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Graph Algorithms. GAM 376 Robin Burke Winter 2006. Outline. Graphs Theory Data structures Graph search Algorithms DFS BFS Project #1 Soccer Break Lab. Admin. Homework #3 due Monday not today Careers in Technology Tomorrow 5 – 7 pm Midway Games will be there. Admin. - PowerPoint PPT Presentation
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Graph Algorithms
GAM 376Robin BurkeWinter 2006
Outline
Graphs Theory Data structures Graph search
Algorithms DFS BFS
Project #1 Soccer
Break Lab
Admin
Homework #3due Monday not today
Careers in TechnologyTomorrow5 – 7 pmMidway Games will be there
Admin
Late policynot spelled out in syllabus10% per day for three days
Meaningtomorrow is the last day to turn in
Homework #230% off though
Grading
Yes, I am way behind in grading
Admin
Final projectfirst build
Same as the exercise from Lab #1but you have to decide what map
you're going to usewhat NPC you're going to create
Deliverablecompiled map
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 graphspath findingtech trees in strategy gamesstate 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
v6 v7
v1, v2, v3, v4, v5, v6, and v7 are vertices. (v1,v2) is an edge in thegraph and thus v2 is adjacent to v1. The graph is directed.
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.
An Example Graph
v1 v2
v3 v4 v5
v6 v7
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
v6 v7
The sequence v1, v2, v5, v4, v3, v1 is a cycle. We had tomake one change to this graph to achieve this cycle. So, thisgraph 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
v6 v7 v1v2v3v4v5v6v7
v1 v2 v3 v4 v5 v6 v70 1 1 1 0 0 00 0 0 1 1 0 0
We can use an “adjacencymatrix” representation.
For each edge (u,v) we set A[u][v] to true;else it is false. If there are weights associatedwith the edges, insert those instead.
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.
Adjacency List
v1 v2
v3 v4 v5
v6 v7 v1 v2 v4 v3v2 v4 v5v3 v6v4 v6 v7 v3v5 v4 v7v6v7 v6
We can use an “adjacencylist” 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
Problemis there a path from v to w?what is the shortest / best path?
• optimalitywhat 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 <i,j> 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 edgesno edges out orall edges point to visited nodes
Then we "backtrack" to the next edge and keep going
DFS
v1 v2
v3 v4 v5
v6 v7
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
Complexity Time: O(E) Space: O(1)
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
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 addeddon't expand "two steps" awayuntil you've expanded all of the "one
step" nodes
BFS
v1 v2
v3 v4 v5
v6 v7
start
target
Characteristics
Will find shortest path Won't get lost in deep trees Can be memory-intensive
frontier can become very largeespecially if branching factor is high
ComplexityTime: O(E)Space: O(E)
Buckland implementation
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
Next week
More graph funDijkstra's algorithmA*
Scripting / Lua
Simple Soccer
Implementation of a 5-player soccer team
Two state machines"Team state""Player state"
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
Player state
defensechase ball if you're the closest
offensemove toward goal with ball
• pass if possiblewithout ball,
• move to support spot• ask for pass
Steering behaviors
chasing the ball steering to support position goalie has special behavior to get in
blocking position
Demo
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
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
AbstractFactory
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
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
Teams
Team 1 Choryan Gilliam Wiemeyer
Team 2 Abero Gantchev McNulty
Team 3 Flaks Hall
Team 4 Chrostowski Hogan Kenley
Break