Glossary

2.2 2D convex hull algorithms

9 min read•august 21, 2024

Convex hulls are the smallest convex shapes enclosing a set of points. They're crucial in computational geometry, forming the foundation for many algorithms. From gift wrapping to divide-and-conquer, various methods compute convex hulls efficiently.

Different approaches balance time complexity, output sensitivity, and special cases. Understanding convex hulls and their algorithms is key to solving geometric problems in graphics, robotics, and machine learning. It's a fundamental concept with wide-ranging applications.

Definition of convex hull

  • Fundamental concept in computational geometry describes the smallest convex set containing all points in a given set
  • Serves as a crucial building block for various geometric algorithms and problem-solving techniques in the field

Geometric interpretation

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  • Visualized as an elastic band stretched around a set of points, enclosing all points within its boundary
  • Represents the of a set, forming a polygon that encloses all other points
  • Includes all line segments between any two points in the set
  • Can be thought of as the "shape" formed by a set of nails on a board when a rubber band is stretched around them

Mathematical formulation

  • Defined as the intersection of all convex sets containing the given set of points
  • Expressed as the set of all convex combinations of points in the original set
  • Formulated mathematically as CH(S)={∑i=1nλipi:pi∈S,λi≥0,∑i=1nλi=1}CH(S) = \{\sum_{i=1}^n \lambda_i p_i : p_i \in S, \lambda_i \geq 0, \sum_{i=1}^n \lambda_i = 1\}
  • Characterized by the property that for any two points in the convex hull, the line segment connecting them lies entirely within the hull

Properties of convex hulls

  • Essential characteristics that make convex hulls useful in computational geometry and related fields
  • Provide insights into the structure and behavior of point sets in geometric spaces

Extreme points

  • Vertices of the convex hull polygon, forming the outermost points of the set
  • Always a subset of the original point set, representing the most "extreme" positions
  • Determine the shape and size of the convex hull
  • Can be used to efficiently represent the entire convex hull, as all other points lie within the polygon formed by these vertices

Convexity and containment

  • Convex hull forms a , meaning all internal angles are less than or equal to 180 degrees
  • Contains all points of the original set within its boundary or on its edges
  • Smallest convex set that contains all points of the original set
  • Preserves convexity under various geometric operations (intersection, union) with other convex sets

Importance in computational geometry

  • Convex hulls serve as fundamental building blocks for many algorithms and problem-solving techniques
  • Enable efficient solutions to complex geometric problems by simplifying the representation of point sets

Applications in various fields

  • Computer graphics uses convex hulls for and bounding volume hierarchies
  • Robotics employs convex hulls for path planning and obstacle avoidance algorithms
  • utilizes convex hulls for shape analysis and feature extraction
  • Geographic Information Systems (GIS) apply convex hulls for spatial analysis and data visualization
  • Machine learning leverages convex hulls for support vector machines and clustering algorithms

Efficiency considerations

  • Convex hull algorithms strive to minimize time and space complexity for large point sets
  • Trade-offs between worst-case and average-case performance influence algorithm selection
  • Preprocessing of input data can significantly impact overall efficiency of convex hull computation
  • Parallelization and GPU acceleration techniques enhance performance for large-scale problems
  • adapt their runtime based on the complexity of the resulting convex hull

Graham scan algorithm

  • Popular and efficient method for computing the convex hull of a planar point set
  • Utilizes a -based approach to construct the convex hull incrementally

Algorithm overview

  • Starts by finding the point with the lowest y-coordinate (and leftmost if tied) as the anchor point
  • Sorts remaining points based on polar angle with respect to the anchor point
  • Processes sorted points, maintaining a stack of potential hull vertices
  • Removes points that create concavities by checking for left turns
  • Constructs the convex hull by iteratively adding points and removing non-convex vertices
  • Terminates when all points have been processed, resulting in the final convex hull

Time complexity analysis

  • Achieves time complexity, where n represents the number of input points
  • Sorting step dominates the runtime, typically using comparison-based sorting algorithms
  • Stack operations and left turn tests contribute O(n) time to the overall complexity
  • Space complexity remains O(n) for storing the sorted points and the stack
  • Performs well in practice for most input distributions and sizes

Jarvis march algorithm

  • Also known as the "gift wrapping" algorithm due to its intuitive approach
  • Constructs the convex hull by simulating the wrapping of a gift with paper

Algorithm description

  • Begins by selecting the leftmost point as the starting point of the hull
  • Iteratively finds the next hull by selecting the point with the smallest polar angle
  • Continues the process until returning to the starting point, completing the hull
  • Uses cross product calculations to determine the next point in the hull
  • Handles collinear points by selecting the farthest point when multiple points have the same angle

Comparison with Graham scan

  • Jarvis march achieves O(nh) time complexity, where h represents the number of hull vertices
  • More efficient than Graham scan when the number of hull vertices is small compared to the total points
  • Does not require a preprocessing sorting step, unlike Graham scan
  • Generally simpler to implement and understand conceptually
  • Can be extended to higher dimensions more easily than Graham scan

Quickhull algorithm

  • Recursive divide-and-conquer approach for computing convex hulls
  • Named after its similarity to the Quicksort algorithm in terms of partitioning strategy

Divide-and-conquer approach

  • Selects two extreme points (leftmost and rightmost) to form an initial line segment
  • Partitions the remaining points into two sets based on their position relative to this line
  • Finds the point farthest from the line segment in each partition
  • Recursively applies the algorithm to the subproblems created by the new triangles
  • Combines the results of subproblems to form the final convex hull

Performance characteristics

  • Average-case time complexity of O(n log n) for uniformly distributed points
  • Worst-case time complexity of when points are arranged in certain degenerate configurations
  • Space complexity of O(n) for storing the recursion stack and intermediate results
  • Performs well in practice, especially for inputs with few points on the hull
  • Susceptible to numerical precision issues in floating-point implementations

Chan's algorithm

  • Combines the strengths of other convex hull algorithms to achieve optimal worst-case complexity
  • Utilizes a clever guessing strategy to determine the output size

Algorithm concept

  • Divides the input points into smaller subsets and computes their convex hulls using Graham scan
  • Applies Jarvis march on the subhulls, guessing the size of the final convex hull
  • Iteratively doubles the guessed size until the correct hull size is found
  • Terminates when the complete convex hull is constructed or the guessed size exceeds the input size
  • Employs a binary search technique to efficiently determine the correct hull size

Optimal worst-case complexity

  • Achieves O(n log h) time complexity, where h represents the number of hull vertices
  • Matches the lower bound for output-sensitive convex hull algorithms
  • Combines the efficiency of Graham scan for small subproblems with the output-sensitivity of Jarvis march
  • Adapts its runtime based on the complexity of the output, performing well for both small and large hulls
  • Provides a theoretical breakthrough in convex hull computation, though rarely used in practice due to implementation complexity

Incremental algorithms

  • Construct the convex hull by adding points one at a time to an existing hull
  • Offer flexibility in handling dynamic point sets and online scenarios

Kallay's algorithm

  • Maintains a triangulation of the convex hull as points are added incrementally
  • Updates the triangulation and hull structure efficiently for each new point
  • Achieves O(n log n) expected time complexity for randomly ordered input points
  • Handles degenerate cases and numerical stability issues effectively
  • Allows for dynamic maintenance of the convex hull as points are inserted or deleted

Online vs offline methods

  • Online algorithms process points as they arrive, without knowledge of future inputs
  • Offline algorithms have access to the entire input set before processing begins
  • Online methods offer flexibility for streaming data and real-time applications
  • Offline methods can often achieve better time complexity by preprocessing the input
  • Trade-off between adaptability to changing inputs and overall efficiency

Output-sensitive algorithms

  • Runtime depends on the size of the output (number of hull vertices) rather than just the input size
  • Particularly efficient for inputs where the convex hull has few vertices compared to the total number of points

Kirkpatrick-Seidel algorithm

  • Also known as the "ultimate planar convex hull algorithm"
  • Achieves O(n log h) worst-case time complexity, where h is the number of hull vertices
  • Uses a divide-and-conquer approach, similar to Quickhull, but with a more sophisticated partitioning strategy
  • Employs a technique called "marriage-before-conquest" to efficiently identify and eliminate interior points
  • Theoretically optimal but complex to implement, limiting its practical use

Relation to instance complexity

  • Output-sensitive algorithms adapt their performance based on the "difficulty" of the input instance
  • Instance complexity considers factors such as point distribution and degeneracies
  • Allows for more fine-grained analysis of algorithm performance across different input types
  • Provides insights into the inherent complexity of convex hull computation for specific problem instances
  • Guides the development of hybrid algorithms that combine multiple approaches based on input characteristics

Special cases and optimizations

  • Tailored algorithms and techniques for specific types of input or problem variations
  • Exploit unique properties of certain point configurations to achieve improved efficiency

Convex hull of a simple polygon

  • Computes the convex hull of a polygon without self-intersections
  • Achieves linear time complexity O(n) using techniques like Melkman's algorithm
  • Exploits the ordering of polygon vertices to efficiently construct the hull
  • Handles both convex and non-convex input polygons
  • Finds applications in shape simplification and polygon decomposition

Dynamic convex hull maintenance

  • Supports efficient updates (insertions and deletions) to the convex hull as points change
  • Utilizes data structures like balanced binary search trees to maintain hull vertices
  • Achieves logarithmic time complexity for point updates in many cases
  • Finds applications in motion planning and computational geometry for moving objects
  • Balances the trade-off between update efficiency and query performance

Lower bounds

  • Theoretical limits on the time and space complexity of convex hull algorithms
  • Provide insights into the fundamental difficulty of the problem and guide algorithm design

Time complexity lower bounds

  • Ω(n log n) lower bound for computing the convex hull in the general case
  • Proof based on the reduction from sorting to convex hull computation
  • Holds in the algebraic decision tree model of computation
  • Implies that comparison-based algorithms cannot achieve sub-O(n log n) worst-case time complexity
  • Motivates the development of output-sensitive algorithms to overcome this bound in certain cases

Space complexity considerations

  • Ω(n) lower bound on space complexity for storing the input points
  • Trade-offs between time and space complexity in various algorithmic approaches
  • In-place algorithms aim to minimize additional space usage beyond the input storage
  • Succinct representations of convex hulls can reduce space requirements in some applications
  • Memory hierarchy considerations impact practical performance, especially for large datasets

Implementation considerations

  • Practical aspects of implementing convex hull algorithms in real-world scenarios
  • Address challenges related to numerical precision, efficiency, and scalability

Numerical stability issues

  • Floating-point arithmetic can lead to rounding errors and inconsistent results
  • Techniques like epsilon comparisons and exact arithmetic libraries mitigate precision problems
  • Robust geometric predicates ensure correct results even with limited precision
  • Careful handling of degenerate cases (collinear points, coincident points) improves stability
  • Trade-offs between numerical robustness and computational efficiency must be considered

Parallel algorithms for convex hulls

  • Exploit multi-core processors and distributed systems to accelerate computation
  • Divide-and-conquer approaches like Quickhull naturally lend themselves to parallelization
  • GPU-accelerated algorithms leverage massive parallelism for large-scale problems
  • Load balancing and communication overhead pose challenges in parallel implementations
  • Hybrid approaches combine sequential and parallel techniques for optimal performance

Convex hull in higher dimensions

  • Extension of the convex hull concept to spaces beyond two dimensions
  • Presents new challenges and opportunities for algorithm design and analysis

Extension to 3D and beyond

  • 3D convex hulls form polyhedra instead of polygons
  • Algorithms like Gift wrapping and Quickhull generalize to higher dimensions
  • Incremental algorithms often perform well in practice for moderate-dimensional problems
  • Applications include computational geometry, computer graphics, and machine learning
  • Visualization and interpretation become more challenging as dimensionality increases

Computational challenges

  • Time and space complexity grow exponentially with the number of dimensions
  • Curse of dimensionality affects both the worst-case and average-case performance
  • Degeneracies and numerical instability become more prevalent in higher dimensions
  • Trade-offs between exact and approximate algorithms become more pronounced
  • Specialized data structures and techniques required for efficient high-dimensional convex hull computation

Key Terms to Review (23)

Binary search tree: A binary search tree (BST) is a data structure that allows for efficient searching, inserting, and deleting of elements. In a BST, each node has at most two children, and for any given node, the left subtree contains only nodes with values less than the node's value, while the right subtree contains only nodes with values greater than the node's value. This structure allows algorithms that rely on it to efficiently manage and retrieve ordered data.
Chan's Algorithm: Chan's Algorithm is an output-sensitive method for computing the convex hull of a set of points in two-dimensional space. It combines aspects of both divide-and-conquer and incremental algorithms, achieving optimal performance in terms of the number of output vertices. This algorithm efficiently handles large sets of points by breaking the problem into smaller subsets and merging their results, which makes it particularly effective for applications where the output size varies significantly.
Collision Detection: Collision detection refers to the computational process of determining whether two or more geometric objects intersect or collide within a defined space. This process is vital in various fields such as computer graphics, robotics, and physics simulations, enabling the accurate modeling of interactions and behaviors among objects. Efficient collision detection algorithms are essential for managing complex environments where many objects may interact simultaneously.
Convex polygon: A convex polygon is a simple polygon in which all interior angles are less than 180 degrees, and no line segment between any two points on the boundary ever goes outside the polygon. This characteristic ensures that the polygon bulges outward, making it easier to work with in various geometric computations and algorithms, such as those involving triangulation, shape analysis, and enclosing properties.
Divide and conquer: Divide and conquer is a fundamental algorithmic paradigm that breaks a problem into smaller, more manageable subproblems, solves each subproblem independently, and then combines the results to solve the original problem. This approach simplifies complex problems by leveraging recursive techniques, making it particularly effective in computational geometry for tasks like triangulation and convex hull generation.
Dominance: In computational geometry, dominance refers to a relationship between points in a multi-dimensional space, where one point is considered 'dominant' over another if it is better in all relevant dimensions. This concept is particularly crucial in understanding 2D convex hull algorithms, as it helps identify which points contribute to the outer boundary of a set of points, ultimately aiding in efficient geometric computations.
Graham's Scan: Graham's Scan is an efficient algorithm used to compute the convex hull of a set of points in the plane. It works by first sorting the points based on their polar angle with respect to a reference point, and then constructing the convex hull by processing these sorted points while maintaining a stack of vertices that form the hull. This method not only establishes the connection between geometric algorithms and convex sets but also showcases practical applications in various fields, such as computer graphics and robotics.
Hull properties: Hull properties refer to the characteristics and attributes of the convex hull, which is the smallest convex set that can enclose a given set of points in a plane. Understanding hull properties is crucial when implementing 2D convex hull algorithms, as they dictate how points interact geometrically and influence the efficiency and outcomes of these algorithms. Key hull properties include aspects like uniqueness, the relationship between points inside and outside the hull, and how they affect computations such as area and perimeter.
Incremental algorithm: An incremental algorithm is a computational approach that constructs a solution step by step, adding one element at a time and updating the existing solution as each new element is introduced. This method is especially useful in scenarios where the input data can change or when only small modifications need to be processed, allowing for efficient updates without starting from scratch.
J. O. Lee: J. O. Lee is a prominent figure in computational geometry, particularly recognized for his contributions to the study and development of algorithms for 2D convex hulls. His work has significantly influenced various algorithms designed to efficiently compute the convex hull of a set of points in a two-dimensional space, impacting fields such as computer graphics, geographic information systems, and pattern recognition.
Jarvis March (Gift Wrapping): Jarvis March, also known as the Gift Wrapping algorithm, is a computational geometry technique used to find the convex hull of a set of points in a two-dimensional space. This algorithm works by 'wrapping' the points in a manner similar to wrapping a gift, starting from an extreme point and iteratively selecting the next point that forms the smallest angle with respect to the previous point, ultimately creating the convex boundary around the set of points.
Kallay's Algorithm: Kallay's Algorithm is a method for computing the convex hull of a set of points in a two-dimensional space. This algorithm focuses on identifying the outermost points that form the boundary of a given set, making it an essential tool in computational geometry. By efficiently processing point coordinates, Kallay's Algorithm contributes to various applications, such as computer graphics, geographic information systems, and pattern recognition.
Kirkpatrick-Seidel Algorithm: The Kirkpatrick-Seidel algorithm is an efficient method for computing the convex hull of a set of points in two-dimensional space, achieving an output-sensitive complexity of O(n log h), where n is the number of input points and h is the number of points on the convex hull. This algorithm stands out because it adapts its performance based on the size of the output, making it particularly useful for cases where the number of output points is significantly smaller than the input size. It uses a divide-and-conquer approach combined with a careful selection of pivot points to ensure optimal performance in varying scenarios.
O(n log n): The notation o(n log n) represents a class of algorithms whose time complexity grows at a rate slower than n log n as the size of the input (n) increases. This complexity is significant in computational geometry and algorithm analysis, indicating efficient performance for large datasets. Algorithms with this complexity are desirable because they can handle larger inputs within a reasonable time frame, particularly in tasks such as sorting, searching, and geometric computations.
O(n^2): The notation o(n^2) describes an upper bound on the growth rate of an algorithm's time complexity, indicating that its performance is significantly better than quadratic time as the input size, n, increases. This means that as n becomes large, the time required by the algorithm grows slower than some constant times n squared. It plays a crucial role in evaluating the efficiency of various algorithms, especially in computational geometry, where optimal performance can be critical.
Outermost Points: Outermost points refer to the set of points that define the boundary of a given set of points in a two-dimensional space. These points are critical for constructing the convex hull, which is the smallest convex polygon that can enclose all the points in the set. Identifying the outermost points helps in understanding the geometric properties and relationships of the point set, as they represent the extreme values in different directions.
Output-sensitive algorithms: Output-sensitive algorithms are a class of algorithms whose running time depends not only on the size of the input but also on the size of the output produced. This means that their efficiency can vary significantly based on how much information is returned, making them particularly useful in scenarios where the output size is much smaller than the input size. This characteristic is crucial in fields where the results can be sparse or where only a subset of the data needs processing, such as in certain computational geometry tasks.
Pattern recognition: Pattern recognition is the process of identifying and classifying patterns in data, often used to make sense of complex information. This involves analyzing data points to determine similarities and differences, which can help in making predictions or decisions. In computational geometry, pattern recognition can be applied to various algorithms and methods, enhancing tasks like shape detection, image processing, and data classification.
Point Location: Point location refers to the problem of determining the region or object in a geometric space that contains a given point. This concept is crucial for various geometric algorithms and applications, allowing for efficient querying of spatial relationships in structures like polygons, Voronoi diagrams, and triangulations.
Quickhull algorithm: The quickhull algorithm is a computational geometry method used to find the convex hull of a set of points in two-dimensional space. This algorithm operates by recursively dividing the point set into smaller subsets and identifying the extreme points that form the boundary, ultimately resulting in a polygon that encapsulates all other points. Its efficiency makes it particularly valuable for applications requiring quick computations of convex hulls, as well as for understanding the properties of convexity and convex sets.
R. l. graham: R. L. Graham is a prominent computer scientist known for his significant contributions to computational geometry, particularly in the development of algorithms for finding convex hulls. His work has greatly influenced the efficiency and effectiveness of geometric computations, laying the groundwork for various algorithms used in computer graphics, geographic information systems, and robotics.
Stack: A stack is a data structure that follows the Last In, First Out (LIFO) principle, meaning that the last element added to the stack is the first one to be removed. This structure is crucial in various algorithms, especially in computational geometry for managing and organizing information efficiently. Stacks are used to maintain order in processes such as determining the convex hull, where the arrangement of points can significantly affect the outcome of the algorithm.
Vertex: A vertex is a fundamental point in geometry that represents a corner or intersection of geometric shapes, such as polygons and polyhedra. In many contexts, vertices are the points where edges meet and can be crucial in defining the structure and properties of shapes, influencing various computational geometry operations and algorithms.
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