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- 1. Advanced DFS & BFS HKOI Training 2004 1 Advanced Depth-First Search and Breadth-First Search
- 2. Advanced DFS & BFS HKOI Training 2004 2 Overview • Depth-first search (DFS) • DFS Forest • Breadth-first search (BFS) • Some variants of DFS and BFS • Graph modeling
- 3. Advanced DFS & BFS HKOI Training 2004 3 Prerequisites • Elementary graph theory • Implementations of DFS and BFS
- 4. Advanced DFS & BFS HKOI Training 2004 4 What is a graph? • A set of vertices and edges vertex edge
- 5. Advanced DFS & BFS HKOI Training 2004 5 Trees and related terms root siblings descendents children ancestors parent
- 6. Advanced DFS & BFS 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
- 7. Advanced DFS & BFS HKOI Training 2004 7 Depth-first search (DFS) • A graph searching method • Algorithm: at any time, go further (depth) if you can; otherwise, retreat
- 8. Advanced DFS & BFS 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
- 9. Advanced DFS & BFS HKOI Training 2004 9 F A B C D E DFS (Demonstration) unvisited visited
- 10. Advanced DFS & BFS 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)
- 11. Advanced DFS & BFS 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
- 12. Advanced DFS & BFS HKOI Training 2004 12 A F B C D E G H DFS forest (Demonstration) unvisited visited visited (dead) A B C D E F G H birth death parent A B C F E D G 1 2 3 13 10 4 14 12 9 8 16 11 5 15 H 6 7 - A B - A C D C
- 13. Advanced DFS & BFS HKOI Training 2004 13 Classification of edges • Tree edge • Forward edge • Back edge • Cross edge • Question: which type of edges is always absent in an undirected graph? A B C F E D G H
- 14. Advanced DFS & BFS 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?
- 15. Advanced DFS & BFS HKOI Training 2004 15 Applications of DFS Forests • Topological sorting (Tsort) • Strongly-connected components (SCC) • Some more “advanced” algorithms
- 16. Advanced DFS & BFS 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
- 17. Advanced DFS & BFS HKOI Training 2004 17 SCC (Illustration)
- 18. Advanced DFS & BFS 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’
- 19. Advanced DFS & BFS HKOI Training 2004 19 A F B C D G H SCC (Demonstration) A F B C D E G H A B C D E F G H birth death parent 1 2 3 13 10 4 14 12 9 8 16 11 5 15 6 7 - A B - A C D C D G A E B F C H
- 20. Advanced DFS & BFS HKOI Training 2004 20 SCC (Demonstration) D G A E B F C H A F B C D G H E
- 21. Advanced DFS & BFS 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
- 22. Advanced DFS & BFS 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
- 23. Advanced DFS & BFS HKOI Training 2004 23 A B C D E F G H I J BFS (Demonstration) unvisited visited visited (dequeued) Queue: A B C F D E H G J I
- 24. Advanced DFS & BFS HKOI Training 2004 24 Applications of BFS • Shortest paths finding • Flood-fill (can also be handled by DFS)
- 25. Advanced DFS & BFS HKOI Training 2004 25 Comparisons of DFS and BFS DFS BFS Depth-first Breadth-first Stack Queue Does not guarantee shortest paths Guarantees shortest paths
- 26. Advanced DFS & BFS HKOI Training 2004 26 Bidirectional search (BDS) • Searches simultaneously from both the start vertex and goal vertex • Commonly implemented as bidirectional BFS start goal
- 27. Advanced DFS & BFS HKOI Training 2004 27 Iterative deepening search (IDS) • Iteratively performs DFS with increasing depth bound • Shortest paths are guaranteed
- 28. Advanced DFS & BFS 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
- 29. Advanced DFS & BFS 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
- 30. Advanced DFS & BFS HKOI Training 2004 30 Simple examples (1) • Given a grid maze with obstacles, find a shortest path between two given points start goal
- 31. Advanced DFS & BFS 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?
- 32. Advanced DFS & BFS 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
- 33. Advanced DFS & BFS 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
- 34. Advanced DFS & BFS 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
- 35. Advanced DFS & BFS HKOI Training 2004 35 Complex examples (1) start goal unused used start goal goal
- 36. Advanced DFS & BFS HKOI Training 2004 36 Complex examples (2) • The famous 8-puzzle • Given a state, find the moves that bring it to the goal state 1 2 3 4 5 6 7 8
- 37. Advanced DFS & BFS 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?
- 38. Advanced DFS & BFS HKOI Training 2004 38 Complex examples (2) 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 7 8 6 1 2 3 4 6 7 5 8 1 2 3 4 5 6 7 8 1 2 4 5 3 7 8 6 1 2 3 4 5 7 8 6
- 39. Advanced DFS & BFS 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.
- 40. Advanced DFS & BFS 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?
- 41. Advanced DFS & BFS 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
- 42. Advanced DFS & BFS 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 *