breadth depth

8/12/2019 Breadth Depth

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State Space Search:

Breadth First and Depth First


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Motivations

Many problems can be viewed as reaching agoal state from a given starting point, e.g., thefarmer-wolf-goat-cabbage problem.

Often there is an underlying state space

successor function to proceed from one stateto the next.

Search strategies are important methods to

explore such a space to obtain the goal state.


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Objectives

1. Top down search2. Bottom up search

3. Breadth first search

4. Depth first search5. Compare breadth first and depth first


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State space for tic-tac-toe game

Represent a problem

as a state space

Nodesrepresentdiscrete 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 ofproblem and search

procedures using

graph theory.


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Am I a descendant of Thomas Jefferson?

The third President of the United States (18011809)

Assume no complete record of Jeffersons family tree. Need to do some research. 2 search strategies

top-down: search from Jefferson to I

bottom-up: search from I to Jefferson

Assume 10 generations has past since Jefferson Depth of family tree is 10

top-down Assume average 3 children per family

branch factor is 3

How many nodes to search?

3 + 32 + 33+ + 310

bottom-up For each child, search 2 parents

branch factor is 2

How many nodes to search?

2 + 22 + 23+ + 210


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Breadth-first searching

A breadth-first search (BFS)explores nodes nearest the

root before exploring nodes

further away

For example, aftersearching A, then B, then C,

the search proceeds with D,

E,F,G

Node are explored in theorder A B C D E F G H I J K LM N O P Q

Jwill be found beforeNL M N O P

G

Q

H JI K

FED

B C

A


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Breadth-first search algorithm

Nodes are lining up to be visited in open.

closed keeps track of all the nodes visited already.


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A trace of

breadth-first algorithm

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Items between redbars are siblings.

B is not the goal.Put his children onto the queue.

Put him in closed. He is done.

goal is reached or open is empty.


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closed contains the nodes that

have been visited, i.e., nodesthat have been opened or

expanded.

open contains the nodes

whose children have not beenlooked at yet. The queue is

also called the frontierof the

search.

Progress at iteration 6


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Breadth-first searching summary

L M N O P

G

Q

H JI K

FED

B C

A


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Depth-first searching

A depth-first search (DFS)explores a path all the way

to a leaf beforebacktracking and exploring

another path

For example, after

searching A, thenB, thenD, the search backtracksand tries another path from

B

Node are explored in theorder A B D E H L M N IO P C F G J K Q

Nwill be found before JL M N O P

G

Q

H JI K

FED

B C

A


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The depth-first search algorithm

This is the only difference betweendepth-first and breadth-first.


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A trace of

depth-first algorithm

top of stack


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Snap shot at iteration 6frontier

visited


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Recursive depth-first search algorithm

DFS(node v) {visit(v)for each child wof v, DFS(w)

}

Recursion uses a stack to store data for previous

calls. The children nodes (ws)are placed on theundeclared (implicit) stack.

Breadth first uses a queue instead of a stack, norecursion.

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 extractedfrom the stack as the recursion unwinds.


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Search sequences of depth-first and breadth-first

Breadth first: A, B, C,

Depth first: 1, 2, 3,


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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-levelfashion

Depth-first search

When a state is examined, all of its children and

their descendants are examined before any ofits siblings

Go deeper into the search space where

possible


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Breadth-first search of the 8-puzzle

Cannot moveblank downward


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Depth-first search of 8-puzzle with a depth bound of 5

Breadth-firsttook46 nodes.


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Summary

Graph representation Nodes are problem solution states Arcs are steps in problem solving

State Space Search

Find a solution path from start state to goalstate.

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

breadth depth

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