4.1 Generic Search

During a search, the fringe (or frontier) is the collection of nodes waiting to be visited, usually stored as a stack, a queue, or a priority queue. Nodes on the fringe are called open. After nodes are deleted from the fringe, they are labeled closed.

Search spaces canbeformulated astrees orgraphs. Below, weformulategeneric search algorithms forboth trees andgraphs; thedifferencebetweenthemisthatthetreealgorithmdoesnotmaintain a list of closed nodes because nodes in a tree are never reencountered during a search.

The remainder ofthe(blind search) algorithmsin thislecture specialize thegeneric search algorithm fortrees(Table 1). Wedeferadetaileddiscussionofgraph search untilthelecturesonheuristic search(Lectures03 and04).

4.2 Breadth-First Search

The mainidea ofbreadth-first search(BFS) isto visit all nodes atdepth i before visiting those at depth i+1: i.e., after visiting a node at the next level, visit its siblings before visiting its children. BFS is implemented by storing the set of open nodes as a queue, and accessing its entries in a first-in-first-out fashion.

BFS is complete: it is guaranteed to find a solution, if one exists. Moreover, BFS is optimal: it always finds a goal node of minimal distance from the start. That’s the good news. The bad news

Search(X,S,G,δ)

Inputs	search problem
Output	(pathto) goal node or failure
Initialize	O = S is the set of open nodes

while (O is not empty) do

1. delete some node n ∈ O
2. if n ∈ G, return(pathto) n
3. insert δ(n)into O

fail

Table 1: Generic Search Algorithm for Trees. Search(X,S,G,δ)

Inputs	search problem
Output	(pathto) goal node or failure
Initialize	O = S is the set of open nodes
	C = ∅ is the queue of closed nodes

while (O is not empty) do

1. delete some node n ∈ O
2. if n ∈ G, return(pathto) n
3. insert δ(n)\ C into O
4. insert n into C

fail

Table 2: Generic Search Algorithm for Graphs.

is, in the worst case, BFS visits every node, which takes time as follows:

d _bd+1 _{− 1}

1+b+b² + ... + b^d = bⁱ == O(b^d) ¹

b− 1

i=0

In terms of space, BFS maintains a list of all nodes at all depths: at depth d the length of this list is b^d . Hence, the space complexity of BFS is exponential in d. In summary, both the time and space complexities of BFS are exponential in d.

4.3 Depth-First Search

LikeBFS,depth-first-search(DFS) can alsobe viewed asproceeding levelby level;however, after visiting a node at the next level, DFS visits its children before visiting its siblings. DFS is imple

1_{+ b}d+1 _bd+1

Let S =1+ b + b² + ... + b^d . Then bS = b + b² + ... + b^d . So(b − 1)S = bS − S = − 1, from which

_bd+1

it follows that S = ⁻¹ .

b−1

BFS(X,S,G,δ)

Inputs	search problem
Output	(pathto) goal node or failure
Initialize	O = S is the queue of open nodes

while (O is not empty) do

1. delete first node n ∈ O
2. if n ∈ G, return(pathto) n
3. append δ(n)to back of O

fail

Table 3: Breadth-First Search.

mented by storing the set of open nodes as a stack, and accessing its entries in a last-in-first-out fashion.

DFS(X,S,G,δ)

Inputs	search problem
Output	(pathto) goal node or failure
Initialize	O = S is the stack of open nodes

while (O is not empty) do

1. delete first node n ∈ O
2. if n ∈ G, return(pathto) n
3. prepend δ(n)to front of O

fail

Table 4: Depth-First Search.

Like BFS, the time complexity of DFS is O(b^d). It is exponential in d because in the worst-case DFS visits every node. The space complexity of DFS, however, is linear in d, where d is the length of longest path. Since at most b nodes are stored at each of the d depths, the space complexity of DFS is O(bd).

DFS is neither complete nor optimal. Given knowledge base {φ → φ,φ} and formula φ, DFS could fail to prove φ by forever visiting φ → φ. Using DFS, a robot that intends to visit its neighbor to the west, but starts out searching to the east, travels all the way around the world before finding its neighbor!

4.4 Iterative Deepening

Iterative deepening (ID) is a search algorithm with the space requirements of DFS—it requires memory linear in d—and theperformanceproperties ofBFS—itis complete and(asymptotically) optimal. The main idea of iterative deepening is to repeatedly search in depth-first fashion, over

ID(X,S,G,δ)

Inputs	search problem
Output	(pathto) goal node
Initialize	c = 0 is the cut-off depth
	O = S is the stack of open nodes

while (1)do

1. while (O is not empty) do

(a)

delete first node n ∈ O

(b)

if n ∈ G, return(pathto) goal n

(c)

if depth(n)≤ c

i. prepend δ(n)to front of O

2. increment c, O = S

Table 5: Iterative Deepening.

ID is optimal and complete: it is guaranteed to find a goal if one exists—specifically, an optimal goalif multiple solutions exist(aslong as c =1). It iteratively performs depth-first searches; thus, its space complexity is that of DFS, namely O(bd).

Like DFS and BFS, the time complexity of ID is O(b^d): i.e., in the worst-case it is exponential in

d. ID visits nodes at depth 0 d+1 times, at depth 1 d times, ...,and atdepth d 1 time. Thus, the total time required is given by:

1+ 1+b +1+ b+ b² +... +1+ b+ b² + ... + b^d = O(1+b+b² + ... + b^d)= O(b^d)

��� � ��� � � �� �

depth 1 depth 2 depth d

4.5 Iterative Broadening

Theidea ofiterativebroadening(IB) is analogoustothat ofiterativedeepening,butIBiteratively broadensits search, whereasIDiterativelydeepensit. IBperformsdepth-firstsearches on subgraphs of breadth 1, breadth 2, breadth 3, and so on, until a goal node is reached.

IBis notcomplete: itis susceptible tofollowinginfinitepaths withinitsbreadthlimit asit conducts depth-first search. Neither is IB optimal.

In the worst-case, its space complexity equals that of DFS, and the time it requires is O(b^d). Specifically, the number of nodes IB visits is given by:

d+1+2+ ... +2^d +... +1+ b+ ... + b^d = O(1^d +2^d + ... + b^d)= O((b+1)^d+1)

	�	��	�	�	��	�
		breadth 2			breadth b
since

_bd+1 � bb � b+1

� (b+1)^d+1 − 1

= x ^ddx ≤ c ^d ≤ x ^ddx =

d+1 01 d+1

c=1

IB(X,S,G,δ)

Inputs	search problem
Output	(pathto) goal node
Initialize	c = 1 is the cut-off breadth
	O = S is the stack of open nodes

while (1)do

1. while (O is not empty) do

(a)

delete first node n ∈ O

(b)

if n ∈ G, return(pathto) goal n

(c)

if breadth(n)≤ c

i. prepend c nodes in δ(n)to front of O

2. increment c, O = S

Table 6: Iterative Broadening.

4.6 Bidirectional Search

A bidirectional search problem is a 5-tuple �X,S,G,δ,γ�, where �X,S,G,δ� is a basic search problem andγ : X → 2^X isan “inverse” transitionfunction,meaning γ(x)is the setofpredecessors of x. Inbidirectionalsearch, as the name suggests, searchproceeds simultaneouslyin twodirections: bothforwardsfrom aninitial state andbackwardsfrom agoal state. Search terminates where and whenthe two searches meet. However, notallbasic searchproblems are easilyposed asbidirectional searchproblems: e.g.,itisnontrivial togeneratepredecessorsof theset of checkmateconfigurations in the game of chess.

Bidirectional search is typically implemented as two breadth-first searches. Like BFS, bidirectional BFS is optimal and complete. Bidirectional search improves upon BFS in terms of complexity, however. The time complexity of bidirectional BFS is exponential in d/2. In the worst case, each direction ofbidirectionalBFSvisits every nodethroughdepth d/2, whichfor twodirections requires O(2b^d/2 )= O(b^d/2). Its space complexity is also O(b^d/2), since it maintains list ofall nodes all at

_bd/2

depths i, and the number of nodes at depth d/2= .

Island-driven search is a generalization of bidirectional search in which a series of intermediate goals, or islands, is identified between the start and goal nodes. The islands reduce the one large search problem to m small search problems. If the islands are spaced evenly between the start and goal nodes, the number of nodes visited is mb^d/m ≪ b^d . An obvious difficulty with this approach is the identification of islands through which an optimal search path traverses. Consequently, it is difficult to guarantee optimality.

4.7 Summary

Criteria

DFS BFS ID IB biBFS

Time

O(b^d) O(b^d) O(b^d) O((b+1)^(d+1) ) O(b^d/2) _O(bd/2₎Space

O(bd) O(b^d) O(bd) O(bd) Completeness

NO YES YES NO YES Optimality

NO YES YES NO YES

1. IBispreferredifthebranchingfactorislarge: thesearch spaceis wide(possiblyinfinite), particularly if goals are known to be deep

1. IDispreferredifthedepthislarge: the search spaceisdeep(possibly infinite),particularly if goals are known to be shallow
2. DFSispreferredif the maximumdepth of thegoal nodesisknown:if thisdepthis n, we can modify DFS to search only to depth n