Notes. Video Game AI: Lecture 5 Planning for Pathfinding. Lecture Overview. Knowledge vs Search. Jonathan Schaeffer this Friday

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1 Notes Video Game AI: Lecture 5 Planning for Pathfinding Nathan Sturtevant COMP 3705 Jonathan Schaeffer this Friday Planning vs localization We cover planning today Localization is just mapping a real-valued coordinate (which is not in the representation) into a coordinate in the representation Lecture Overview Breadth-first search Best-first search Dijkstra A* Heuristics Knowledge vs Search Search can be on of the most expensive parts of AI Difficult to avoid the need for search Used in all types of games FPS, RPG, RTS If we know the distance between all locations (all-pairs shortest path), don t need any search Floyd-Warshall algorithm / Johnson s algorithm O(V 2 ) space complexity, O(V 3 ) time Space complexity too high for many games

2 Ad-hoc approaches Most early games used ad-hoc approaches: Walk straight towards the goal Follow right-hand rule to get around obstacles How well will this work? State Spaces Formal search methods require a state space: start state goal state (goal test) successor function (cost of actions) Assumptions In AI, often exponential state spaces are used: Branching factor b, depth d, b d nodes in tree In games, maps are usually 2d (or 2.5d) At radius r, r 2 states Successor functions Grids: Octile movement (Can move to 8 neighbors) Waypoint Graphs: Follow edges in graph Navmesh / Delaunay Triangulation One move for each unblocked polygon edge

3 Algorithm Properties Complete: Will it find a path if one exists Optimal: Will it find the optimal path Time: How many node expansions are required Space: How many states are kept in memory Breadth-first search Look at all possible moves at successive depths: BFS(start, goal) queue start while (queue is not empty) next pop from queue if (next == goal) return path; add successors of next to queue return no path; BFS demo BFS analysis Complete: Yes, if all edges/actions have cost 1 Optimal: Yes; paths are explored by increasing cost When the goal is found, there cannot be a path to the goal of lesser cost Time & space: O(r 2 ) Very simple data structure

4 Terminology g-cost of a state s The cost of the path found from the start state to s Dijkstra BFS cannot handle non-uniform weights Some games have made all movement uniform cost In a FPS like this, turn diagonal and run diagonally to cover ground faster Need better ordering of states to ensure optimality OPEN: priority queue sorted by g-cost CLOSED: hash table Dijkstra Dijkstra(start, goal) OPEN start while (OPEN is not empty) next pop from queue if (next == goal) return path; expand successors of next return no path; Adding successors Check to see if state is on CLOSED: Ignore Check to see if state is on OPEN: Update g-cost Or, just add to OPEN and allow duplicates When removing an item, check if already on CLOSED Leads to larger OPEN list

5 Dijkstra Example Review: Heap Heap property: every parent is larger than its children S 3 5 A 5 B 2 3 D 1 C 4 1 E F 2 G 3 2 H A binary heap has log(n) remove and insert operations Can use an array as the underlying representation Insert: Add state to the end of the array Heapify up if key value is lower than parent Remove: Take top item off of heap, replace with last item Assume undirected edges Heapify down, swapping with smallest child Heap complications Might need to look items up inside the heap Often requires a secondary data structure for lookups Hash table is efficient (but may allocate memory) Can avoid if data structure is stored directly in states Means that you can only perform one search at a time Might change key values in the heap requires extra heap up/down to maintain heap property Dijkstra Analysis Complete & Optimal: Yes; paths are explored by increasing cost Proof by induction over the optimal path When the goal is found, there cannot be a path to the goal of lesser cost Time & space: O(r 2 ) More complicated data structure log overhead in size of OPEN

6 Dijkstra drawbacks Demo of performance What does it need for better performance? Terminology g-cost of a state s The cost of the path found from the start state to s h-cost of a state s An estimate of the cost from s to the goal Heuristic An estimate of the cost to get from some state s to the goal: h(s) h*(s) is the perfect heuristic; exact cost An admissible heuristic never overestimates the cost to the goal For all states, s, h(s)! h*(s) A consistent heuristic never changes more than the cost of an edge: h(a)-h(b)! c(a, b) [for an undirected graph] Two simple heuristics Euclidean distance: sqrt((x1-x2) 2 + (y1-y2) 2 ) Octile distance (for grids) max( x1-x2, y1-y2 ) + ("2-1) min( x1-x2, y1-y2 )

7 A* Like Dijkstra, except use an estimate of the full cost of the path [g+h cost] Uses f-costs: f(n) = g(n) + h(n) Otherwise, nearly identical to Dijkstra, assuming a consistent heuristic Implies that f-costs will be monotonically increasing Possible A* Enhancements (1/4) Real-valued g/h-costs make the priority queue more expensive Making diagonals 1.5 simplifies things greatly Limited number of f-costs Instead of general sorting, perform bucketing Put all states with the same f-cost in the same bucket Constant time access Possible A* Enhancements (2/4) How do you sort states with the same f-cost? Choose those closest to the goal Highest g-cost There may be hundreds of states with the same f-cost as the goal Makes sure that we look near the goal first Possible A* Enhancements (3/4) When using a priority queue, often the next state to be expanded is a child of the previous state Don t put on queue and the immediately remove Often the parent of a state is a successor of the child Don t generate the parent as a successor of the child

8 Possible A* Enhancements (4/4) Modify search to be faster, but possibly return suboptimal paths Weighted A* f(n) = g(n) + w h(n) Solutions are at most w times worse than optimal Can perform much less work