white)03 u.setparent(NIL) 04 time ← 0 05 for each vertex u ∈ G.V() 06 If u.color() = white then DFS-Visit(u) DFS-Visit(u) 01 u.setcolor(gray) 02 time ← time + 1 03 u.setd(time) 04 for each v ∈ u.adjacent() 05 if v.color() = white then 06 v.setparent(u) 07 DFS-Visit(v) 08 u.setcolor(black) 09 time ← time + 1 10 u.setf(time)

Init all vertices Make sure all vertices are visited

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DFS Algorithm (cont’)

v  Initialize – color all vertices white

v Visit each and every white vertex using DFS-Visit

v Each call to DFS-Visit(u) roots a new tree of the depth-first forest at vertex u

v When DFS returns, every vertex u is assigned Ø  a discovery time d[u], and a finishing time f[u]

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DFS Timestamping

v Vertex u is Ø white before time d[u] Ø gray between time d[u] and time f[u], and Ø black thereafter

v Notice the structure througout the algorithm Ø gray vertices form a linear chain Ø correponds to a stack of vertices that have not

been exhaustively explored (DFS-Visit started but not yet finished)

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An Example

v Let’s do an example of DFS:

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DFS Algorithm Running Time

v Running time Ø  the loops in DFS take time Θ(V) each, excluding the time to

execute DFS-Visit Ø  DFS-Visit is called once for every vertex

§  its only invoked on white vertices, and §  paints the vertex gray immediately

Ø  for each DFS-visit a loop interates over all v.adjacent()

Ø  the total cost for DFS-Visit is Θ(E) Ø  the running time of DFS is Θ(V+E)

. () ( )v Vv adjacent E

∈

=Θ∑

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Generic Graph Search

v  BFS, when GrayVertices is a Queue v  DFS, when GrayVertices is a Stack

GenericGraphSearch(G,s) 01 for each vertex u ∈ G.V() 02 u.setcolor(white) 03 u.setparent(NIL) 04 s.setcolor(gray) 05 GrayVertices.init() 06 GrayVertices.add(s) 07 while not GrayVertices.isEmpty()08 u ← GrayVertices.remove() 09 for each v ∈ u.adjacent() do 10 if v.color() = white then 11 v.setcolor(gray) 12 v.setparent(u) 13 GrayVertices.add(v) 14 u.setcolor(black)

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DFS Parenthesis Theorem

v Discovery and finish times have parenthesis structure Ø  represent discovery of u with left parenthesis "(u" Ø  represent finishin of u with right parenthesis "u)" Ø  history of discoveries and finishings makes a well-formed

expression (parenthesis are properly nested)

v  Intuition for proof: any two intervals are either disjoint or enclosed Ø  Overlaping intervals would mean finishing ancestor, before

finishing descendant or starting descendant without starting ancestor

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DFS Parenthesis Theorem (cont’)

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DFS Edge Classification

v Tree edge (gray to white) Ø Edges in depth-first forest

v Back edge (gray to gray) Ø  from descendant to ancestor in depth-first tree

Edge (u,v) is tree edge if v was first discovered by exploring edge (u,v)

Edge (u,v) connecting u to an ancestor v in depth first tree.

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DFS Edge Classification (cont’)

v Forward edge (gray to black) Ø Nontree edge from ancestor to descendant in

depth-first tree v Cross edge (gray to black)

Ø remainder – between trees or subtrees

Between vertices in same depth first tree (one vertex is not ancestor of the other) or ...

Lecture 25