Shortest Path Algorithms By: Nick Bielik Aaron Staranowicz Mike Knadle

Content Objective Node Expansion A* Uniform Cost Dijkstra’s Bi-directional Dijkstra’s Demo

Objective Uniform Cost Search (UCS) vs Dijkstra’s A* vs Dijkstra’s vs Bi-directional Dijkstra’s – Differences between each – Node expansion

Node Expansion Diagram A* Dijkstra’s Bi-directional Dijkstra’s UCS Most Efficient Least Efficient

Node Expansions Start Node: Luebeck Goal Node: Munich Node Expansions: – A* - 21 – Dijkstra’s – 244 – Bidirectional Dijkstra’s – 16 – UCS - 19

A* Finds the shortest path from 1 start node to 1 goal node Uses a distance-plus-cost heuristic function The heuristic function includes 2 parts: – the path-cost function, which is the cost from the starting node to the current node – and an admissible "heuristic estimate" of the distance to the goal

A* Admissible "heuristic estimate“ – considers fewer nodes – Estimate must less than or equal to total cost in all cases

Uniform Cost Search Begins at a root node and will continually expand nodes, taking the node with the smallest total cost from the root until it reaches the goal state. Special Case of A* if the heuristic is a constant function Similar to Dijkstra’s

Dijkstra’s Same as Uniform Cost Search but instead of stopping at the goal node, it continues until all nodes are covered

Bi-directional Dijkstra’s Same as Dijkstra’s but with the search algorithm starting from both sides – Idea: If Dijkstra’s is given 1 start node and 1 goal node, then another Dijkstra’s is given the inverse then the 2 should reach the middle node at the same time

Questions?