GRAPHS .

GRAPHS

http://collectionsonline.nmsi.ac.uk/detail.php?t=objects&type=related&kv=8083246

BASIC TERMINOLOGY

• A Graph is composed of:• Nodes (aka Vertices)• Edges (aka Arcs)• Directed or Undirected (aka bi-directional)• May have an associated cost

• Representation• Adjacency Matrix• Adjacency List

Alice

Calvin

Dee

Bob

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SOME COMMON APPLICATIONS / ALGORITHMS

• Visiting every node in some order:• Depth-first and Breadth-first search

• Finding the shortest path from each node to another• Network packet routing• Floyd-Warshall algorithm

• Visiting each node in the most efficient path possible and return to start (travelling salesman)• UPS delivery• [No polynomial-time solution!] Several greedy / heuristic

algorithms give good solutions.

• **Find the shortest path from A to B.• Google maps• Djikstra's algorithm (and A*)

SEARCH TREES

• If we are doing some kind of search of a graph, we often need to construct a search tree.• Note: often multiple search trees for one graph.

Search Tree

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class SearchNode{ GraphNode mTwin; SearchNode mParent; float mTotalCost; [float mHeuristic;] [int mPQueuePos;]}

BREADTH-FIRST SEARCH

• Maintain a Queue of SearchNode's: frontier• Initially with just the starting node's s.node.

• Maintain a HashSet of visited SearchNodes: visited• At each update:

1. Create an empty Queue of SearchNodes: new_frontier.2. Iterate through the frontier for each S.Node C:

a. If C is the goal, construct the solution path.b. For each neighbor, N, of C:

• If N isn't on visited, add it to new_frontier (and make the parent = C).

3. Replace frontier with new_frontier.

• Problem: we will get to the goal, but it won't (necessarily) be the optimal path.

BREADTH-FIRST SEARCHS

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DEPTH-FIRST SEARCH

• Often done recursively.• However, it is often more efficient to use a Stack

(of SearchNode's)• Also maintain a HashSet (like in B.F.S.): visited• At each update:

1. Peek at the top search node from the stack: C.2. If C is the goal, construct the solution path.3. Else:

a. Look at each Neighbor, N, of C:i. If C isn't in visited, add it to visited and push onto the stack.

b. Pop C off the stack.

DEPTH-FIRST SEARCH

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Total=17

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A* ALGORITHM

• A* is often called a "best-first" search.• At each stage, we pick the most promising node:• Ordered by the sum of cost-so-far and heuristic• Heuristic: the estimate of how far to the goal.

• Often the straight-line distance.

• We store the nodes on a PriorityQueue (OpenList) for fast-access.

1. Create a single search node for the start and put on p.queue

2. Repeat as long as p.queue is not empty:a. Pull the best node, C, off the p.queue.b. For each Neighbor, N, of C:

i. if N hasn't been explored, create a search node and add to p.queue.ii. If N is on open, see if the path through C is better. If so:

a. Update the parent and cost-so-farb. Reheapify that node on the OPEN list.