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ECE 242 Data Structures Lecture 29 Graph Traversal November 23, - PDF document

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ECE 242 Data Structures Lecture 29 Graph Traversal November 23, 2009 ECE242 L29: Graph Traversal Overview Problem: How can we efficiently visit all vertices in the graph? Depth first search Quickly follow paths between vertices

  1. ECE 242 Data Structures Lecture 29 Graph Traversal November 23, 2009 ECE242 L29: Graph Traversal Overview ° Problem: How can we efficiently visit all vertices in the graph? ° Depth first search • Quickly follow paths between vertices ° Breadth first search • Visit all neighbors of a vertex ° Compact algorithms available to perform traversals • Note Galaxy example and cheap flights example ° Useful for many applications November 23, 2009 ECE242 L29: Graph Traversal

  2. Graph Traversal ° List out all cities that United Airline can reach from Hartford Airport CHI Hartford SF NYC LA W. DC November 23, 2009 ECE242 L29: Graph Traversal Graph Traversal ° From vertex u, list out all vertices that u can reach via graph G ° Set of nodes to expand ° Each node has a flag to indicate visited or not ° Keep visiting until all vertices are visited ° Some nodes may not be connected November 23, 2009 ECE242 L29: Graph Traversal

  3. Demos For Traversal Algorithm ° Step 1: { Hartford } • find neighbors of Hartford • { Hartford, NYC, CHI } CHI Hartford SF NYC LA W. DC November 23, 2009 ECE242 L29: Graph Traversal Demos For Traversal Algorithm ° Step 2: { Hartford, NYC, CHI } • find neighbors of NYC, CHI • { Hartford, NYC, CHI, LA, SF } CHI Hartford SF NYC LA W. DC November 23, 2009 ECE242 L29: Graph Traversal

  4. Demos For Traversal Algorithm ° Step 3: {Hartford, NYC, CHI, LA, SF } • find neighbors of LA, SF • no other new neighbors CHI Hartford SF NYC LA W. DC November 23, 2009 ECE242 L29: Graph Traversal Demos For Traversal Algorithm ° Finally we get all cities that United Airline can reach from Hartford Airport • {Hartford, NYC, CHI, LA, SF } CHI Hartford SF NYC LA W. DC November 23, 2009 ECE242 L29: Graph Traversal

  5. Algorithm of Graph Traversal 1. Mark all nodes as unvisited 2. Pick a starting vertex u, add u to probing list 3. While ( probing list is not empty) { Remove a node v from probing list Mark node v as visited For each neighbor w of v, if w is unvisited, add w to the probing list } November 23, 2009 ECE242 L29: Graph Traversal Graph Traversal ° Source vertex • We must choose an arbitrary vertex to start from. ° Depth-first traversal • Follows edges until it reaches a dead end, then backtracks to the last branching point. ° Breadth-first • Visits one vertex, then its neighbors, then vertices two edges away, and so on. November 23, 2009 ECE242 L29: Graph Traversal

  6. Graph Traversal ° Depth-first traversal Start Note that all 8 nodes are visited eventually November 23, 2009 ECE242 L29: Graph Traversal Depth First Traversal ° Probing List is implemented as stack (LIFO) ° Example • A’s neighbor: B, C, E • B’s neighbor: A, C, F A • C’s neighbor: A, B, D • D’s neighbor: E, C, F • E’s neighbor: A, D B E C • F’s neighbor: B, D • start from vertex A D F November 23, 2009 ECE242 L29: Graph Traversal

  7. Depth First Traversal (Cont) – A’s neighbor: B C E A – B’s neighbor: A C F – C’s neighbor: A B D – D’s neighbor: E C F B E C – E’s neighbor: A D – F’s neighbor: B D D F ° Initial State • Visited Vertices { } • Probing Vertices { A } • Unvisited Vertices { A, B, C, D, E, F } stack A November 23, 2009 ECE242 L29: Graph Traversal Depth First Traversal (Cont) – A’s neighbor: B C E A – B’s neighbor: A C F – C’s neighbor: A B D – D’s neighbor: E C F B E C – E’s neighbor: A D – F’s neighbor: B D D F ° Peek a vertex from stack, it is A, mark it as visited ° Find A’s first unvisited neighbor, push it into stack • Visited Vertices { A } • Probing vertices { A, B } B • Unvisited Vertices { B, C, D, E, F } A A stack November 23, 2009 ECE242 L29: Graph Traversal

  8. Depth First Traversal (Cont) – A’s neighbor: B C E A – B’s neighbor: A C F – C’s neighbor: A B D – D’s neighbor: E C F B E C – E’s neighbor: A D – F’s neighbor: B D D F ° Peek a vertex from stack, it is B, mark it as visited ° Find B’s first unvisited neighbor, push it in stack C • Visited Vertices { A, B } • Probing Vertices { A, B, C } B B • Unvisited Vertices { C, D, E, F } A A stack November 23, 2009 ECE242 L29: Graph Traversal Depth First Traversal (Cont) – A’s neighbor: B C E A – B’s neighbor: A C F – C’s neighbor: A B D – D’s neighbor: E C F B E C – E’s neighbor: A D – F’s neighbor: B D D F ° Peek a vertex from stack, it is C, mark it as visited ° Find C’s first unvisited neighbor, push it in stack D • Visited Vertices { A, B, C } C C • Probing Vertices { A, B, C, D } B B • Unvisited Vertices { D, E, F } A A stack November 23, 2009 ECE242 L29: Graph Traversal

  9. Depth First Traversal (Cont) – A’s neighbor: B C E A – B’s neighbor: A C F – C’s neighbor: A B D – D’s neighbor: E C F B E C – E’s neighbor: A D – F’s neighbor: B D D F ° Peek a vertex from stack, it is D, mark it as visited E ° Find D’s first unvisited neighbor, push it in stack D D C C • Visited Vertices { A, B, C, D } • Probing Vertices { A, B, C, D, E } B B • Unvisited Vertices { E, F } A A stack November 23, 2009 ECE242 L29: Graph Traversal Depth First Traversal (Cont) – A’s neighbor: B C E A – B’s neighbor: A C F – C’s neighbor: A B D – D’s neighbor: E C F B E C – E’s neighbor: A D – F’s neighbor: B D D F ° Peek a vertex from stack, it is E, mark it as visited E ° Find E’s first unvisited neighbor, no vertex found, Pop E D D C C • Visited Vertices { A, B, C, D, E } • Probing Vertices { A, B, C, D } B B • Unvisited Vertices { F } A A stack November 23, 2009 ECE242 L29: Graph Traversal

  10. Depth First Traversal (Cont) – A’s neighbor: B C E A – B’s neighbor: A C F – C’s neighbor: A B D – D’s neighbor: E C F B E C – E’s neighbor: A D – F’s neighbor: B D D F ° Peek a vertex from stack, it is D, mark it as visited F ° Find D’s first unvisited neighbor, push it in stack D D C C • Visited Vertices { A, B, C, D, E } • Probing Vertices { A, B, C, D, F} B B • Unvisited Vertices { F } A A stack November 23, 2009 ECE242 L29: Graph Traversal Depth First Traversal (Cont) – A’s neighbor: B C E A – B’s neighbor: A C F – C’s neighbor: A B D – D’s neighbor: E C F B E C – E’s neighbor: A D – F’s neighbor: B D D F ° Peek a vertex from stack, it is F, mark it as visited F ° Find F’s first unvisited neighbor, no vertex found, Pop F D D C C • Visited Vertices { A, B, C, D, E, F } • Probing Vertices { A, B, C, D} B B • Unvisited Vertices { } A A stack November 23, 2009 ECE242 L29: Graph Traversal

  11. Depth First Traversal (Cont) – A’s neighbor: B C E A – B’s neighbor: A C F – C’s neighbor: A B D – D’s neighbor: E C F B E C – E’s neighbor: A D – F’s neighbor: B D D F ° Peek a vertex from stack, it is D, mark it as visited ° Find D’s first unvisited neighbor, no vertex found, Pop D D C C • Visited Vertices { A, B, C, D, E, F } • Probing Vertices { A, B, C } B B • Unvisited Vertices { } A A stack November 23, 2009 ECE242 L29: Graph Traversal Depth First Traversal (Cont) – A’s neighbor: B C E A – B’s neighbor: A C F – C’s neighbor: A B D – D’s neighbor: E C F B E C – E’s neighbor: A D – F’s neighbor: B D D F ° Peek a vertex from stack, it is C, mark it as visited ° Find C’s first unvisited neighbor, no vertex found, Pop C C • Visited Vertices { A, B, C, D, E, F } • Probing Vertices { A, B } B B • Unvisited Vertices { } A A stack November 23, 2009 ECE242 L29: Graph Traversal

  12. Depth First Traversal (Cont) – A’s neighbor: B C E A – B’s neighbor: A C F – C’s neighbor: A B D – D’s neighbor: E C F B E C – E’s neighbor: A D – F’s neighbor: B D D F ° Peek a vertex from stack, it is B, mark it as visited ° Find B’s first unvisited neighbor, no vertex found, Pop B • Visited Vertices { A, B, C, D, E, F } • Probing Vertices { A } B • Unvisited Vertices { } A A stack November 23, 2009 ECE242 L29: Graph Traversal Depth First Traversal (Cont) – A’s neighbor: B C E A – B’s neighbor: A C F – C’s neighbor: A B D – D’s neighbor: E C F B E C – E’s neighbor: A D – F’s neighbor: B D D F ° Peek a vertex from stack, it is A, mark it as visited ° Find A’s first unvisited neighbor, no vertex found, Pop A • Visited Vertices { A, B, C, D, E, F } • Probing Vertices { } • Unvisited Vertices { } A stack November 23, 2009 ECE242 L29: Graph Traversal

  13. Depth First Traversal (Cont) – A’s neighbor: B C E A – B’s neighbor: A C F – C’s neighbor: A B D – D’s neighbor: E C F B E C – E’s neighbor: A D – F’s neighbor: B D D F ° Now probing list is empty ° End of Depth First Traversal • Visited Vertices { A, B, C, D, E, F } • Probing Vertices { } • Unvisited Vertices { } stack November 23, 2009 ECE242 L29: Graph Traversal Graph Traversal ° Breadth-first traversal Start Visit ALL neighbors at each step November 23, 2009 ECE242 L29: Graph Traversal

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