CS171 Introduction to Computer Science II
Graphs
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
Graphs
• Simple graphs • Directed graphs • Weighted graphs • Shortest path in weighted digraphs
– Dijkstra Algorithm – Implementation – A* Algorithm
• Project discussion • Symbol graphs • Minimum spanning trees
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)
Dijkstra’s algorithm
MapQuest
10
15
20 ?
20
15
5
18
25
33
• Shortest path for a single source-target pair
• Dijkstra algorithm can be used
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”
The A* Search
• A* is an algorithm that:
– Uses heuristic to guide search
– While ensuring that it will compute a path with minimum cost
• A* computes the function f(n) = g(n) + h(n)
“actual cost”
“estimated cost”
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”
10
15
20
20
15
5
18
25
33
n g(n)
h(n)
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
n’
d
h(n)
c(n,n’) h(n’)
• If h(n) is consistent then h(n) is admissible • Frequently when h(n) is admissible, it is also consistent
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
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)
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)
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
Graphs
• Simple graphs
• Directed graphs
• Weighted graphs
• Shortest path
• Symbol graphs
• Project discussion
• Minimum spanning trees
Applications
• Phone/cable network design – minimum cost
• Approximation algorithms for NP-hard problems
Assumptions
• The graph is connected
• If the graph is not connected: find minimum spanning trees of each connected components, i.e. minimum spanning forest
Quiz 4
