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Advanced Graph Modelling and Searching HKOI Training 2010

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Graph A graph is a set of vertices and a set of edges G = (V, E) Number of vertices = |V| Number of edges = |E| We assume simple graph, so |E| = O(|V| 2 )

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Trees in graph theory In graph theory, a tree is an acyclic, connected graph – Acyclic means “ without cycles ”

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Properties of trees |E| = |V| - 1 – |E| = (|V|) Between any pair of vertices, there is a unique path Adding an edge between a pair of non-adjacent vertices creates exactly one cycle Removing an edge from the tree breaks the tree into two smaller trees

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Definition? The following four conditions are equivalent: – G is connected and acyclic – G is connected and |E| = |V| - 1 – G is acyclic and |E| = |V| - 1 – Between any pair of vertices in G, there exists a unique path G is a tree if at least one of the above conditions is satisfied

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Trees and related terms root siblings descendants children ancestors parent

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Representation of Graph Adjacency Matrix Adjacency list Edge list

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Representation of Graph Adjacency Matrix Adjacency Linked List Edge List Memory Storage O(V 2 )O(V+E) Check whether (u,v) is an edge O(1)O(deg(u)) Find all adjacent vertices of a vertex u O(V)O(deg(u)) deg(u): the number of edges connecting vertex u

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Graph Traversal Given: a graph Goal: visit all (or some) vertices and edges of the graph using some strategy (the order of visit is systematic) DFS, BFS are examples of graph traversal algorithms Some shortest path algorithms and spanning tree algorithms have specific visit order

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Idea of DFS and BFS This is a brief idea of DFS and BFS DFS: continue visiting next vertex whenever there is a road, go back if no road (ie. visit to the depth of current path) – Example: a human want to visit a place, but do not know the path BFS: go through all the adjacent vertices before going further (ie. spread among next vertices) – Example: set a house on fire, the fire will spread through the house

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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

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F A B C D E DFS (Demonstration) unvisited visited

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“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

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DFS spanning 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 Edges are those in forward direction of DFS, ie. when visiting vertices that are not visited before If some vertices are not reachable from the start vertex, those vertices will form other spanning trees (1 or more) The collection of the trees are called forest

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DFS (pseudo code) DFS (vertex u) { mark u as visited time time+1; birth[u]=time; for each vertex v directly reachable from u if v is unvisited parent[v]=u DFS (v) time time+1; death[u]=time; }

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A F B C D E G H DFS forest (Demonstration) ABCDEFGH birth death parent unvisited visited visited (dead) A B C F E D G H 6 7 -AB-ACDC

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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

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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?

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Determination of edge types Tree edge Forward EdgeBack EdgeCross Edge parent [v] = u not a tree edge birth[v] > birth[u] death[v] < death[u] birth[v] < birth[u] death[v] > death[u] birth[v] < birth[u] death[v] < death[u]

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Applications of DFS Forests Topological sorting (Tsort) Strongly-connected components (SCC) Some more “advanced” algorithms

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Example: Tsort Topological order: A numbering of the vertices of a directed acyclic graph such that every edge from a vertex numbered i to a vertex numbered j satisfies i

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Tsort Algorithm If the graph has more then one vertex that has indegree 0, add a vertice to connect to all indegree-0 vertices Let the indegree 0 vertice be s Use s as start vertice, and compute the DFS forest The death time of the vertices represent the reverse of topological order

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Tsort (Demonstration) SABCDEFG birth death S D B E F C G A G C F B A E D D E A B F C G

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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

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SCC (Illustration)

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SCC (Algorithm) Compute the DFS forest of the graph G to get the death time of the vertices 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’

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A F B C D G H SCC (Demonstration) A F B C D E G H ABCDEFGH birth death parent AB-ACDC D G AEB F C H

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SCC (Demonstration) D G AEB F C H A F B C D G H E

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DFS Summary DFS spanning tree / forest We can use birth time and death time in DFS spanning tree to do varies things, such as Tsort, SCC Notice that in the previous slides, we related birth time and death time. But in the discussed applications, birth time and death time can be independent, ie. birth time and death time can use different time counter

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Breadth-first search (BFS) In order to “spread”, we need to makes use of a data structure, queue,to remember just visited vertices Revised: DFS: continue visiting next vertex whenever there is a road, go back if no road (ie. visit to the depth of current path) BFS: go through all the adjacent vertices before going further (ie. spread among next vertices)

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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

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A B C D E F G H I J BFS (Demonstration) unvisited visited visited (dequeued) Queue: ABCFDEHGJI

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Applications of BFS Shortest paths finding Flood-fill (can also be handled by DFS)

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Comparisons of DFS and BFS DFSBFS Depth-firstBreadth-first StackQueue Does not guarantee shortest paths Guarantees shortest paths

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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

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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

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Simple examples (1) Given a grid maze with obstacles, find a shortest path between two given points start goal

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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?

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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

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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

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Complex examples (1) A vertex is in the form (position, used) The vertices are divided into two groups – trick used – trick not used

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Complex examples (1) start goal unused used start goal

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Complex examples (1) How about you can walk through obstacles for k times?

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Complex examples (1) k k-1 start goal k-2

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Complex examples (1) k k-1 start goal k-4 k-2k-3

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Complex examples (2) The famous 8-puzzle Given a state, find the moves that bring it to the goal state

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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?

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Complex examples (2)

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Complex examples (3) Theseus and Minotaur – – 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.

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Complex examples (3) What does a vertex represent? – Theseus’ position – Minotaur’s position – Both

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Some more examples How can the followings be modeled? – Tilt maze (Single-goal mazes only) – Double tilt maze – No-left-turn maze – Same as complex example 1, but you can use the trick for k times

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Teacher’s Problem Question: A teacher wants to distribute sweets to students in an order such that, if student u tease student v, u should not get the sweet before v Vertex: student Edge: directed, (v,u) is a directed edge if student v tease u Algorithm: Tsort

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OI Man Question: OIMan (O) has 2 kinds of form: H- and S-form. He can transform to S-form for m minutes by battery. He can only kill monsters (M) and virus (V) in S-form. Given the number of battery and m, find the minimum time needed to kill the virus. Vertex: position, form, time left for S-form, number of batteries left Edge: directed – Move to P in H-form (position) – Move to P/M/V in S-form (position, S-form time) – Use battery and move (position, form, S-form time, number of batteries) PWWP PPPP PWWP OMMV

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What you have learnt: Graph Modeling

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Variations of BFS and DFS Bidirectional Search (BDS) Iterative Deepening Search(IDS)

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BDS(BI-DIRECTIONAL SEARCH) BFS eats up memory Let it waste more Start BFS at both start and goal Searching becomes faster startgoal

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BDS(BI-DIRECTIONAL SEARCH)

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BDS Example: Bomber Man (1 Bomb) find the shortest path from the upper-left corner to the lower-right corner in a maze using a bomb. The bomb can destroy a wall. S E

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Bomber Man (1 Bomb) S E Shortest Path length = What will happen if we stop once we find a path?

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Example S E

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Iterative deepening search (IDS) Iteratively performs DFS with increasing depth bound Shortest paths are guaranteed

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IDS

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IDS (pseudo code) DFS (vertex u, depth d) { mark u as visited if (d>0) for each vertex v directly reachable from u if v is unvisited DFS (v,d-1) } i=0 Do { DFS(start vertex,i) Increment i }While (target is not found)

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IDS The complexity of IDS is the same as DFS if the search tree is balanced

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Summary of DFS, BFS We learned some variations of DFS and BFS – Bidirectional search (BDS) – Iterative deepening search (IDS)

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Other topics Euler circuit Articulation Point Bridge

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