Search Problems: Concepts and Examples

Exploring different options or possibilities to find a solution.

Represents the current situation or configuration of a problem. For example, in an 8-puzzle, it would be the arrangement of the tiles on the board.

A function that takes a state as input and outputs a set of possible next states. It describes how you can move from one state to another in a problem.

The desired outcome or final arrangement in a problem. In the 8-puzzle, it's the correct order of the numbered tiles.

Using knowledge to explore alternatives and find the best solution. It involves defining the problem, choosing a search strategy, and evaluating the explored options.

A set of all possible states or configurations that a problem can be in.

The starting point of a search problem.

A function that determines if a given state satisfies the goal condition.

The cost of transitioning from one state to another.

A sequence of states that starts from the initial state and ends at a goal state.

A solution that has the lowest cost among all possible solutions.

A mathematical representation of how many tiles are out of order compared to the goal state. Calculated by counting inversions and the row number of the blank tile.

The difference between the state space of a puzzle and the set of reachable states. Only states with the same parity (even or odd) as the goal state are reachable.

A property of a state space that shows that any legal move of the blank tile in the (n²-1)-puzzle will either leave 'N' unchanged or increase/decrease it by an even number.

Used to determine if a puzzle configuration is solvable. If the N values of the starting state and the goal state have the same parity (both even or both odd), then the goal state is reachable.

The (n²-1)-puzzle state space is divided into two equally sized sets of reachable configurations, determined by the parity of N.

A specific, challenging puzzle involving 15 numbered tiles that highlights the importance of state space and solvability conditions.

A famous puzzle that offered $1,000 to anyone who could solve it, highlighting the difficulty and principles involved in solving puzzles.

The process of navigating through the state space to find a path from the initial state to the goal state.

A problem that cannot be solved due to an unreachable goal state, meaning there's no solution within the possible arrangements of the problem.

A diagram used in search algorithms to visualize the process of exploring states and finding a solution by branching out from the initial state to other potential states.

When exploring an entire state space is computationally too expensive or impossible, search algorithms must focus on a smaller subset of states within the space.

An abstract representation of a collection of possible worlds with shared important properties, ignoring non-important details.

The set of all possible states that a problem can be in.

A function that takes a state as input and returns a set of possible next states, along with their associated costs.

A positive number representing the cost of transitioning from one state to another.

The initial state is where you begin the problem solving process.

A sequence of states starting from the initial state and ending at a goal state, representing a solution to the problem.

The desired outcome or the final arrangement in a problem.

The process of exploring different paths in the state space to find the best solution.

The cost of moving between states when considering only horizontal, vertical, and diagonal movements, with diagonal movements being the square root of 2 times more expensive than the others.

Involves discretizing the space into a grid and finding the shortest path on this grid. It then involves smoothing the path to make it more efficient for the actual continuous space.

A method of planning that utilizes a sweep-line to define the workspace or environment. The sweep-line moves across the space and helps in defining the states and decisions in the problem.

Represent the different possible configurations or arrangements of the problem at a given point in time. These states are defined based on the sweep-line's movement through the workspace.

Describes the possible transitions between states in a problem. In this case, each state represents a position and the successor function describes possible movements of the sweep-line, leading to new states.

The shortest path found in the discretized state space is further optimized by smoothing it out to ensure it is also the shortest path in the original continuous space.

Defines the cost of moving between states as the length of the line segment connecting these states. This formulation directly considers the real distances involved.

A graph that represents all the possible paths a moving object can take. The vertices are points in the workspace, and the edges are the possible connections between them.

The shortest path found in the visibility graph is guaranteed to be the shortest path in the original continuous space as it takes into account the actual distances.

A planning approach where the goal is to assemble a product, defined by the stages of assembling subassemblies from individual parts. The goal state is a fully assembled product.