M*

Multi-agent pathfinding algorithm that finds collision-free paths for multiple agents by resolving conflicts and coordinating their movements.

Family: Planning Algorithms Status: ✅ Complete

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Overview

M is a multi-agent pathfinding algorithm that extends A search to handle multiple agents simultaneously.

The algorithm finds collision-free paths for multiple agents by detecting conflicts between their planned paths and resolving them through coordination and replanning.

The key insight is that agents can plan their paths independently initially, but when conflicts arise (agents occupying the same location at the same time), the algorithm expands the search space to include coordinated movements that resolve these conflicts.

Mathematical Formulation¶

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Mathematical Properties

Multi-Agent State

s = (s₁, s₂, ..., sₙ) where sᵢ is state of agent i

Joint state of all agents

Conflict Detection

conflict(s, s') if ∃i,j: sᵢ = s'ⱼ and i ≠ j

Agents occupy same location

Coordination Graph

G = (V, E) where V is agents and E is conflicts

Graph showing which agents need coordination

Expanded Search Space

S' = S ∪ {coordinated_moves}

Search space including coordinated actions

Key Properties¶

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Multi-Agent

Handles multiple agents simultaneously
Conflict Resolution

Detects and resolves agent conflicts
Coordination

Coordinates agent movements when needed
Scalable

Efficient for moderate numbers of agents

Implementation Approaches¶

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Standard M*

Classic M* implementation with conflict detection and resolution

Complexity:

Time: O(b^d * n²)
Space: O(b^d * n)

Advantages

Handles multiple agents simultaneously
Finds collision-free paths
Coordinates agent movements
Extends A* to multi-agent domain
Good for moderate numbers of agents

Disadvantages

Exponential complexity with number of agents
May not scale to many agents
Requires coordination graph computation
Can be memory intensive

Use Cases & Applications¶

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Application Categories

Multi-Robot Systems

warehouse robots: warehouse robots
swarm robotics: swarm robotics
autonomous vehicles: autonomous vehicles

Game AI

strategy games: strategy games
real-time strategy: real-time strategy
multi-agent games: multi-agent games

Logistics

delivery systems: delivery systems
traffic management: traffic management
supply chain: supply chain

Simulation

crowd simulation: crowd simulation
traffic simulation: traffic simulation
evacuation planning: evacuation planning

Educational Value

Understanding multi-agent systems: Understanding multi-agent systems
Conflict resolution strategies: Conflict resolution strategies
Coordination algorithms: Coordination algorithms
Extension of single-agent algorithms: Extension of single-agent algorithms

References & Further Reading¶

:material-book: Core Textbooks

:material-file-document:

Interactive Learning

Try implementing the different approaches yourself! This progression will give you deep insight into the algorithm's principles and applications.

Pro Tip: Start with the simplest implementation and gradually work your way up to more complex variants.

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Related Algorithms in Planning Algorithms:

A* Search - Optimal pathfinding algorithm that uses heuristics to efficiently find the shortest path from start to goal in weighted graphs.
Depth-First Search - Graph traversal algorithm that explores as far as possible along each branch before backtracking, using stack-based or recursive implementation.
Breadth-First Search - Graph traversal algorithm that explores all nodes at the current depth level before moving to the next level, guaranteeing shortest path in unweighted graphs.

M*