Better Bounds for Semi-Streaming Single-Source Shortest Paths

TL;DR

This work advances the study of single-source shortest paths in the semi-streaming model by delivering a near-optimal two-fold advance: a simple, practical multi-pass, space-efficient algorithm achieving a (1+ε)-approximation for SSSP from a source, and a new lower bound proving that any constant-factor approximation requires at least Ω(log n / log log n) passes. The upper bound leverages a sample-and-solve approach with an O(log n)-spanner and a multiplicative weight update scheme across O(k^2/ε) rounds, yielding the semi-streaming bound when choosing k ≈ log n / log log n; the method is adaptable to dynamic streams and can be derandomized with poly-log factors. The lower bound introduces a correlated distribution for paired pointer chasing and uses information-theoretic techniques to show that even constant-factor approximations demand ω(1) passes, narrowing the gap between upper and lower bounds from polylog n to a quadratic gap in passes. Collectively, the results illuminate the pass complexity landscape for semi-streaming SSSP, present a conceptually accessible algorithm, and establish a robust lower bound framework based on pointer-chasing and information complexity that may inform future streaming lower bounds.

Abstract

In the semi-streaming model, an algorithm must process any $n$-vertex graph by making one or few passes over a stream of its edges, use $O(n \cdot \text{polylog }n)$ words of space, and at the end of the last pass, output a solution to the problem at hand. Approximating (single-source) shortest paths on undirected graphs is a longstanding open question in this model. In this work, we make progress on this question from both upper and lower bound fronts: We present a simple randomized algorithm that for any $ε> 0$, with high probability computes $(1+ε)$-approximate shortest paths from a given source vertex in \[ O\left(\frac{1}ε \cdot n \log^3 n \right)~\text{space} \quad \text{and} \quad O\left(\frac{1}ε \cdot \left(\frac{\log n}{\log\log n} \right) ^2\right) ~\text{passes}. \] The algorithm can also be derandomized and made to work on dynamic streams at a cost of some extra $\text{poly}(\log n, 1/ε)$ factors only in the space. Previously, the best known algorithms for this problem required $1/ε\cdot \log^{c}(n)$ passes, for an unspecified large constant $c$. We prove that any semi-streaming algorithm that with large constant probability outputs any constant approximation to shortest paths from a given source vertex (even to a single fixed target vertex and only the distance, not necessarily the path) requires \[ Ω\left(\frac{\log n}{\log\log n}\right) ~\text{passes}. \] We emphasize that our lower bound holds for any constant-factor approximation of shortest paths. Previously, only constant-pass lower bounds were known and only for small approximation ratios below two. Our results collectively reduce the gap in the pass complexity of approximating single-source shortest paths in the semi-streaming model from $\text{polylog } n$ vs $ω(1)$ to only a quadratic gap.

Figures (9)

• Figure 1: Pointer Chasing modeled as a layered graph. The path from vertex $u$ is shown using the wiggly line (brown).
• Figure 2: Equality of Pointer Chasing. When the final vertex is $x$ for both the instances, there is an $(s, t)$-path of length $2k+1$. Otherwise, the two vertices are not connected. The players are $P_1, P_2 \ldots, P_{2k}$, and which matching is given to each player is specified below the matching.
• Figure 3: Multiple equality of $\textnormal{PC}$ instances embedded in the same graph. Each color corresponds to one instance. There may be paths of multi-colored edges from $u$ and $v$ to the same vertex in the middle layer, so that the shortest path length between $u,v$ is $2k+1$, but the parameters are chosen so that this is unlikely.
• Figure 4: Three paths from three equality of $\textnormal{PC}$ instances, with different colors used to represent edges from different instances. Together, these three paths form an $(u, v)$-path of length $O(k)$.
• Figure 5: Paired pointer chasing instances. The player whom the matching is given to is specified below the edges. A reminder that here, Bob speaks first.

Theorems & Definitions (92)

• Proposition 3.2: FeigenbaumKMSZ05jin2024streaming
• Theorem 1
• Lemma 4.1: Sampling Lemma
• proof
• Claim 4.2
• proof
• Lemma 4.3
• proof
• Lemma 4.4
• proof