New Algorithms and Lower Bounds for Streaming Tournaments

TL;DR

This work studies streaming problems on directed graphs, focusing on tournaments where every pair of vertices is connected by a single directed edge. It provides a deterministic single-pass semi-streaming SCC-decomposition algorithm that works for tournaments and extends to graphs that are $k$-close to tournaments, achieving $\tilde{O}(n)$ or $\tilde{O}(n+k)$ space bounds, respectively, and yields improved upper bounds for reachability, strong connectivity, Hamiltonian cycle/path, and FAS. The paper also proves near-tight lower bounds, including $\Omega(n^2)$ space for exact FAST and $s,t$-distance in a single pass, and $\Omega(n/\sqrt{\varepsilon})$ space for $(1+\varepsilon)$-approximate FAST, along with $\Omega(n/p)$ space for reach-t and str-conn-t across $p$ passes; it further shows a smooth complexity transition as graphs move away from being tournaments. In addition to algorithmic advances, the work connects streaming complexity to communication complexity and reveals combinatorial graph-theoretic insights, such as indegree sequences determining SCCs and structural characterizations of reachability in tournaments. The results significantly advance understanding of streaming digraphs, offering new sublinear-space methods and tight bounds for a broad set of problems on tournaments and near-tournament graphs with practical implications for large directed networks.

Abstract

We study fundamental directed graph (digraph) problems in the streaming model. An initial investigation by Chakrabarti, Ghosh, McGregor, and Vorotnikova [SODA'20] on streaming digraphs showed that while most of these problems are provably hard in general, some of them become tractable when restricted to the well-studied class of tournament graphs where every pair of nodes shares exactly one directed edge. Thus, we focus on tournaments and improve the state of the art for multiple problems in terms of both upper and lower bounds. Our primary upper bound is a deterministic single-pass semi-streaming algorithm (using $\tilde{O}(n)$ space for $n$-node graphs, where $\tilde{O}(.)$ hides polylog$(n)$ factors) for decomposing a tournament into strongly connected components (SCC). it improves upon the previously best-known algorithm by Baweja, Jia, and Woodruff [ITCS'22] in terms of both space and passes: for $p\geq 1$, they used $(p+1)$-passes and $\tilde{O}(n^{1+1/p})$-space. We further extend our algorithm to digraphs that are close to tournaments and establish tight bounds demonstrating that the problem's complexity grows smoothly with the "distance" from tournaments. Applying our framework, we obtain improved tournament algorithms for $s,t$-reachability, strong connectivity, Hamiltonian paths and cycles, and feedback arc set. On the other hand, we prove the first $Ω(n^2)$-space lower bounds for this class, exhibiting that some well-studied problems -- such as (exact) feedback arc set on tournaments (FAST) and $s,t$-distance -- remain hard here. We obtain a generalized lower bound on space-approximation tradeoffs for FAST: any single-pass $(1\pm \varepsilon)$-approximation algorithm requires $Ω(n/\sqrt{\varepsilon})$ space. As a whole, our collection of results contributes significantly to the growing literature on streaming digraphs.

Figures (8)

• Figure 1: The graph constructed in the lower bound for reach-t for $N=4$ when Alice has the vector $\mathbf{x}=0110$ and Bob has the vector $\mathbf{y}=1010$. Edges that Alice (resp. Bob) adds are colored red (resp. blue), and input-independent edges are colored black. Other edges independent of the input have been excluded for clarity, and always go from right to left.
• Figure 2: The graph constructed in the lower bound for reach and str-conn where $x_{1, 1} = 1$ and $x_{1, 2} = 0$. Edges that Alice (resp. Bob) adds are colored red (resp. blue). The dotted lines represent the non-edges. All other edges have been excluded for clarity; edges incident on $s$ or $t$ go from right to left, and edges between $u_a$'s where $a \neq i$ are oriented arbitrarily, as are edges between $v_b$'s for $b \neq j$.
• Figure 3: The graph $G_\mathbf{x}$ constructed in the query lower bound for str-conn where bits 0, $k'+1$ and $n$ of the index $i$ are 0 and bit $k'$ of $i$ is 1. The dashed lines represent non-edges. Most edges are excluded for clarity, and go from left to right. Note that this is the construction for the $\Omega(2^n)$ case, but the $\Omega(2^k)$ case is very similar.
• Figure 4: The graph constructed in the lower bound for fas-t. Edges Alice adds are represented in red, and edges Bob adds are represented in blue. The '$?$' marks the edge Alice and Bob are trying to determine.
• Figure 5: The graph $G_1$ constructed in the lower bound for fas-size-t. Edges Alice adds are represented in red, and edges Bob adds are represented in blue. The '$?$' marks the edge Alice and Bob are trying to determine. Note that, in $G_2$, $u$ and $v$ are swapped.

Theorems & Definitions (70)

• Proposition 2.4
• proof
• Theorem 3.1
• Definition 3.2: SCC-cut
• Lemma 3.2
• proof
• Lemma 3.2
• proof
• Lemma 3.2
• proof