Advanced DFS BFS Advanced D F S and

Advanced DFS & BFS Advanced D -F S and B F S epth irst earch readth- irst earch 1

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

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

Advanced DFS & BFS What is a graph? • A set of vertices and edges vertex edge HKOI Training 2004 4

Advanced DFS & BFS Trees and related terms ancestors root parent siblings descendents children HKOI Training 2004 5

Advanced DFS & BFS 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 6

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

Advanced DFS & BFS 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 8

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

Advanced DFS & BFS “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 10

Advanced DFS & BFS 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 11

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

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

Advanced DFS & BFS 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 14

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

Advanced DFS & BFS 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 16

Advanced DFS & BFS SCC (Illustration) HKOI Training 2004 17

Advanced DFS & BFS 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 18

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

Advanced DFS & BFS SCC (Demonstration) F A E B H D D G F C B H C G HKOI Training 2004 20

Advanced DFS & BFS 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 21

Advanced DFS & 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 22

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

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

Advanced DFS & BFS Comparisons of DFS and BFS Depth-first BFS Breadth-first Stack Queue Does not guarantee shortest paths Guarantees shortest paths HKOI Training 2004 25

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

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

Advanced DFS & BFS 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 28

Advanced DFS & BFS 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 29

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

Advanced DFS & BFS 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 31

Advanced DFS & BFS 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 32

Advanced DFS & BFS 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 33

Advanced DFS & BFS 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 34

Advanced DFS & BFS Complex examples (1) unused start goal HKOI Training 2004 35

Advanced DFS & BFS 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 HKOI Training 2004 36

Advanced DFS & BFS 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 37

Advanced DFS & BFS Complex examples (2) 1 2 3 4 5 6 7 8 1 2 3 4 6 7 5 8 1 2 3 4 5 7 8 6 1 2 4 5 3 7 8 6 1 2 3 4 5 6 7 8 1 2 3 4 5 7 8 6 HKOI Training 2004 38

Advanced DFS & BFS 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 39

Advanced DFS & BFS 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 40

Advanced DFS & BFS Some more examples • How can the followings be modeled? – Tilt maze (Single-goal mazes only) • http: //www. clickmazes. com/newtilt/ixtilt 2 d. 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 41

Advanced DFS & BFS Competition problems • • • HKOI 2000 S – Wormhole Labyrinth HKOI 2001 S – A Node Too Far HKOI 2004 S – Teacher’s Problem * TFT 2001 – OIMan * TFT 2002 – Bomber Man * NOI 2001 – cung 1 ming 4 dik 7 daa 2 zi 6 jyun 4 IOI 2000 – Walls * IOI 2002 – Troublesome Frog IOI 2003 – Amazing Robots * HKOI Training 2004 42