CS171 Introduction to Computer Science II. Graphs

Transcription

1 CS171 Introduction to Computer Science II Graphs

2 Announcements/Reminders Median grades of midterms Midterm 2: 88 Midterm 1: 90 Median grades of quizzes Quiz 3: 91 Quiz 2: 81 Quiz 1: 89 Quiz 4 on Friday (priority queue, graphs) Project demo due Sunday night (optional) Project workshop on Monday

3 Graphs Simple graphs Directed graphs Weighted graphs Shortest path in weighted digraphs Dijkstra Algorithm Implementation A* Algorithm Project discussion Symbol graphs Minimum spanning trees

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5 Shortest Paths Dijkstra s Algorithm Assign a distance value to every node zero for source node infinity for all other nodes. Add source node to a queue (open set) While the queue is not empty Remove the node with the smallest distance from the source node as the "current node, mark it as visited (visited set) For current node, consider all its unvisited neighbors and calculate their tentative distance. Add/update them in the queue: if the distance is less than the previously recorded distance, overwrite the distance (edge relaxation)

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8 Dijkstra s algorithm

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11 MapQuest Shortest path for a single source-target pair Dijkstra algorithm can be used ? 33

12 Better Solution: Make a hunch! Use heuristics to guide the search Heuristic: estimation or hunch of how to search for a solution We define a heuristic function: h(n) = estimate of the cost of the cheapest path from the starting node to the goal node

13 The A* Search A* is an algorithm that: Uses heuristic to guide search While ensuring that it will compute a path with minimum cost estimated cost A* computes the function f(n) = g(n) + h(n) actual cost

14 A* f(n) = g(n) + h(n) g(n) = cost from the starting node to reach n h(n) = estimate of the cost of the cheapest path from n to the goal node g(n) n h(n) 33

15 Properties of A* A* generates an optimal solution if h(n) is an admissible heuristic and the search space is a tree: h(n) is admissible if it never overestimates the cost to reach the destination node A* generates an optimal solution if h(n) is a consistent heuristic and the search space is a graph: h(n) is consistent if for every node n and for every successor node n of n: h(n) c(n,n ) + h(n ) n h(n) d c(n,n ) If h(n) is consistent then h(n) is admissible Frequently when h(n) is admissible, it is also consistent n h(n )

16 Admissible Heuristics A heuristic is admissible if it is optimistic, estimating the cost to be smaller than it actually is. MapQuest: h(n) = Earth distance to destination is admissible as normally cities are not connected by roads that make straight lines

17 Shortest Paths A* Algorithm Assign a distance value to every node zero for source node infinity for all other nodes. Add source node to a queue (open set) While the queue is not empty Remove the node with the smallest estimated cost from the source node to the target node as the "current node, mark it as visited (visited set) For current node, consider all its unvisited neighbors and calculate their tentative distance. Add/update them in the queue: if the distance is less than the previously recorded distance, overwrite the distance (edge relaxation)

18 Project Option 1: MapQuest Weighted digraph Shortest path (Dijkstra) Shortest path (A* for bonus points) Option 2: FaceSpace Digraph Search by user profiles Add profiles Remove profiles Add friendships Shortest path (BFS)

19 Symbol Graph Typical applications involve graphs using strings, not integer indices, to define and refer to vertexes User name in FaceSpace City name for MapQuest

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22 Graphs Simple graphs Directed graphs Weighted graphs Shortest path Symbol graphs Project discussion Minimum spanning trees

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26 Applications Phone/cable network design minimum cost Approximation algorithms for NP-hard problems

27 Assumptions The graph is connected If the graph is not connected: find minimum spanning trees of each connected components, i.e. minimum spanning forest

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35 Quiz 4