1. Breadth First and Depth First
State Space Search:
Breadth First and Depth First

2. Motivations
Many problems can be viewed as reaching a goal state from a given starting point, e.g., the farmer-wolf-goat-cabbage problem.
Often there is an underlying state space successor function to proceed from one state to the next.
Search strategies are important methods to explore such a space to obtain the goal state.

3. Objectives
Top down search
Bottom up search
Breadth first search
Depth first search
Compare breadth first and depth first

4. State space for tic-tac-toe game
Represent a problem as a state space
Nodes represent discrete states, e.g., configuration of game board.
Arcs (links, edges) represent transition between states e.g., move in a game.
Analyze the structure and complexity of problem and search procedures using graph theory.

5. Breadth-first searching
A breadth-first search (BFS) explores nodes nearest the root before exploring nodes further away
For example, after searching A, then B, then C, the search proceeds with D, E, F, G
Node are explored in the order A B C D E F G H I J K L M N O P Q
J will be found before N L M N O P G Q H J I K F E D B C A

6. Breadth-first search algorithm
Nodes are lining up to be visited in open. closed keeps track of all the nodes visited already.

7. breadth-first algorithm
A trace of
breadth-first algorithm
B is not the goal. Put his children onto the queue. Put him in closed. He is done. | | |
Items between red bars are siblings. | | | | | | | | | | | |
goal is reached or open is empty.

8. Progress at iteration 6
open contains the nodes whose children have not been looked at yet. The queue is also called the frontier of the search.
closed contains the nodes that have been visited, i.e., nodes that have been opened or expanded. | | | | | | | | | | | | | | |

9. Breadth-first searching summaryL M N O P G Q H J I K F E D B C A

10. Depth-first searching
A depth-first search (DFS) explores a path all the way to a leaf before backtracking and exploring another path
For example, after searching A, then B, then D, the search backtracks and tries another path from B
Node are explored in the order A B D E H L M N I O P C F G J K Q
N will be found before J L M N O P G Q H J I K F E D B C A

11. The depth-first search algorithm
This is the only difference between
depth-first and breadth-first.

12. depth-first algorithm
A trace of
depth-first algorithm
top of stack

13. Snap shot at iteration 6
frontier
visited

14. Recursive depth-first search algorithm
DFS(node v) { visit(v) for each child w of v, DFS(w) }
Recursion uses a stack to store data for previous calls. The children nodes (w’s)are placed on the undeclared (implicit) stack.
Breadth first uses a queue instead of a stack, no recursion.
At any given time, the top of the stack contains only the node on a path from the root to a goal.
The stack only needs to be large enough to hold the deepest search path.
When a goal node is found, the path can be extracted from the stack as the recursion unwinds.

15. Search sequences of depth-first and breadth-first
Breadth first: A, B, C, …
Depth first: 1, 2, 3, …
Insert fig 3.15

16. Comparing the ordering of search sequences
Determine the order of nodes (states) to be examined
Breadth-first search
When a state is examined, all of its children are examined, one after another
Explore the search space in a level-by-level fashion
Depth-first search
When a state is examined, all of its children and their descendants are examined before any of its siblings
Go deeper into the search space where possible

17. Summary Graph representation State Space Search
Nodes are problem solution states
Arcs are steps in problem solving
State Space Search
Find a solution path from start state to goal state.
Determine complexity of problem
Many problems (TSP) have exponential complexity, impossible to search exhaustively
Strategies to search large space need heuristic to reduce space and time complexity