Efficient Sorting and Ranking Algorithms for Self-Delimiting Numbers with Applications to Outerplanar Graph Isomorphism

Core Concepts

This paper presents space-efficient algorithms for sorting, dense ranking, and competitive ranking of self-delimiting numbers in the word RAM model. The algorithms run in O(k + N/log N) time and use O(N) bits, where k is the number of self-delimiting numbers and N is the total size of the input. These algorithms are then applied to develop linear-time and linear-space algorithms for tree, forest, and outerplanar graph isomorphism.

Abstract

The paper focuses on developing space-efficient algorithms for sorting and ranking self-delimiting numbers, which are numbers represented using a variable-length encoding scheme.
For sorting, the authors first divide the input sequence S into two parts: S≤q containing numbers up to size q = 2^(N/τ), and S>q containing the remaining larger numbers. They sort S≤q using a stable counting sort approach that first groups the numbers by their length and then sorts each group. For S>q, they interpret the numbers as strings over an alphabet of size 2^(τ/2) and sort them by length and then lexicographically.
For dense ranking and competitive ranking, the authors again handle small and large numbers differently. For small numbers in S≤N, they use a bit vector B to mark the distinct numbers and an array P to store prefix sums. This allows them to compute the dense/competitive rank of a number in constant time. For large numbers in S>N, they store the rank information in an additional data structure.
As an application, the authors present linear-time and linear-space algorithms for tree isomorphism, forest isomorphism, and outerplanar graph isomorphism. These improve upon the previous best algorithms that used Θ(n log n) bits of space.

Stats

The self-delimiting number of an integer x can be stored within 2⌈log x⌉+ 1 bits.
The authors assume a word RAM model with word size Ω(log N) bits, where N is the input size in bits.
The sorting algorithm runs in O(k + N/log N) time and uses O(N) bits.
The dense/competitive ranking algorithms run in O(k + N/log N) time and use O(N) bits.

Quotes

"Assume that an N-bit sequence S of k numbers encoded as Elias gamma codes is given as input. We present space-efficient algorithms for sorting, dense ranking and competitive ranking on S in the word RAM model with word size Ω(log N) bits."
"Our algorithms run in O(k + N/log N) time and use O(N) bits."

Key Insights Distilled From

Sorting and Ranking of Self-Delimiting Numbers with Applications to Outerplanar Graph Isomorphism

by Frank Kammer... at arxiv.org 05-03-2024

https://arxiv.org/pdf/2002.07287.pdf

Deeper Inquiries

How can the proposed algorithms be extended to handle other types of self-delimiting number representations beyond Elias gamma codes

The proposed algorithms can be extended to handle other types of self-delimiting number representations by modifying the data structures and operations to accommodate the specific encoding format. For example, if we want to handle Elias delta codes or Elias omega codes instead of Elias gamma codes, we would need to adjust the way we interpret and manipulate the self-delimiting numbers within the algorithms.
To extend the algorithms to support different types of self-delimiting number representations, we would need to:

Update the parsing and decoding mechanisms to correctly interpret the new encoding format.
Modify the data structures used for sorting and ranking to work with the specific characteristics of the new representation.
Adjust any lookup tables or auxiliary structures to match the requirements of the alternative encoding scheme.
Ensure that the time and space complexities of the algorithms remain within acceptable bounds for the new representation.

By making these adaptations, the algorithms can effectively handle a variety of self-delimiting number representations beyond Elias gamma codes, providing flexibility and versatility in processing different types of encoded data.

What are the potential trade-offs between the time and space complexities of the sorting and ranking algorithms, and how can they be adjusted based on the specific requirements of the application

The potential trade-offs between the time and space complexities of the sorting and ranking algorithms depend on the specific requirements of the application and the size of the input data. Here are some considerations for adjusting the algorithms based on these trade-offs:

Time Complexity vs. Space Complexity:

If the application prioritizes faster processing speed, the algorithms can be optimized for lower time complexity even if it results in higher space usage.
Conversely, if minimizing memory usage is critical, the algorithms can be adjusted to reduce space complexity, even if it leads to slightly longer processing times.

Input Size and Distribution:

For datasets with a large number of elements but limited memory availability, the algorithms can be fine-tuned to balance time and space usage efficiently.
If the input data is skewed or has specific patterns, the algorithms can be optimized to exploit these characteristics for better performance.

Algorithmic Techniques:

Utilizing advanced data structures, such as succinct data structures or compressed representations, can help reduce space complexity without significantly impacting time complexity.
Implementing parallel processing or distributed computing techniques can improve processing speed while managing memory resources effectively.

By carefully analyzing the requirements of the application and considering these trade-offs, the sorting and ranking algorithms can be adjusted to meet the specific needs of the problem at hand.

Can the techniques used in the tree, forest, and outerplanar graph isomorphism algorithms be applied to develop space-efficient algorithms for other graph-related problems

The techniques used in the tree, forest, and outerplanar graph isomorphism algorithms can be applied to develop space-efficient algorithms for a wide range of graph-related problems. Here's how these techniques can be extended to address other graph problems:

Graph Isomorphism Variants:

The tree isomorphism algorithm can be adapted for isomorphism testing in other graph types, such as directed graphs, acyclic graphs, or planar graphs.
The forest isomorphism technique can be extended to handle isomorphism in more complex graph structures like hypergraphs or bipartite graphs.

Graph Connectivity and Components:

The methods for identifying connected components in forests can be applied to general graphs to determine connectivity and component structures efficiently.
Techniques for detecting cycles in outerplanar graphs can be utilized for cycle detection in various graph types.

Graph Matching and Subgraph Isomorphism:

The algorithms for outerplanar graph isomorphism can be adapted for matching subgraphs within larger graphs, enabling efficient subgraph isomorphism testing.
The tree isomorphism approach can be used for matching specific tree patterns or motifs within larger graphs.

By leveraging the principles and methodologies employed in the tree, forest, and outerplanar graph isomorphism algorithms, space-efficient solutions can be developed for a wide array of graph-related problems, offering scalability and performance optimization.