AI - Lecture 1b: Route Finding

Introduction

The focus is particularly on replicating thought processes and knowledge representation.

Other approaches, such as Alife, neural networks and robotics, focus more on replication of behaviour.

AI approaches are informed by an

• introspective understanding of thought processes,

• concepts of symbolic computation and

• principles of mathematics

Search as a key concept

1. identifies ways in which possible actions can be arranged into sequences

2. finds a `route' through the sequences which achieves a desired result.

Most AI methods use search in one way or another.

Route finding

Toy rail map of Europe

Each (blob-connecting) line represents a direct rail connection somewhere in western europe.

Route finding task

• Given knowledge of direct rail connections, what's the shortest rail itinery which gets you from A to B?

At some given point, certain actions are possible.

These actions take you to new points.

At each of those new points, more actions are possible.

And so on.

At each point the possible actions form a branch.

Joining up the branches gives you a tree.

Search by generation

But since the tree doesn't actually exist to begin with, we will have to generate it first.

If we are going to do this, we may as well inspect nodes as we are going along.

So, in practice, `tree generation' and `search' are merged into one process.

Search strategies

• Depth-first search (DFS) always expand nodes at a deeper level of the tree whenever there is a choice.

• Breadth-first search (BFS): always expand every node at the present level of the tree before moving to any deeper level.

The two strategies can be illustrated by showing how they generate the search tree for the route-finding problem.

Breadth-first search from `Brussels': step 1

Step 2

Step 3

Step 4

Step 5

Depth-first search from `Brussels': step 1

Step 2

Step 3

Step 4

Step 5

Node expansion

• To generate the nodes in a particular tree, we first generate a node to represent the starting point. This is the start node.

• We then work out the possible transitions from the start node, creating one node for each point that can be reached. This is expanding the start node.

• The new nodes are the children or successors of the original or parent node.

• To continue the search we expand each of the children in turn, creating more nodes at the next level down.

• Nodes which cannot be expanded are terminal nodes or tip nodes

• The search tree may also be called the search space.

Search completion

When a node is about to be expanded, a check should be made to see if it is the node we're searching for, i.e., if it's a node representing the goal location.

This is the target or goal node.

As soon as we identify the goal node, a solution to the problem can be generated by listing out the sequence of nodes connecting the start node to the goal node.

Any such sequence of nodes is a path.

A path connecting the start node to the goal node is a solution path.

How much work is involved?

Even simple problems can create search trees which are extremely large.

If we don't want to waste a lot of time using trial-and-error, we need a way of estimating how much work is going to be involved in a particular search.

• We want to know how much time it's going to take. This is known as the time complexity of the process.

• We want to know how much memory it's going to need. This is known as the space complexity of the process.

Branching factor

To estimate this, we need to calculate the branching factor, which is just the average number of children of each node.

Next we calculate the depth of the tree, i.e., the expected number of levels.

To estimate the total number of nodes at a particular level, we then raise the branching factor to the relevant degree, i.e., we calculate

where b is the branching factor and d is the depth. This gives the number of nodes at depth d.

Example

The number of states to be processed at level 1 is then 3.

The number to be processed at level 2 is 3 x 3, or

The number to be processed at level 3 is 3 x 3 x 3, or

And so on.

Time and space complexity

For many purposes, deriving this value is sufficient to decide whether or not search is a practical option.

If the expected depth is 8 and the branching factor is 5, a lower bound on the total number of nodes in the space is

To estimate time complexity, we would multiply this by the time it takes to check out a single node.

To estimate space complexity, we would multiply this by the amount of memory it takes to represent a single node.

Again these values would in fact be lower bounds.

.

Summary

• The concept of search

• Route finding

• Tree-structure of possible action sequences

• Search by generation

• Node expansion

• Search strategies (DFS v BFS)

• Branching factor

• Time and space complexity

Questions

• Why do route-finding problems produce tree-structured searches?

• A particular search tree has a branching factor of 2 and a depth of 4. What is the total number of nodes?

• What additional information is required in order to be able to identify the space complexity of a search, other than the branching factor and depth of the tree?

• In what circumstances will the expansion of a search-tree node produce no children?

• AI is just one of several approaches which seek to replicate intelligent behaviour. What is distinctive about it?

• Why does AI tend to rely on search-based methods?

More questions

• Depth-first search and breadth-first search are the two basic search methods. Which one will solve a route-finding problem quickest?

• What are the main components of a search tree?

• What information is needed (i.e., must be saved by the search process) in order to be able to list a solution path?

• What degree of space complexity implies that a search is intractible?

• What degree of time complexity implies that a search is intractible?

• Define what these terms mean in the context of search: node, start node, goal node, children, depth, branching factor, path, solution path.

Exercises

This schematic map of Brighton shows bus routes between a number of locations. A valid bus route is simply a connected sequence of locations.

Exercises cont.

• Using the map, draw out the first four levels in the search tree for bus routes starting from `Clocktower'.

• Add numeric labels (1, 2, 3...) to the nodes in your search tree to indicate the order in which they would be expanded in a valid depth-first search.

• Add alphabetical labels (a, b, c...) to the nodes in your tree to indicate the order in which they would be expanded in a valid breadth-first search.

• Identify the shortest route (i.e., the route with the smallest number of legs) connecting WestPier with Steine.

• In the worst case, how many nodes would a depth-first search process expand in order to identify the shortest route between Clocktower and StPeter.

Exercises cont.

• Estimate the branching factor for the search tree for this map, stating any assumptions made.

• Using your estimate of branching factor, estimate the space complexity of a breadth-first search carried out to a depth of five levels.

• Using your estimate of branching factor, estimate the space complexity of an exhaustive, loop-avoiding depth-first search of this space, stating any assumptions made.

Resources

• Russell and Norvig have a good section on basic search and time and space complexity; see pp. 74-75