Graph Algorithms

1. Graph Algorithms GAM 376 Robin Burke Winter 2006

2. Outline • Graphs • Theory • Data structures • Graph search • Algorithms • DFS • BFS • Project #1 • Soccer • Break • Lab

3. Admin • Homework #3 • due Monday not today • Careers in Technology • Tomorrow • 5 – 7 pm • Midway Games will be there

4. Admin • Late policy • not spelled out in syllabus • 10% per day for three days • Meaning • tomorrow is the last day to turn in Homework #2 • 30% off though

5. Grading • Yes, I am way behind in grading

6. Admin • Final project • first build • Same as the exercise from Lab #1 • but you have to decide what map you're going to use • what NPC you're going to create • Deliverable • compiled map

7. Graph Algorithms • Very important for real world problems: • The airport system is a graph. What is the best flight from one city to another? • Class prerequisites can be represented as a graph. What is a valid course order? • Traffic flow can be modeled with a graph. What are the shortest routes? • Traveling Salesman Problem: What is the best order to visit a list of cities in a graph?

8. Graph Algorithms in Games • Many problems reduce to graphs • path finding • tech trees in strategy games • state space search • problem solving • "game trees"

9. What is a Graph? • A graph G = (V,E) consists of a set of vertices V and a set of edges E. Each edge is a pair (v,w) where v and w are vertices. • If the edges are ordered (indicated with arrows in a picture of a graph), the graph is “directed” and (v,w) != (w,v). • Edges can also have weights associated with them. • Vertex w is “adjacent” to v if and only if (v,w) is an edge in E.

10. An Example Graph v2 v1 v5 v4 v3 v6 v7 v1, v2, v3, v4, v5, v6, and v7 are vertices. (v1,v2) is an edge in the graph and thus v2 is adjacent to v1. The graph is directed.

11. Definitions • A “path” is a sequence of vertices w1, w2,w3, …, wn such that (wi, wi+1) are edges in the graph. • The “length” of the path is the number of edges (n-1). • A “simple” path is one where all vertices are distinct, except perhaps the first and last.

12. An Example Graph v2 v1 v5 v4 v3 v6 v7 The sequence v1, v2, v5, v4, v3, v6 is a path. The length is 5. It is a simple path.

13. More Definitions • A “cycle” in a directed graph is a path such that the first and last vertices are the same. • A directed graph is “acyclic” if it has no cycles. This is sometimes referred to as a DAG (directed acyclic graph). • The previous graph is a DAG (convince yourself of this!).

14. A Modified Graph v2 v1 v5 v4 v3 v6 v7 The sequence v1, v2, v5, v4, v3, v1 is a cycle. We had to make one change to this graph to achieve this cycle. So, this graph is not acyclic.

15. More Definitions… • An undirected graph is “connected” if there is a path from every vertex to every other vertex. • A directed graph with this property is called “strongly connected”. If the directed graph is not strongly connected, but the underlying undirected graph is connected, then the graph is “weakly connected”. • A “complete” graph is a graph in which there is an edge between every pair of vertices. • The prior graphs have been weakly connected and have not been complete.

16. Graph Representation v2 We can use an “adjacency matrix” representation. v1 v5 v4 v3 v1 v2 v3 v4 v5 v6 v7 0 1 1 1 0 0 0 v6 v7 v1 0 0 0 1 1 0 0 v2 v3 For each edge (u,v) we set A[u][v] to true; else it is false. If there are weights associated with the edges, insert those instead. v4 v5 v6 v7

17. Representation • The adjacency matrix representation requires O(V2) space. This is fine if the graph is complete, or nearly complete. • But what if it is sparse (has few edges)? • Then we can use an “adjacency list” representation instead. This will require O(V+E) space.

18. Adjacency List v2 We can use an “adjacency list” representation. v1 v5 v4 v3 v1  v2  v4  v3 v6 v7 v2  v4  v5 v3  v6 v4  v6  v7  v3 For each vertex we keep a list of adjacent vertices. If there are weights associated with the edges, that information must be stored as well. v5  v4  v7 v6 v7  v6

19. Graph search • Problem • is there a path from v to w? • what is the shortest / best path? • optimality • what is a plausible path that I can compute quickly? • bounded rationality

20. General search algorithm • Start with • "frontier" = { (v,v) } • Until frontier is empty • remove an edge (n,m) from the frontier set • mark n as parent of m • mark m as visited • if m = w, • return • otherwise • for each edge <i,j> from m • add (i, j) to the frontier • if j not previously visited

21. Note • We don't say how to pick a node to "expand" • We don't find the best path, some path

22. Depth First Search • Last-in first-out • We continue expanding the most recent edge until we run out of edges • no edges out or • all edges point to visited nodes • Then we "backtrack" to the next edge and keep going

23. DFS v2 v1 start v5 v4 v3 v6 v7 target

24. Characteristics • Can easily get side-tracked into non-optimal paths • Very sensitive to the order in which edges are added • Guaranteed to find a path if one exists • Low memory costs • only have to keep track of current path • nodes fully explored can be discarded • Complexity • Time: O(E) • Space: O(1)

25. Optimal DFS • Really expensive • Start with • bestPath = { } • bestCost =  • "frontier" = { <{ }, (v,v)>} • Repeat until frontier is empty • remove a pair <P, > from the frontier set • if n = w • Add w to P • If cost of P is less than bestCost • bestPath = P • record n as "visited" • add n to the path P • for each edge <n,m> from n • add <P, m> to the frontier • if m not previously visited • or if previous path to m was longer

26. Iterative Deepening DFS • Add a parameter k • Only search for path of lengths <= k • Start with k = 1 • while solution not found • do DFS to depth k • Sounds wasteful • searches repeated over and over • but actually not too bad • more nodes on the frontier • finds optimal path • less memory than BFS

27. Buckland's implementation

28. Breadth-first search • First-in first-out • Expand nodes in the order in which they are added • don't expand "two steps" away • until you've expanded all of the "one step" nodes

29. BFS v2 v1 start v5 v4 v3 v6 v7 target

30. Characteristics • Will find shortest path • Won't get lost in deep trees • Can be memory-intensive • frontier can become very large • especially if branching factor is high • Complexity • Time: O(E) • Space: O(E)

31. Buckland implementation

32. What if edges have weight? • If edges have weight • then we might want the lowest weight path • a path with more nodes might have lower weight • Example • a path around the lava pit has more steps • but you have more health at the end • compared to the path that goes through the lava pit • We will cover this next week

33. Next week • More graph fun • Dijkstra's algorithm • A* • Scripting / Lua

34. Simple Soccer • Implementation of a 5-player soccer team • Two state machines • "Team state" • "Player state"

35. Team state • kickoff • everybody go to default position • offense • look for opportunities to get a pass upfield from the player with the ball • defense • go to defensive position • transition • offense / defense based on possession of ball

36. Player state • defense • chase ball if you're the closest • offense • move toward goal with ball • pass if possible • without ball, • move to support spot • ask for pass

37. Steering behaviors • chasing the ball • steering to support position • goalie has special behavior to get in blocking position

38. Demo

39. SteeringSoccerLab • Not the same as Buckland's • Allows multiple team implementations • Records the CPU time used by each AI implementation • Don't use Buckland's code

40. How to allow different opponents? • Need students to make their own soccer teams • need to run tournament in which teams compete • don't want to recompile when adding a team • How to make extensible code that doesn't need recompilation? • In particular • how can I create an instance if I don't know the name of the class

41. AbstractFactory

42. Registration • How to know which factory object to use? • Static instance that registers a name on instantiation • Table associating factories with names • Result • dynamic object creation • A bit easier in Java using reflection

43. Tournament rules • Round-robin • 3 game matches • 5 minutes / match • Scoring • Lower scoring team • get a bonus if they used less CPU time • 20% less CPU = 1 point • Ties go to the most efficient team • Degenerate strategies disqualified

44. Teams • Team 1 • Choryan • Gilliam • Wiemeyer • Team 2 • Abero • Gantchev • McNulty • Team 3 • Flaks • Hall • Team 4 • Chrostowski • Hogan • Kenley

45. Break