# Graph Algorithms GAM 376 Robin Burke Winter 200. Homework #2 No 10s Most common mistake not handling the possibilities associated with damage Big no-no.

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Graph Algorithms GAM 376 Robin Burke Winter 200

Homework #2 No 10s Most common mistake not handling the possibilities associated with damage Big no-no States with no exit conditions Handling explosion without state blip

My solution States Patrol Combat-Normal Combat-HideSolid Combat-HideWeak Combat-Close Combat-Closer Combat-Damaged Combat-DamagedHide Combat- DamagedCloser Hit Explode Dead Conditions Player enters tank's field of view Player fires Player hides behind a non-destructible object Player hides behind a dest. object Player is not hidden Tank / player distance > 50 meters Tank / player distance 10 Tank / player distance < 10 meters Player hits with grenade or RPG Player health < 0 Animation complete Tank health < 0 Tank health < 25%

P CNCHS CHW CCCC2 H E (S or F) and D50 F and Hs F and Hw D50 DMed D50D10 DMed V and D50 Hs Hw from any state A Aout and Dead D Aout Aout and ~Dead return to previous from any state except E KO CD from any state Dmg CDH CDC D10 DMed Hs or Hw V and (DMed or D50) V and DMed V and D10 V and DMed

Homework #3 Buckland’s API yes, it isn’t very well documented very, very typical of production game code Cannot wait for the world to become better documented investigate the code and its usage find clues apply logic

Example PointToLocalSpace look at a function call (Obstacle avoidance) //calculate this obstacle's position in local space Vector2D LocalPos = PointToLocalSpace((*curOb)->Pos(), m_pVehicle->Heading(), m_pVehicle->Side(), m_pVehicle->Pos()); look at the function itself inline Vector2D PointToLocalSpace(const Vector2D &point, Vector2D &AgentHeading, Vector2D &AgentSide, Vector2D &AgentPosition) { //make a copy of the point Vector2D TransPoint = point; //create a transformation matrix C2DMatrix matTransform; double Tx = -AgentPosition.Dot(AgentHeading); double Ty = -AgentPosition.Dot(AgentSide); //create the transformation matrix matTransform._11(AgentHeading.x); matTransform._12(AgentSide.x); matTransform._21(AgentHeading.y); matTransform._22(AgentSide.y); matTransform._31(Tx); matTransform._32(Ty); //now transform the vertices matTransform.TransformVector2Ds(TransPoint); return TransPoint; }

Clues Calling convention first the data being converted then information about the vehicle local space Return value converted point Vehicle API includes heading, side (?) and position

Outline Graphs Theory Data structures Graph search Algorithms DFS BFS

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?

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

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.

An Example Graph v1 v2 v3 v4 v5 v6v7 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.

Definitions A “path” is a sequence of vertices w 1, w 2, w 3, …, w n such that (w i, w i+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.

An Example Graph v1 v2 v3 v4 v5 v6v7 The sequence v1, v2, v5, v4, v3, v6 is a path. The length is 5. It is a simple path.

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!).

A Modified Graph v1 v2 v3 v4 v5 v6v7 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.

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.

Graph Representation v1 v2 v3 v4 v5 v6v7 v1 v2 v3 v4 v5 v6 v7 v1 v2 v3 v4 v5 v6 v7 0 1 1 1 0 0 0 0 0 0 1 1 0 0 We can use an “adjacency matrix” representation. For each edge (u,v) we set A[u][v] to true; else it is false.

Representation The adjacency matrix representation requires O(V 2 ) 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.

Adjacency List v1 v2 v3 v4 v5 v6v7 v1  v2  v4  v3 v2  v4  v5 v3  v6 v4  v6  v7  v3 v5  v4  v7 v6 v7  v6 We can use an “adjacency list” representation. 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.

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

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 from m add (i, j) to the frontier if j not previously visited

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

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

DFS v1 v2 v3 v4 v5 v6v7 start target

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 Typical Complexity Time: O(E/2) Space: O(1) assuming paths are short relative to the size of the graph

Optimality DFS does not find the shortest path returns the first path it encounters If we want the shortest path we have to keep going until we have expanded everything

Optimal DFS Really expensive Start with bestPath = { } bestCost =  "frontier" = { } Repeat until frontier is empty remove a pair 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 from n add to the frontier if m not previously visited or if previous path to m was longer

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

Buckland's implementation

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

BFS v1 v2 v3 v4 v5 v6v7 start target

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 Typical Complexity Time: O(p*b) Space: O(E)

Buckland implementation

Exercise NodesEdges 11-4, 1-3, 1-2 2 33-4, 3-5 44-6 55-2, 5-6 66-3 1 6 5 432 Path from node1 to node6 depth-first breadth-first iterative deepening dfs

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

Weighted graph v1 v2 v3 v4 v5 v6v7 1 1 1 2 2 1 53 3 2 3 1 v1 v2 v3 v4 v5 v6 v7 v1 v2 v3 v4 v5 v6 v7 0 1 1 5 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 3 2

Uniformed algorithms Can use DFS and BFS but how to know when the shortest path found? Problem condition long paths of cheap links must examine whole network

Midterm review Midterm topics Finite state machines Steering behaviors Graph search

Tuesday Soccer Lab

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