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Advanced D epth - F irst S earch and B readth- F irst S earch

Advanced D epth - F irst S earch and B readth- F irst S earch. Overview. Depth-first search (DFS) DFS Forest Breadth-first search (BFS) Some variants of DFS and BFS Graph modeling. Prerequisites. Elementary graph theory Implementations of DFS and BFS. vertex. edge. What is a graph?.

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Advanced D epth - F irst S earch and B readth- F irst S earch

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  1. AdvancedDepth-First Search and Breadth-First Search

  2. Overview • Depth-first search (DFS) • DFS Forest • Breadth-first search (BFS) • Some variants of DFS and BFS • Graph modeling HKOI Training 2004

  3. Prerequisites • Elementary graph theory • Implementations of DFS and BFS HKOI Training 2004

  4. vertex edge What is a graph? • A set of vertices and edges HKOI Training 2004

  5. ancestors root parent siblings descendents children Trees and related terms HKOI Training 2004

  6. What is graph traversal? • Given: a graph • Goal: visit all (or some) vertices and edges of the graph using some strategy • Two simple strategies: • Depth-first search • Breadth-first search HKOI Training 2004

  7. Depth-first search (DFS) • A graph searching method • Algorithm: at any time, go further (depth) if you can; otherwise, retreat HKOI Training 2004

  8. DFS (Pseudo code) DFS (vertex u) { mark u as visited for each vertex v directly reachable from u if v is unvisited DFS (v) } • Initially all vertices are marked as unvisited HKOI Training 2004

  9. F E unvisited visited A D B C DFS (Demonstration) HKOI Training 2004

  10. “Advanced” DFS • Apart from just visiting the vertices, DFS can also provide us with valuable information • DFS can be enhanced by introducing: • birth time and death time of a vertex • birth time: when the vertex is first visited • death time: when we retreat from the vertex • DFS tree • parent of a vertex (see next slide) HKOI Training 2004

  11. DFS tree / forest • A rooted tree • The root is the start vertex • If v is first visited from u, then u is the parent of v in the DFS tree HKOI Training 2004

  12. A D B E G E F C A D F H unvisited B visited visited (dead) C G H DFS forest (Demonstration) 1 2 3 13 10 4 14 6 12 9 8 16 11 5 15 7 - A B - A C D C HKOI Training 2004

  13. A D B E G C F H Classification of edges • Tree edge • Forward edge • Back edge • Cross edge • Question: which type of edges is always absent in an undirected graph? HKOI Training 2004

  14. Determination of edge types • How to determine the type of an arbitrary edge (u, v) after DFS? • Tree edge • parent [v] = u • Forward edge • not a tree edge; and • birth [v] > birth [u]; and • death [v] < death [u] • How about back edge and cross edge? HKOI Training 2004

  15. Applications of DFS Forests • Topological sorting (Tsort) • Strongly-connected components (SCC) • Some more “advanced” algorithms HKOI Training 2004

  16. Example: SCC • A graph is strongly-connected if • for any pair of vertices u and v, one can go from u to v and from v to u. • Informally speaking, an SCC of a graph is a subset of vertices that • forms a strongly-connected subgraph • does not form a strongly-connected subgraph with the addition of any new vertex HKOI Training 2004

  17. SCC (Illustration) HKOI Training 2004

  18. SCC (Algorithm) • Compute the DFS forest of the graph G • Reverse all edges in G to form G’ • Compute a DFS forest of G’, but always choose the vertex with the latest death time when choosing the root for a new tree • The SCCs of G are the DFS trees in the DFS forest of G’ HKOI Training 2004

  19. F A D B C G H SCC (Demonstration) 1 2 3 13 10 4 14 6 12 9 8 16 11 5 15 7 - A B - A C D C E F A E B H D A D G F B C C G H HKOI Training 2004

  20. A E B H E D F G F C A D B C G H SCC (Demonstration) HKOI Training 2004

  21. Breadth-first search (BFS) • A graph searching method • Instead of searching “deeply” along one path, BFS tries to search all paths at the same time • Makes use of a data structure - queue HKOI Training 2004

  22. BFS (Pseudo code) while queue not empty dequeue the first vertex u from queue for each vertex v directly reachable from u if v is unvisited enqueue v to queue mark v as visited • Initially all vertices except the start vertex are marked as unvisited and the queue contains the start vertex only HKOI Training 2004

  23. I G D C H unvisited visited A E J visited (dequeued) F B BFS (Demonstration) Queue: A B C F D E H G J I HKOI Training 2004

  24. Applications of BFS • Shortest paths finding • Flood-fill (can also be handled by DFS) HKOI Training 2004

  25. Comparisons of DFS and BFS HKOI Training 2004

  26. start goal Bidirectional search (BDS) • Searches simultaneously from both the start vertex and goal vertex • Commonly implemented as bidirectional BFS HKOI Training 2004

  27. Iterative deepening search (IDS) • Iteratively performs DFS with increasing depth bound • Shortest paths are guaranteed HKOI Training 2004

  28. What is graph modeling? • Conversion of a problem into a graph problem • Sometimes a problem can be easily solved once its underlying graph model is recognized • Graph modeling appears almost every year in NOI or IOI HKOI Training 2004

  29. Basics of graph modeling • A few steps: • identify the vertices and the edges • identify the objective of the problem • state the objective in graph terms • implementation: • construct the graph from the input instance • run the suitable graph algorithms on the graph • convert the output to the required format HKOI Training 2004

  30. start goal Simple examples (1) • Given a grid maze with obstacles, find a shortest path between two given points HKOI Training 2004

  31. Simple examples (2) • A student has the phone numbers of some other students • Suppose you know all pairs (A, B) such that A has B’s number • Now you want to know Alan’s number, what is the minimum number of calls you need to make? HKOI Training 2004

  32. Simple examples (2) • Vertex: student • Edge: whether A has B’s number • Add an edge from A to B if A has B’s number • Problem: find a shortest path from your vertex to Alan’s vertex HKOI Training 2004

  33. Complex examples (1) • Same settings as simple example 1 • You know a trick – walking through an obstacle! However, it can be used for only once • What should a vertex represent? • your position only? • your position + whether you have used the trick HKOI Training 2004

  34. Complex examples (1) • A vertex is in the form (position, used) • The vertices are divided into two groups • trick used • trick not used HKOI Training 2004

  35. start goal Complex examples (1) unused start goal used goal HKOI Training 2004

  36. 1 2 3 4 5 6 7 8 Complex examples (2) • The famous 8-puzzle • Given a state, find the moves that bring it to the goal state HKOI Training 2004

  37. Complex examples (2) • What does a vertex represent? • the position of the empty square? • the number of tiles that are in wrong positions? • the state (the positions of the eight tiles) • What are the edges? • What is the equivalent graph problem? HKOI Training 2004

  38. 1 2 3 4 5 6 1 2 3 7 8 4 5 6 7 8 1 2 3 4 6 7 5 8 1 1 1 2 2 2 3 3 4 4 4 5 5 5 3 7 7 7 8 8 8 6 6 6 1 2 3 4 5 6 7 8 Complex examples (2) HKOI Training 2004

  39. Complex examples (3) • Theseus and Minotaur • http://www.logicmazes.com/theseus.html • Extract: • Theseus must escape from a maze. There is also a mechanical Minotaur in the maze. For every turn that Theseus takes, the Minotaur takes two turns. The Minotaur follows this program for each of his two turns: • First he tests if he can move horizontally and get closer to Theseus. If he can, he will move one square horizontally. If he can’t, he will test if he could move vertically and get closer to Theseus. If he can, he will move one square vertically. If he can’t move either horizontally or vertically, then he just skips that turn. HKOI Training 2004

  40. Complex examples (3) • What does a vertex represent? • Theseus’ position • Minotaur’s position • Both • How long do you need to solve the last maze? • How long does a well-written program take to solve it? HKOI Training 2004

  41. Some more examples • How can the followings be modeled? • Tilt maze (Single-goal mazes only) • http://www.clickmazes.com/newtilt/ixtilt2d.htm • Double title maze • http://www.clickmazes.com/newtilt/ixtilt.htm • No-left-turn maze • http://www.clickmazes.com/noleft/ixnoleft.htm • Same as complex example 1, but you can use the trick for k times HKOI Training 2004

  42. Competition problems • HKOI2000 S – Wormhole Labyrinth • HKOI2001 S – A Node Too Far • HKOI2004 S – Teacher’s Problem * • TFT2001 – OIMan * • TFT2002 – Bomber Man * • NOI2001 – cung1 ming4 dik7 daa2 zi6 jyun4 • IOI2000 – Walls * • IOI2002 – Troublesome Frog • IOI2003 – Amazing Robots * HKOI Training 2004

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