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Parameterized Linear Time Transitive Closure

Giorgos Kritikakis, Ioannis G Tollis

TL;DR

This work addresses fast reachability queries on directed graphs by targeting the transitive closure problem through a chain-decomposition–based reachability index. It provides a parameterized linear-time construction in terms of the number of chains $k_c$, with constant-time query capability and space $O(k_c|V|)$, while leveraging a transitive-edge reduction that bounds non-transitive edges by $|E_{red}|\\\le w|V|$ where $w$ is the DAG width. The authors introduce efficient chain-decomposition heuristics (notably NH_conc) and demonstrate experimentally that most edges are transitive in dense graphs, enabling substantial practical speedups and even accelerating classical methods such as Fulkerson’s approach for computing DAG width. The results highlight the framework’s potential for real-world systems requiring fast reachability queries and point to interesting extensions for dynamic graph scenarios.

Abstract

Inquiries such as whether a task A depends on a task B, whether an author A has been influenced by a paper B, whether a certain protein is associated with a specific biological process or molecular function, or whether class A inherits from class B, are just a few examples of inquiries that can be modeled as reachability queries on a network (Directed Graph). Digital systems answer myriad such inquiries every day. In this paper, we discuss the transitive closure problem. We focus on applicable solutions that enable us to answer queries fast, in constant time, and can serve in real-world applications. In contrast to the majority of research on this topic that revolves around the construction of a two-dimensional adjacency matrix, we present an approach that builds a reachability indexing scheme. This scheme enables us to answer queries in constant time and can be built in parameterized linear time. In addition, it captures a compressed data structure. Our approach and algorithms are validated by extensive experiments that shed light on the factors that play a key role in this problem. To stress the efficiency of this solution and demonstrate the potential to apply our approach to important problems, we use it to speed up Fulkerson's method for finding the width of a DAG. Our results challenge the prevailing belief, reiterated over the last thirty years, regarding the efficiency of this method. Our approach is based on the concept of chain decomposition. Before we delve into its description, we introduce, analyze, and utilize a chain decomposition algorithm. Furthermore, we explore how chain decomposition can facilitate transitive closure solutions introducing a general purpose linear time reduction technique that removes a large subset of transitive edges given any chain decomposition.

Parameterized Linear Time Transitive Closure

TL;DR

This work addresses fast reachability queries on directed graphs by targeting the transitive closure problem through a chain-decomposition–based reachability index. It provides a parameterized linear-time construction in terms of the number of chains , with constant-time query capability and space , while leveraging a transitive-edge reduction that bounds non-transitive edges by where is the DAG width. The authors introduce efficient chain-decomposition heuristics (notably NH_conc) and demonstrate experimentally that most edges are transitive in dense graphs, enabling substantial practical speedups and even accelerating classical methods such as Fulkerson’s approach for computing DAG width. The results highlight the framework’s potential for real-world systems requiring fast reachability queries and point to interesting extensions for dynamic graph scenarios.

Abstract

Inquiries such as whether a task A depends on a task B, whether an author A has been influenced by a paper B, whether a certain protein is associated with a specific biological process or molecular function, or whether class A inherits from class B, are just a few examples of inquiries that can be modeled as reachability queries on a network (Directed Graph). Digital systems answer myriad such inquiries every day. In this paper, we discuss the transitive closure problem. We focus on applicable solutions that enable us to answer queries fast, in constant time, and can serve in real-world applications. In contrast to the majority of research on this topic that revolves around the construction of a two-dimensional adjacency matrix, we present an approach that builds a reachability indexing scheme. This scheme enables us to answer queries in constant time and can be built in parameterized linear time. In addition, it captures a compressed data structure. Our approach and algorithms are validated by extensive experiments that shed light on the factors that play a key role in this problem. To stress the efficiency of this solution and demonstrate the potential to apply our approach to important problems, we use it to speed up Fulkerson's method for finding the width of a DAG. Our results challenge the prevailing belief, reiterated over the last thirty years, regarding the efficiency of this method. Our approach is based on the concept of chain decomposition. Before we delve into its description, we introduce, analyze, and utilize a chain decomposition algorithm. Furthermore, we explore how chain decomposition can facilitate transitive closure solutions introducing a general purpose linear time reduction technique that removes a large subset of transitive edges given any chain decomposition.
Paper Structure (13 sections, 9 theorems, 11 figures, 4 tables, 5 algorithms)

This paper contains 13 sections, 9 theorems, 11 figures, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Width of a DAG and Decomposition into Paths/Chains
  3. Path/Chain Concatenation
  4. A Better Chain Decomposition Heuristic
  5. DAG Decomposition: Experimental Results
  6. DAG Reduction for Faster Transitivity
  7. Reachability Indexing Scheme
  8. The Indexing Scheme
  9. Sorting the Adjacency lists
  10. Creating the Indexing Scheme
  11. Experimental Results
  12. Speeding up the Method of Fulkerson
  13. Conclusions and Extensions

Key Result

Theorem 1

The time complexity of Algorithm alg:conc is $O({\mid}E{\mid}+(k_p-k_c)*l)$, where $l$ is the length of a longest path in $G$.

Figures (11)

  • Figure 1: Path and chain decomposition of an example graph.
  • Figure 2: The width curve on graphs of 5000 and 10000 nodes using three different models.
  • Figure 3: A comparison of width between ER model and PB model.
  • Figure 4: The efficiency of the chain decomposition algorithm in the BA model.
  • Figure 5: The efficiency of our chain decomposition algorithm in the ER model.
  • ...and 6 more figures

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Theorem 7
  • ...and 3 more