Uniform Cost Search Algorithm

Sriniketh J 26 Jun, 2024
12 min read

Introduction

Whoever it may be (humans or machine learning models) need to think of all possible ways to reach the goal state(if it exists) from the initial state or current state, all the consequences, etc. Similarly, AI systems or python programming uses various search algorithms in artificial intelligence for a particular goal state(if it exists) or for some problem-solving. ‘Uniform Cost Search’, as the name suggests, means the machine blindly follows the algorithm regardless of whether right or wrong, efficient or in-efficient.

These algorithms are brute force operations, and they don’t have additional information about the search space; the only information they have is on how to traverse or visit the nodes in the tree. Thus uniform cost search algorithms are also called blind search algorithms in artificial intelligence. The search algorithm produces the search tree without using any domain knowledge, which is a brute force in nature. They don’t have any background information like informed search techniques algorithms on how to approach the goal or whatsoever. But these are the basics of search algorithms in AI.

Learning Objectives

  • Get acquainted with the different types of informed search techniques algorithms, their uses, advantages, and disadvantages.
  • Learn to compare between them and choose the most befitting one for your model.

This article was published as a part of the Data Science Blogathon.

Uninformed search, also known as blind search, is a search algorithm that explores a problem space without any specific knowledge or information about the problem other than the initial state and the possible actions to take. It lacks domain-specific heuristics or prior knowledge about the problem. Uninformed search algorithms, such as breadth-first search and depth-first search, systematically explore the search space by applying predefined rules to generate successor states until a goal state is found or the search is exhausted. These algorithms are typically less efficient than informed search techniques algorithms but can be useful in certain scenarios or as a basis for more advanced search techniques.

Types of Uninformed Search Algorithms

The different types of uninformed search algorithms used in AI are as follows:

  • Depth First Search
  • Breadth-First Search
  • Depth Limited Search
  • Uniform Cost Search
  • Iterative Deepening Depth First Search
  • Bidirectional Search (if applicable)

But before we go into these search types and you go a step further wandering into any Artificial Intelligence course, let’s get to know the few terms which will be frequently used in the upcoming sections.

State: It provides all the information about the environment.

Goal State: The desired resulting condition in a given problem and the kind of uniform cost search algorithm we are looking for.

Goal Test: The test to determine whether a particular state is a goal state.

Path/Step Cost: These are integers that represent the cost to move from one node to another node.

Space Complexity: A function describing the amount of space(memory) an algorithm takes in terms of input to the algorithm.

Time Complexity: A function describing the amount of time the algorithm takes in terms of input to the algorithm.

Optimal: Extent of preference of the algorithm

b‘ – maximum branching factor in a tree.

d‘ – the depth of the least-cost solution.

m‘ – maximum depth state space(maybe infinity)

Now let’s dive deep into each type of algorithm.

Depth First Search (DFS)

It is a search algorithm where the search tree will be traversed from the root node. It will be traversing, searching for a key at the leaf of a particular branch. If the key is not found, the searcher retraces its steps back (backtracking) to the point from where the other branch was left unexplored, and the same procedure is repeated for that other branch.

Uninformed Search Algorithms DFS

The above image clearly explains the DFS Algorithm. First, the search technique starts from the root node A and then goes to the branch where node B is present (lexicographical order). Then it goes to node D because of DFS, and from D, there is only one node to traverse, i.e., node H. But after node H  does not have any child nodes, we retrace the path in which we traversed earlier and again reach node B, but this time, we traverse through in the untraced path a traverse through node E.

There are two branches at node E, but let’s traverse node I (lexicographical order) and then retrace the path as we have no further number of nodes after E to traverse. Then we traverse node J as it is the untraced branch and then again find we are at the end and retrace the path and reach node B and then we will traverse the untraced branch, i.e., through node C, and repeat the same process. This is called the DFS Algorithm.

Advantages

  • DFS requires very little memory as it only needs to store a stack of the nodes on the path from the root node to the current node.
  • It takes less time to reach the goal node than the BFS algorithm [which is explained later](if it traverses in the right path).

Disadvantages

  • There is the possibility that many states keep reoccurring, and there is no guarantee of finding a solution.
  • The DFS algorithm goes for deep-down searching, and sometimes it may go to the infinite loop.

Verdict

It occupies a lot of memory space and time to execute when the solution is at the bottom or end of the tree and is implemented using the LIFO Stack data structure[DS].

  • Complete: No
  • Time Complexity: O(bm)
  • Space complexity: O(bm)
  • Optimal: Yes

Breadth-First Search (BFS)

This is another graph search algorithm in AI that traverses breadthwise to search for the goal in a tree. It begins searching from the root node and expands the successor node before expanding further along breadthwise and traversing those nodes rather than searching depth-wise.

Uninformed Search Algorithms BFS

The above figure is an example of a BFS Algorithm. It starts from the root node A and then traverses node B. Till this step, it is the same as DFS. But here, instead of expanding the children of B as in the case of DFS, we expand the other child of A, i.e., node C because of BFS, and then move to the next level and traverse from D to G and then from H to K in this typical example. To traverse here, we have only taken into consideration the lexicographical order. This is how the BFS Algorithm is implemented.

Advantages

  • BFS will provide a solution if any solution exists.
  • If there is more than one solution for a given problem, then BFS will provide the minimal solution which requires the least number of steps.

Disadvantages

  • It requires lots of memory since each level of the tree must be saved in memory to expand to the next level.
  • BFS needs lots of time if the solution is far away from the root node.

Verdict

It requires a lot of memory space and is time-consuming if the goal state is at the bottom or end. It uses a FIFO queue DS to implement.

  • Complete: Yes (assuming b is finite)
  • Time Complexity: O(bd)
  • Space complexity: O(bd)
  • Optimal: Yes, if step cost = 1 (i.e., no cost/all step costs are same)

Uniform Cost Search Algorithm (UCS)

Uniform Cost Search (UCS) explores graphs by expanding nodes from the start to the goal based on edge costs. It finds the lowest-cost path, essential when step costs vary for optimal solutions.

Uninformed Search Algorithms UCS

In the above figure, consider S to be the start node and G to be the goal state. From node S we look for a node to expand, and we have nodes A and G, but since it’s a uniform cost search, it’s expanding the node with the lowest step cost, so node A becomes the successor rather than our required goal node G. From A we look at its children nodes B and C. Since C has the lowest step cost, it traverses through node C. Then we look at the successors of C, i.e., D and G. Since the cost to D is low, we expand along with node D. Since D has only one child G which is our required goal state we finally reach the goal state D by implementing UFS Algorithm. If we have traversed this way, definitely our total path cost from S to G is just 6 even after traversing through many nodes rather than going to G directly where the cost is 12 and 6<<12(in terms of step cost). But this may not work with all cases.

Advantages

  • Uniform cost search is an optimal search method because at every state, the path with the least cost is chosen.

Disadvantages

  • It does not care about the number of steps or finding the shortest path involved in the search problem, and it is only concerned about path cost. This algorithm may be stuck in an infinite loop.

Verdict

  • Complete: Yes (if b is finite and costs are stepped, costs are zero)
  • Time Complexity: O(b(c/ϵ))   where, ϵ -> is the lowest cost, c -> optimal cost
  • Space complexity: O(b(c/ϵ))
  • Optimal: Yes (even for non-even cost)

Depth Limited Search (DLS)

DLS is an uninformed search algorithm. This is similar to DFS but differs only in a few ways. The sad failure of DFS is alleviated by supplying a depth-first search with a predetermined depth limit. That is, nodes at depth are treated as if they have no successors. This approach is called a depth-limited search. The depth limit solves the infinite-path problem. Depth-limited search can be halted in two cases:

  1. Standard Failure Value (SFV): The SFV tells that there is no solution to the problem.
  2. Cutoff Failure Value (CFV): The Cutoff Failure Value tells that there is no solution within the given depth limit.
DLS

The above figure illustrates the implementation of the DLS algorithm. Node A is at Limit = 0, followed by nodes B, C, D, and E at Limit = 1 and nodes F, G, and H at Limit = 2. Our start state is considered to be node A, and our goal state is node H. To reach node H, we apply DLS. So in the first case, let’s set our limit to 0 and search for the goal.

Since Limit 0

The algorithm will assume that there are no children after limit 0 even if nodes exist further. Now, if we implement it, we will traverse only node A as there is only one node in limit 0, which is basically our goal state. If we use SFV, it says there is no solution to the problem at limit 0, whereas LCV says there is no solution for the problem until the set depth limit. Since we could not find the goal, let’s increase our limit to 1 and apply DFS till limit 1, even though there are further nodes after limit 1. But those nodes aren’t expanded as we have set our limit as 1.

Hence nodes A, followed by B, C, D, and E, are expanded in the mentioned order. As in our first case, if we use SFV, it says there is no solution to the problem at limit 1, whereas LCV says there is no solution for the problem until the set depth limit 1. Hence we again increase our limit from 1 to 2 in the notion to find the goal.

Till limit 2

DFS will be implemented from our start node A and its children B, C, D, and E. Then from E, it moves to F, similarly backtracks the path, and explores the unexplored branch where node G is present. It then retraces the path and explores the child of C, i.e., node H, and then we finally reach our goal by applying DLS Algorithm. Suppose we have further successors of node F but only the nodes till limit 2 will be explored as we have limited the depth and have reached the goal state.

DLS

This image explains the DLS implementation and could be referred to for better understanding.

Depth-limited search can be terminated with two Conditions of failure:

  1. Standard Failure: it indicates that the problem does not have any solutions.
  2. Cutoff Failure Value: It defines no solution for the problem within a given depth limit.

Advantages

  • Depth-limited search is Memory efficient.

Disadvantages

  • The DLS has disadvantages of completeness and is not optimal if it has more than one goal state.

Verdict 

  • Complete: Complete (if solution > depth-limit)
  • Time Complexity: O(bl)  where, l -> depth-limit
  • Space complexity: O(bl)
  • Optimal: Yes (only if l > d)

Iterative Deepening Depth First Search (IDDFS)

It is a search algorithm that uses the combined power of the BFS and DFS algorithms. It is iterative in nature. Searches for the best depth in each iteration. It performs the Algorithm until it reaches the goal node. The algorithm is set to search until a certain depth and the depth keeps increasing at every iteration until it reaches the goal state.

IDDFS

In the above figure, let’s consider the goal node to be G and the start state to be A. We perform our IDDFS from node A. In the first iteration, it traverses only node A at level 0. Since the goal is not reached, we expand our nodes, go to the next level, i.e., 1 and move to the next iteration. Then in the next iteration, we traverse the node A, B, and C.

Even in this iteration, our goal state is not reached, so we expand the node to the next level, i.e., 2, and the nodes are traversed from the start node or the previous iteration and expand the nodes A, B, C, and D, E, F, G. Even though the goal node is traversed, we go through for the next iteration, and the remaining nodes A, B, D, H, I, E, C, F, K, and G(BFS & DFS) too are explored, and we find the goal state in this iteration. This is the implementation of the IDDFS Algorithm.

Advantages

  • It combines the benefits of BFS and DFSsearch algorithms in artificial intelligence in terms of fast search and memory efficiency.

Disadvantages

  • The main drawback of IDDFS is that it repeats all the work from the previous phase.

Verdict

  • Complete: Yes (by limiting the depth)
  • Time Complexity: O(bd)
  • Space complexity: O(bd)
  • Optimal: Yes (if step cost is uniform)
  • Systematic: It’s not systematic.

Bidirectional Search (BS)

Before moving into bidirectional search, let’s first understand a few terms.

Forward Search: Looking in front of the end from the start.

Backward Search: Looking from end to the start backward.

So Bidirectional Search, as the name suggests, is a combination of forwarding and backward search. Basically, if the average branching factor going out of node / fan-out, if fan-out is less, prefer forward search. Else if the average branching factor going into a node/fan-in is less (i.e., fan-out is more), prefer backward search. We must traverse the tree from the start node and the goal node, and wherever they meet, the path from the start node to the goal through the intersection is the optimal solution. The BS Algorithm is applicable when generating predecessors is easy in both forward and backward directions, and there exist only 1 or fewer goal states.

BS

This figure provides a clear-cut idea of how BS is executed. We have node 1 as the start/root node and node 16 as the goal node. The algorithm divides the search tree into two sub-trees. So from the start of node 1, we do a forward search, and at the same time, we do a backward search from goal node 16. The forward search traverses nodes 1, 4, 8, and 9, whereas the backward search traverses through nodes 16, 12, 10, and 9. We see that both forward and backward search meets at node 9, called the intersection node. So the total path traced by forwarding search and the path traced by backward search is the optimal solution. This is how the BS Algorithm is implemented.

Advantages

  • Since BS uses various techniques like DFS, BFS, DLS, etc., it is efficient and requires less memory.

Disadvantages

  • Implementation of the bidirectional search tree is difficult.
  • In bidirectional search, one should know the goal state in advance.

Verdict

  • Complete: Yes
  • Time Complexity: O(bd/2)
  • Space complexity: O(bd/2)
  • Optimal: Yes (if step cost is uniform in both forward and backward directions)

Final Interpretations

The Uninformed Search strategy for searching is a multipurpose strategy that combines the power of unguided search and works in a brute-force way. The algorithms of this strategy can be applied to a variety of problems in computer science as they don’t have the information related to state space and target problems, and they do not know how to traverse trees.

Final Interpretation

Conclusion

This is the complete analysis of all the Uninformed Search Strategies. Each search algorithm is no less than the other, and we can use any one of the search strategies based on the problem. The term ‘uninformed’ means that they do not have any additional information about states or state space. Thus we conclude “uninformed algorithm” is an algorithm that doesn’t use any prior knowledge or heuristics to solve a problem.

Key Takeaways

  • Uninformed algorithms are used in search problems, where the goal is to find a solution by exploring a large search space.
  • Uninformed algorithms are often simple to implement and can be effective in solving certain problems, but they may also be less efficient than informed algorithms that use heuristics to guide their search.

Frequently Asked Questions

Q1. What is a uniform cost search example?

A. Uniform cost search finds the cheapest path in a weighted graph. For example, in a road network, it identifies the path with the lowest travel cost, considering distances between nodes.

Q2. What is uniform cost search tool?

A. Uniform cost search is a graph traversal algorithm used in AI and computer science to find the lowest cost path from a starting node to a goal node in a weighted graph.

Q3. Why is it called uniform cost search?

A. Uniform cost search is named because it explores paths in order of increasing path cost from the start node, ensuring the cheapest path is found first.

Q4. What is uniform cost search in uninformed search?

A. Uniform cost search is an uninformed search algorithm that expands nodes with the lowest path cost g(n) first, making it suitable for finding the shortest path in graphs with non-uniform edge costs.

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Sriniketh J 26 Jun, 2024

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Responses From Readers

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Women Dame
Women Dame 26 Aug, 2022

thanks for this post its very useful for me.

Susmita Jana
Susmita Jana 12 Jun, 2023

In UCS there exist a path S to A to C to G. Whose path cost is 4. Why we could not consider this path.. Please reply . Thank you

Nandhu Kishore
Nandhu Kishore 25 Mar, 2024

In DLS, Complete: Complete (if solution > depth-limit). I think it should be complete only if solution < depth limit not greater than. please correct me if I am wrong.