$\tilde{O}(1)$-Depth Parallel Reachability Faster than Transitive Closure

Abstract: A $d$-shortcut of a directed graph $G=(V,E)$ is a subset of edges drawn from the transitive closure $TC(G)$ whose addition reduces the graph diameter to at most $d$. In the special case $d=1$, computing a $1$-shortcut is \emph{equivalent} to computing the transitive closure. For larger values of $d$, a lower bound of [Hesse, SODA 2003] shows that $n<sup>δ$-shortcuts, for small constants $δ&gt;0$, may still contain a large fraction of the edges of $TC(G)$, suggesting that shortcut construction may remain as hard as transitive closure even in this regime. Consequently, since $\widetilde{O}(d)$-depth parallel reachability algorithms rely on computing $d$-shortcuts, achieving $\widetilde{O}(1)$ depth by this approach has so far required computing the full transitive closure. Assuming $ω=2$, the PS-AE-Triangle hypothesis of [Abboud, Bringmann, Fischer, and Künnemann, SODA 2024] yields a conditional $T<sup>{4/3-o(1)}$ time barrier for computing transitive closure when $T\leq n<sup>{3/2}$, where $T=|TC(G)|$. In this work, we bypass the transitive-closure barrier for $\widetilde{O}(1)$-depth parallel reachability. We introduce randomized $d$-shortcut constructions that already circumvent this barrier for $d=3$ and, more generally, for every even $d\geq4$ up to $O(\log n)$. Our approach yields a randomized $\widetilde{O}(1)$-depth parallel reachability algorithm with total work $\widetilde{O}(T<sup>{ω/2})$, which becomes $\widetilde{O}(T)$ when $ω=2$, falling below this conditional $T<sup>{4/3-o(1)}$ barrier throughout that regime. Under the current bound of $ω$, this gives $\widetilde{O}(T<sup>{1.186})$ work, improving on the current $T<sup>{1.3459+o(1)}$ sequential-time bound for transitive closure due to Abboud et al. Thus, although $\widetilde{O}(1)$-shortcuts might be almost as dense as the full transitive closure, they can nevertheless be computed substantially faster.

• The paper presents a parallel algorithm that solves Single-Source Reachability (SSR) with polylogarithmic depth and $\tilde{O}(T^{\omega/2})$ work, where $\omega$ is the matrix-multiplication exponent and $T = |TC(G)|$.
• The algorithm achieves this by using balanced TC-hitting sets and a hierarchy of even-$d$ shortcut constructions, avoiding the need to materialize the full transitive closure $TC(G)$.
• Under the current matrix-multiplication exponent bound ($\omega < 2.371339$), the work is $\tilde{O}(T^{1.186})$, improving on the best known sequential transitive-closure time of $T^{1.3459+o(1)}$ and restricting work to nearly linear.

Context and problem

The transitive closure bottleneck is a long-standing obstacle in parallel graph algorithms: solving single-source reachability (SSR) in polylogarithmic depth has conventionally required computing the full transitive closure $TC(G)$, typically via iterated matrix squaring. Thorup's notion of a $d$-shortcut — a subset of edges of $TC(G)$ whose addition reduces the diameter of $G$ to at most $d$ — was introduced precisely to circumvent this bottleneck: given a $d$-shortcut $H$, SSR can be solved with $\widetilde O(d)$ depth and $\widetilde O(m+|H|)$ work. However, Hesse's lower bound shows that even for diameter targets as large as $n^\delta$, every shortcut may contain nearly $d$0 edges, and Hesse himself remarked that this forces polylogarithmic-time algorithms to precompute essentially the entire transitive closure. Prior work on parallel reachability therefore focused on near-linear work with sublinear depth ($d$1 by Fineman, improved to $d$2 by Liu–Jambulapati–Sidford, and recently to $d$3 depth with $d$4 work on dense digraphs by Ashvinkumar et al.).

This paper, by Kogan and Parter (2608.13231), asks whether SSR can be solved in polylogarithmic depth with asymptotically less work than computing the full transitive closure. The answer is affirmative in the subquadratic-output regime, conditionally on standard fine-grained hypotheses.

Main results

Let $d$5 and let $d$6 denote the current matrix-multiplication exponent bound. The central theorem states:

There is a randomized parallel algorithm that solves SSR using $d$7 work and polylogarithmic depth.

When $d$8, the work becomes $d$9. This beats the conditional barrier $TC(G)$0 that Abboud, Bringmann, Fischer, and Künemann derive from their PS-AE-Triangle hypothesis (via the three-layer reduction from sparse Boolean matrix multiplication to transitive closure), throughout the regime $TC(G)$1; more generally, for every fixed $TC(G)$2 and $TC(G)$3, the barrier is at least $TC(G)$4, giving a polynomial separation from the paper's $TC(G)$5 work. Under the current bound on $TC(G)$6, the work is $TC(G)$7, improving on the best known sequential transitive-closure time of $TC(G)$8.

The result rests on a family of shortcut constructions:

• Small-diameter shortcuts: for every even $TC(G)$9, a randomized algorithm computes a $G$0-shortcut in $G$1 time, with a parallel implementation at polylogarithmic depth. A specialized construction handles $G$2 in $G$3 time ($G$4 when $G$5). No matching bound is currently obtained for odd $G$6, and the complexity of $G$7-shortcuts remains open.
• Large-diameter shortcuts: for every $G$8, there is a parallel algorithm computing a $G$9-shortcut with $d$0 work and $d$1 depth. With $d$2 this is $d$3 work. Setting $d$4 yields $d$5 work and $d$6 depth for dense graphs, breaking the previous $d$7 depth barrier — though the independent algorithm of Ashvinkumar et al. achieves the stronger $d$8 depth for dense graphs.

Separating shortcuts from transitive closure

A conceptual contribution is a combined hard instance showing that near-maximal shortcut size does not imply shortcut-construction hardness. For every $d$9 with $d$0, there is a family of $d$1-vertex digraphs with $d$2 such that every $d$3-shortcut contains $d$4 edges, yet — assuming $d$5 and the PS-AE-Triangle hypothesis — computing the full transitive closure requires $d$6 time while an $d$7-shortcut is computable in randomized $d$8 time. The construction splices together a Hesse shortcut-hard component, an ABFK transitive-closure-hard three-layer DAG, and a directed path fixing the closure size, joined only through outgoing edges from a new source so that neither hardness property is destroyed. This directly refutes the size-based intuition drawn from Hesse's remark: although polylogarithmic-diameter shortcuts may be almost as dense as the full transitive closure, they can be computed substantially faster.

Techniques

The algorithms are built around balanced TC-hitting sets: a set $d$9 that intersects every predecessor and successor list of TC-degree at least $H$0, while having total TC-degree $H$1. Such sets exist by a simple sampling argument (each vertex sampled with probability $H$2, with Markov and union bounds over candidates) and are computable in randomized near-linear time using Cohen's size-estimation framework, which approximates all TC-in/out-degrees within constant factors via $H$3 reachability computations.

The core primitive, TCLargeDeg, given a graph with maximum TC-indegree at most $H$4 and threshold $H$5, computes the induced transitive closure over all vertices of TC-outdegree at least $H$6 in time $H$7. It works by computing a hitting set $H$8, taking incoming BFS trees to obtain each predecessor set, and running naive matrix-multiplication-based transitive closure on each induced predecessor subgraph. A key structural fact is that the returned vertex set is predecessor-closed, so deleting it preserves distances among surviving vertices and never increases the diameter — this monotonicity underlies both correctness and the parallel implementation.

The even-$H$9 hierarchy performs $\widetilde O(d)$0 peeling steps with geometrically decreasing thresholds $\widetilde O(d)$1, alternating between the graph and its reversal so that each step bounds one side of the TC-degree; stitching edges handle the final boundary between degree classes. Each path decomposes into at most $\widetilde O(d)$2 blocks, each compressed to a single edge, yielding diameter at most $\widetilde O(d)$3. The per-step cost telescopes to $\widetilde O(d)$4.

The $\widetilde O(d)$5-depth parallel implementation first runs the sequential-style algorithm in $\widetilde O(d)$6 depth, then reduces depth by repeatedly applying the algorithm to $\widetilde O(d)$7-layered graphs: a $\widetilde O(d)$8-shortcut on the layered graph projects to a subset of $\widetilde O(d)$9 shortening every length-$\widetilde O(m+|H|)$0 path by one edge, so $\widetilde O(m+|H|)$1 iterations shrink the diameter geometrically to $\widetilde O(m+|H|)$2. Since the layered graph has diameter $\widetilde O(m+|H|)$3 and only $\widetilde O(m+|H|)$4 reachable pairs, each iteration costs the same work bound.

For large $\widetilde O(m+|H|)$5, two additional ideas are needed. First, since estimating global quantities or topological orderings would require SSR calls costing $\widetilde O(m+|H|)$6 depth, the algorithm restricts attention to radius-$\widetilde O(m+|H|)$7 balls, using truncated Cohen estimators and dynamically decreasing the ball radius across iterations to compensate for the loss of the monotonicity property that holds for unrestricted TC-degrees. Second, rather than computing topological orders, it uses parallel directed low-diameter decomposition (LDD): running LDD $\widetilde O(m+|H|)$8 times produces topologically ordered subgraphs that collectively preserve every length-$\widetilde O(m+|H|)$9 shortest path, after which a simple block-based construction computes a $n^\delta$0-shortcut per block in $n^\delta$1 time. A diameter-halving phase iterates over $n^\delta$2 degree thresholds, shortening prefixes and suffixes of every length-$n^\delta$3 shortest path until the residual path is shorter than $n^\delta$4; repeating the phase $n^\delta$5 times yields the $n^\delta$6-shortcut.

The $n^\delta$7 case uses a distinct argument: low-TC-degree vertices (below $n^\delta$8) are handled by adding their full predecessor/successor lists; the remaining $n^\delta$9 vertices pass through two peeling steps plus a bounded-degree closure, and a rectangular-multiplication subroutine (Shortcut2LenPaths) stitches the boundary between the outermost blocks, achieving overall $d$00 time.

Limitations and open questions

Several gaps remain explicit in the paper. The complexity of computing $d$01-shortcuts is unresolved, and no matching bound is given for odd $d$02 — the hierarchy covers only $d$03 and even $d$04. All algorithms are randomized; derandomization is not addressed. The work bound $d$05 matches the $d$06 ideal only if $d$07; under the current exponent the improvement over the ABFK barrier, while polynomial, is smaller. For dense graphs, the large-$d$08 tradeoff does not match the $d$09 depth achieved independently by Ashvinkumar et al., so the paper's advantage there is confined to the output-sensitive regime. Finally, the separation results are conditional on the PS-AE-Triangle hypothesis and on $d$10; unconditional separations between shortcut construction and transitive closure remain open.

Conclusion

This paper establishes that single-source reachability admits randomized polylogarithmic-depth parallel algorithms with $d$11 work — $d$12 when $d$13 — strictly below the conditional $d$14 transitive-closure barrier in the subquadratic-output regime. The key insight is that balanced TC-hitting sets allow iterative peeling of high-TC-degree vertices without ever materializing the full closure, and that layered-graph reductions convert shallow-diameter shortcut constructions into $d$15-depth algorithms. The accompanying hard-instance family demonstrates unconditionally that shortcut density does not dictate construction complexity, resolving the question posed by Hesse's lower bound in the negative for the computational (as opposed to combinatorial) sense.