Michael Walker

From Maps to Graphs  Make Intersections Vertices  Make Roads Edges  Result is Weighted Directed Graph  Weights: Speed Limit Length Tolls, Number of Lanes, Traffic Information

Data Structures Queue Stack Priority Queue

Queue  First In First Out (FIFO)  Think checkout line at a store  Efficient Addition and removal are constant time

Stack  First in Last Out (FILO) Alternatively Last in First out (LIFO)  Think stack of plates  Also efficient Addition and removal are constant time

Priority Queue  Each element has priority  Highest priority removed first

Where to Start?  Start with finding ANY path  Find a spanning tree  Algorithm Idea: From start, create tree going outwards Depth First Search Breadth First Search Shortest Edge First (Prim’s Algorithm)

Depth First Search (DFS)  Continue down one path until reaching a “dead end”  Then backtrack and choose a different path  Programmatically: Remove vertex from stack, set as “current vertex” Add current vertex to spanning tree if it not present (if present skip next step) Add adjacent vertices to stack if not in tree

Breadth First Search (DFS)  Explore all paths in order of # of edges  Programmatically: Remove vertex from queue, set as “current vertex” Add current vertex to spanning tree if it not present (if present skip next step) Add adjacent vertices to queue if not in tree

Prim’s Algorithm  Shortest Edge First  Creates Minimal Spanning Tree Spanning tree with the lowest possible sum of the edge weights  Uses Priority Queue Instead of highest removed, lowest is removed

Kruskal’s Algorithm  Also finds minimum spanning tree  Builds a forest that forms a tree  Algorithm: Select smallest unprocessed edge If two vertices are not part of the same tree, then add edge to forest Repeat until all vertices are in forest

Dijkstra’s Algorithm  Resembles Prim’s Algorithm Total cost is used instead of edge cost Tentative costs are maintained and updated  Algorithm Loop: Vertex with next lowest tentative distance is “current” vertex. For vertices adjacent to current vertex ○ Compare tentative distance to current vertex distance + edge ○ If new distance is lower, replace tentative distance Add current vertex to spanning tree.

Board Example

A* Algorithm  Improvement to Dijkstra’s Algorithm  Roadmaps are not abstract mathematical constructions  Possible to add heuristic from real world  Heuristic Conditions: Must only underestimate, never overestimate Overestimation runs faster, but produces non optimal paths

A* Algorithm  Algorithm Loop: Vertex with next lowest tentative distance + heuristic distance is “current” vertex. For vertices adjacent to current vertex ○ Compare tentative distance to current vertex distance + edge ○ If new distance is lower, replace tentative distance Add current vertex to spanning tree.

A* Web Examples  ar/AStar.html ar/AStar.html  /AStarRouter.html /AStarRouter.html  baur.de/cs.web.mashup.pathfinding.html baur.de/cs.web.mashup.pathfinding.html

Further Exploration  Bi-Directional Search Simultaneously run A* from both start and end  Reduce Dataset: Multi-Leveled Maps One graph for large cities One graph for county/city level One graph for town/borough level Google seamlessly does this with shortcuts  Multi-point and simultaneous path finding  Incomplete data path finding (D*)

References  Johnsonbaugh, Richard, and Marcus Schaefer. Algorithms. Upper Saddle River, NJ: Pearson Education, Print.  Khuller, Samir, Balaji Raghavachari, and Neal Young. "Balancing Minimum Spanning and Shortest Path Trees." Web. 29 Apr  Kruskal, Joseph B. "On The Shortest Spanning Subtree Of A Graph And The Traveling Salesman Problem." AMS Journal (1955). Web. 29 Apr  Patel, Amit J. "Amit’s Game Programming Information." Amit’s Patel's Personal Website. Stanford, Web. 29 Apr  Prim, R. C. "Shortest Connection Networks And Some Generalizations." (1957): Web. 29 Apr  Sanders, Peter. "Fast Route Planning." Google Tech Talk. 23 Mar Lecture.