Succinct Graph Representations and Algorithmic Applications

Abstract: We propose new graph representations that exploit dense local structure to improve time and space simultaneously. Given an undirected graph $G$, we define a dual clique cover (DCC) representation of $G$ to be the pair $(C, L)$, where $C$ is a collection of cliques that covers the edges of $G$ and $L$ is the incidence dual of $C$. We identify classes of polynomial-time constructible DCC representations that are compact and call them succinct DCC representations. We then develop representation-aware algorithms for several fundamental graph problems. We show that graph primitives such as connected components, breadth-first search forests, depth-first search forests, and maximal matchings can be computed in time proportional to the size of a DCC representation rather than the number of edges. Combined with our succinct DCC representations, these results give a class of algorithms that either match or improve the time and space bounds of their counterparts on standard graph representations. Furthermore, we design several algorithms for constructing succinct DCC representations and establish provable guarantees on their efficiency. We evaluate several graph algorithms on DCC representations against adjacency-list-based implementations on a large collection of real-world and synthetic graphs. All evaluated applications show substantial execution memory savings and total-time speedups; for example, the connected components algorithm achieves about $9\times$ execution memory savings on average, with a maximum of $35\times$, and about $6.5\times$ total-time speedups on average, with a maximum of $35\times$. We also evaluate several DCC construction algorithms and find that the succinctness property plays a key role in making DCC representations effective for algorithmic applications.

• The paper introduces succinct DCC representations that compress dense subgraphs to improve both time and space complexities.
• It presents polynomial-time construction algorithms, including Succinct-Peeling, that yield near-optimal clique covers.
• Empirical results demonstrate significant runtime and memory savings, validating the compression-informed processing approach.

Succinct Graph Representations and Algorithmic Applications

Introduction and Problem Statement

Conventional graph representations such as adjacency lists and adjacency matrices treat all graph regions uniformly—representing both sparse and dense localities at the edge level. This uniformity impedes the exploitation of redundancy in locally dense graphs, often rendering joint improvements in time and space complexity elusive. The paper "Succinct Graph Representations and Algorithmic Applications" (2604.28096) introduces an alternative representation grounded in clique covers, specifically dual clique cover (DCC) representations, and demonstrates how algorithms designed atop these representations can achieve significant improvements in both computational and space requirements.

DCC Representations: Formulation and Minimality

A DCC representation for an undirected graph $G = (V, E)$ consists of a clique cover $\mathcal{C}$ (a set of cliques covering all edges) and its incidence dual $\mathcal{L}$ (which, for each $v \in V$, tracks the cliques it belongs to). The space requirement, termed $\text{size}(G)$, is the sum of vertex-clique assignments in $\mathcal{C}$ and $\mathcal{L}$.

Multiple minimality properties for clique covers are analyzed—such as inclusion-minimality, support-minimality, composition-minimality—as well as two NP-hard optimization objectives: minimizing the number of cliques (cardinality-optimal) and minimizing assignments (assignment-optimal). Key to practical DCC representations are succinct clique covers: composition-minimal covers with $O(m)$ total assignment size (i.e., $|\mathcal{C}| = O(m)$, $\text{size}(\mathcal{C}) = O(m)$), with further refinements via the notion of $\mathcal{C}$0-succinctness, where $\mathcal{C}$1 is the clique distance (number of non-edges plus one).

The representation design advances over prior clique cover approaches by innovating succinctness-enforcing constructions that not only retain minimality guarantees but are also polynomial-time constructible for broad graph classes.

Construction Algorithms and Complexity

Several algorithmic schemes for constructing succinct DCC representations are developed and rigorously analyzed:

• Succinct-Peeling: An efficient algorithm using degeneracy ordering and greedy coloring of the complement graph, with complexity $\mathcal{C}$2, where $\mathcal{C}$3 is graph degeneracy.
• Global-Admissibility and Local-Admissibility: Algorithms using admissibility sets and local neighborhood subgraphs, respectively, balancing construction speed and succinctness.

These strategies guarantee composition-minimality while keeping the representation size close to optimal relative to inherent clique and non-edge structure in the input.

Notably, the constructions enforce that each new clique or assignment corresponds directly to an as-yet-uncovered edge, guaranteeing injective assignment mapping and thereby providing strong worst-case energy bounds (see Theorem 2 from the paper).

Algorithmic Applications and Representation-Awareness

The authors develop DCC-aware algorithms for core graph primitives—connected components, BFS and DFS forests, maximal matchings, maximal independent sets, $\mathcal{C}$4-core decompositions, greedy/coloring, and maximal cliques. The distinguishing feature is that each algorithm's work scales with $\mathcal{C}$5 rather than $\mathcal{C}$6 or $\mathcal{C}$7, often yielding a runtime equal to or better than the best edge- or vertex-based implementations.

For instance:

• BFS/DFS: Combination of the clique cover $\mathcal{C}$8 and its dual $\mathcal{C}$9 supports efficient vertex discovery and adjacency queries, with BFS/DFS forests computed in $\mathcal{L}$0 time.
• Connected Components: Using the clique cover, a union-find data structure computes the components in $\mathcal{L}$1, where $\mathcal{L}$2 is the inverse Ackermann function.
• Maximal Matchings and Independent Sets: The algorithms only require the DCC clique cover and thus benefit directly from the compressed representation.
• Additional Problems: Diameter, radius, coloring, $\mathcal{L}$3-cores, and maximal clique computations are shown possible with running time proportional to graph parameters such as clique number $\mathcal{L}$4 and DCC representation size.

For all DCC-aware algorithms, the per-step cost is directly linked to DCC succinctness, enabling performance that rivals or surpasses adjacency-list-based routines, especially on locally dense graphs.

Empirical Evaluation

Comprehensive benchmarking was conducted on both real-world and synthetic graphs—covering large-scale sparse matrices, brain networks, and random/dense graph models. Key findings include:

• Memory and Storage Improvements: For tasks like connected components, DCC representations provided geometric mean memory savings of $\mathcal{L}$5 (up to $\mathcal{L}$6), with storage savings closely matching these factors.
• Runtime: DCC-based algorithms achieved $\mathcal{L}$7 median and up to $\mathcal{L}$8 maximal total time speedups compared to standard list-based implementations. Even though compute time alone could be higher for certain tasks with large clique numbers (due to $\mathcal{L}$9-factor overhead), the reduction in disk or representation read-time outweighed this for nearly all queries.
• Comparison to WebGraph: DCC applications outperformed WebGraph [boldi2004webgraph] in both run time (up to $v \in V$0) and memory (up to $v \in V$1 less) for fundamental algorithms such as matching and component analysis on large sparse and dense graphs.
• Construction Overhead: Imposing succinctness during clique cover construction—although incurring higher computational cost—yielded $v \in V$2 better compression compared to naive constructions.

Theoretical and Practical Implications

The introduction and thorough analysis of succinct DCC representations provide a new paradigm for handling graphs with dense local structure. This approach not only simplifies the joint optimization of time and space but also introduces a suite of algorithms that can natively exploit the redundancy present in real-world networks.

Major implications include:

• Algorithm-Representation Co-Design: The work demonstrates how representation-aware algorithms—designed with knowledge of the representation's structure—can surpass black-box or query-only algorithms in both asymptotic and practical performance.
• Compression-Informed Processing: DCCs provide storage formats that are self-indexing; that is, a single compact representation suffices for both storage, transmission, and direct algorithmic use without explicit decompression.
• Future Directions: The authors highlight open directions such as faster construction for dynamic/streaming scenarios, DCC-based parallelization, assignment-optimal clique covers and approximability, integration into mainstream graph repositories, and weighted/generalized graph algorithms.

Conclusion

This work establishes succinct DCC representations and their associated algorithmic ecosystem as competitive—if not superior—alternatives to standard graph representations for a variety of applications, particularly where local density can be harnessed for compression. By explicitly linking data representation with algorithmic pipeline design and providing both theoretical analysis and empirical evidence, the paper advances the state-of-the-art in compact graph processing and sets an agenda for future AI and graph theory research on data structure and algorithm co-design (2604.28096).