Search in Algorithm in artificial intelligence

11/30/2024
1
Search Algorithms in
AI
Ahmad Baryal
Saba Institute of Higher Education
Computer Science Faculty
Oct 8, 2024

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2 Outline:
1. Search Algorithms Terminologies
2. Types of Search Algorithms
3. Uninformed/Blind Search
4. Informed Search

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3 Search Algorithms Terminologies
o Search: Searching is a step by step procedure to solve a search-problem in a given
search space. A search problem can have three main factors:
a. Search Space: Search space represents a set of possible solutions, which a system may
have.
b. Start State: It is a state from where agent begins the search.
c. Goal test: It is a function which observe the current state and returns whether the goal
state is achieved or not.
o Search tree: A tree representation of search problem is called Search tree. The root of the
search tree is the root node which is corresponding to the initial state.
o Actions: It gives the description of all the available actions to the agent.
o Transition model: A description of what each action do, can be represented as a transition
model.
o Path Cost: It is a function which assigns a numeric cost to each path.
o Solution: It is an action sequence which leads from the start node to the goal node.
o Optimal Solution: If a solution has the lowest cost among all solutions.

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4 Properties of Search Algorithms:
Following are the four essential properties of search algorithms to compare the
efficiency of these algorithms:
• Completeness: A search algorithm is said to be complete if it guarantees to return a
solution if at least any solution exists for any random input.
• Optimality: If a solution found for an algorithm is guaranteed to be the best solution
(lowest path cost) among all other solutions, then such a solution for is said to be an
optimal solution.
• Time Complexity: Time complexity is a measure of time for an algorithm to complete
its task.
• Space Complexity: It is the maximum storage space required at any point during the
search, as the complexity of the problem.

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5 Types of search algorithms
Based on the search problems we can classify the search algorithms into uninformed
(Blind search) search and informed search (Heuristic search) algorithms.

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6 Uninformed/Blind Search:
• The uninformed search does not contain any domain knowledge such as closeness, the
location of the goal.
• It operates in a brute-force way as it only includes information about how to traverse the
tree and how to identify leaf and goal nodes.
• Uninformed search applies a way in which search tree is searched without any
information about the search space like initial state operators and test for the goal, so it is
also called blind search.
• It examines each node of the tree until it achieves the goal node.
• It can be divided into five main types:
o Breadth-first search
o Uniform cost search
o Depth-first search
o Iterative deepening depth-first search
o Bidirectional Search

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7 Informed Search
• Informed search algorithms use domain knowledge. In an informed search, problem
information is available which can guide the search.
• Informed search strategies can find a solution more efficiently than an uninformed
search strategy.
• Informed search is also called a Heuristic search.
• A heuristic is a way which might not always be guaranteed for best solutions but
guaranteed to find a good solution in reasonable time.
• Informed search can solve much complex problem which could not be solved in
another way.
• An example of informed search algorithms is a traveling salesman problem.
1. Greedy Search
2. A* Search

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8 1. Breadth-first Search:
o Breadth-first search is the most common search strategy for traversing a tree or graph.
This algorithm searches breadthwise in a tree or graph, so it is called breadth-first search.
o BFS algorithm starts searching from the root node of the tree and expands all successor
node at the current level before moving to nodes of next level.
o The breadth-first search algorithm is an example of a general-graph search algorithm.
o Breadth-first search implemented using FIFO queue data structure.
Advantages:
o BFS will provide a solution if any solution exists.
o If there are more than one solutions for a given problem, then BFS will provide the
minimal solution which requires the least number of steps.
Disadvantages:
o It requires lots of memory since each level of the tree must be saved into memory to
expand the next level.
o BFS needs lots of time if the solution is far away from the root node.

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9 Example:
o In the below tree structure, we have shown the traversing of the tree using BFS algorithm
from the root node S to goal node K. BFS search algorithm traverse in layers, so it will
follow the path which is shown by the dotted arrow, and the traversed path will be:
S---> A--->B---->C--->D---->G--->H--->E---->F---->I---->K

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10 Features of breadth-first Search
• Time Complexity: Time Complexity of BFS algorithm can be obtained by the number of
nodes traversed in BFS until the shallowest Node. Where the d= depth of shallowest
solution and b is a node at every state.
• T (b) = 1+b2
+b3
+.......+ bd
= O (bd
)
• Space Complexity: Space complexity of BFS algorithm is given by the Memory size O(bd
).
• Completeness: BFS is complete, which means if the shallowest goal node is at some finite
depth, then BFS will find a solution.
• Optimality: BFS is optimal if path cost is a non-decreasing function of the depth of the
node.

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11 2. Depth-first Search
o Depth-first search is a recursive algorithm for traversing a tree or graph data structure.
o It is called the depth-first search because it starts from the root node and follows each
path to its greatest depth node before moving to the next path.
o DFS uses a stack data structure for its implementation.
o The process of the DFS algorithm is similar to the BFS algorithm.
o Note: Backtracking is an algorithm technique for finding all possible solutions using
recursion.

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12 2. Depth-first Search
Advantage:
o DFS requires very less memory as it only needs to store a stack of the nodes on the path
from root node to the current node.
o It takes less time to reach to the goal node than BFS algorithm (if it traverses in the right
path).
Disadvantage:
o There is the possibility that many states keep re-occurring, and there is no guarantee of
finding the solution.
o DFS algorithm goes for deep down searching and sometime it may go to the infinite
loop.

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13 Example:
o In the below search tree, we have shown the flow of depth-first search, and it will follow
the order as:
o Root node--->Left node ----> right node.
o It will start searching from root node S, and traverse A, then B, then D and E, after
traversing E, it will backtrack the tree as E has no other successor and still goal node is not
found. After backtracking it will traverse node C and then G, and here it will terminate as
it found goal node.

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14 Features of DFS
o Completeness: DFS search algorithm is complete within finite state space as it will
expand every node within a limited search tree.
o Time Complexity: Time complexity of DFS will be equivalent to the node traversed
by the algorithm. It is given by:
o T(n)= 1+ n2
+ n3
+.........+ nm
=O(nm
)
o Where, m= maximum depth of any node and this can be much larger than d
(Shallowest solution depth)
o Space Complexity: DFS algorithm needs to store only single path from the root
node, hence space complexity of DFS is equivalent to the size of the fringe set,
which is O(bm).
o Optimal: DFS search algorithm is non-optimal, as it may generate a large number
of steps or high cost to reach to the goal node.

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15
3. Depth-Limited Search Algorithm:
o A depth-limited search algorithm is similar to depth-first search with a
predetermined limit.
o Depth-limited search can solve the drawback of the infinite path in the Depth-first
search. In this algorithm, the node at the depth limit will treat as it has no successor
nodes further.
Depth-limited search can be terminated with two Conditions of failure:
• Standard failure value: It indicates that problem does not have any solution.
• Cutoff failure value: It defines no solution for the problem within a given depth limit.
Advantages:
• Depth-limited search is Memory efficient.
Disadvantages:
• Depth-limited search also has a disadvantage of incompleteness.
• It may not be optimal if the problem has more than one solution.

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17 Features of DLS
o Completeness: DLS search algorithm is complete if the solution is above
the depth-limit.
o Time Complexity: Time complexity of DLS algorithm is O().
o Space Complexity: Space complexity of DLS algorithm is O(b×ℓ).
o Optimal: Depth-limited search can be viewed as a special case of DFS,
and it is also not optimal even if ℓ>d.

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18 4. Uniform-cost Search Algorithm:
o Uniform-cost search is a searching algorithm used for traversing a weighted tree or graph.
o This algorithm comes into play when a different cost is available for each edge.
o The primary goal of the uniform-cost search is to find a path to the goal node which has the
lowest cumulative cost.
o Uniform-cost search expands nodes according to their path costs from the root node.
o It can be used to solve any graph/tree where the optimal cost is in demand. A uniform-cost
search algorithm is implemented by the priority queue.
o It gives maximum priority to the lowest cumulative cost. Uniform cost search is equivalent to
BFS algorithm if the path cost of all edges is the same.
o Advantages:
o Uniform cost search is optimal because at every state the path with the least cost is
chosen.
o Disadvantages:
o It does not care about the number of steps involve in searching and only concerned
about path cost. Due to which this algorithm may be stuck in an infinite loop.

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20 Features of UCS:
Completeness:
o Uniform-cost search is complete, such as if there is a solution, UCS will find
it.
Time Complexity:
o Let C* is Cost of the optimal solution, and ε is each step to get closer to the
goal node. Then the number of steps is = C*/ε+1. Here we have taken +1,
as we start from state 0 and end to C*/ε.
o Hence, the worst-case time complexity of Uniform-cost search isO(b1 +
[C*/ε])/.
Optimal:
o Uniform-cost search is always optimal as it only selects a path with the
lowest path cost.

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21 5. Bidirectional Search Algorithm:
o Bidirectional search algorithm runs two simultaneous searches, one form initial
state called as forward-search and other from goal node called as
backward-search, to find the goal node.
o Bidirectional search replaces one single search graph with two small
subgraphs in which one starts the search from an initial vertex and other starts
from goal vertex. The search stops when these two graphs intersect each
other.
o Bidirectional search can use search techniques such as BFS, DFS, DLS, etc.
Advantages:
o Bidirectional search is fast.
o Bidirectional search requires less memory
Disadvantages:
o Implementation of the bidirectional search tree is difficult.
o In bidirectional search, one should know the goal state in advance.

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22 Example:
o In the below search tree, bidirectional search algorithm is applied. This algorithm
divides one graph/tree into two sub-graphs. It starts traversing from node 1 in the
forward direction and starts from goal node 16 in the backward direction.
o The algorithm terminates at node 9 where two searches meet

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23 Features of Bidirectional Search :
• Completeness: Bidirectional Search is complete if we use BFS in both
searches.
• Time Complexity: Time complexity of bidirectional search using BFS
is O(bd
).
• Space Complexity: Space complexity of bidirectional search is O(bd
).
• Optimal: Bidirectional search is Optimal.

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