Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs

Abstract: We give the first polynomial improvements over the textbook algorithms for $3$SUM and All-Pairs Shortest Paths (APSP): we show how to deterministically solve $3$SUM on $n$ integers of polynomial size in $O(n<sup>{1.9992})$ time and APSP on directed $n$-vertex graphs with polynomially bounded integer weights in $O(n<sup>{2.9995})$ time. This refutes the $3$SUM and APSP hypotheses. Using known reductions, we also refute the real-valued versions of the $3$SUM and APSP hypotheses, the Exact Triangle hypothesis, the Zero-Weight $k$-Clique hypotheses, and the three rectangular hinted Online Matrix--Vector conjectures of van den Brand, Nanongkai, and Saranurak, and we give polynomial speedups for a variety of other problems. All of these results follow from a single new algorithm for thin matrix products. Let $X$ be an $N\times D$ integer matrix and $Y$ a $D\times N$ integer matrix with $D\le N<sup>{1/18}$, and let $W$ be any set of at most $N^2/\sqrt D$ positions. We compute the entries $(XY)[I,J]$, $(I,J)\in W$, in $O(N<sup>2/D<sup>{0.063})$ operations, which is polynomially less than the time needed to write down $XY$ or to compute $N^2/\sqrt D$ inner products one by one. We design this algorithm by modifying a variant of Coppersmith's rectangular matrix multiplication algorithm, built from a ten-multiplication identity of Schönhage, to perform only the operations needed for the entries in $W$, and show that few operations are needed. Interpreted as a graph algorithm, this solves the All-Edges Sparse Triangle problem in truly subquadratic time on sparse lopsided tripartite graphs where two parts have $n$ vertices but one part has $n<sup>{\varepsilon}$ vertices for $\varepsilon&lt;0.12$. By known reductions, Exact Triangle, and hence $3$SUM and APSP, reduce to this problem. We also give a data structure version that answers queries for single entries of $XY$, not known in advance.

• The paper introduces a deterministic algorithm for computing sparse entries of a thin matrix product X * Y with D=O(N^ε), achieving O(N^2/log²D) complexity.
• It enables truly subquadratic computation for All-Edges Sparse Triangle, leveraging Schönhage's ten-multiplication identity to prune redundant computations.
• This breakthrough offers polynomial improvements for integer-based 3SUM, APSP, and Exact Triangle, with exponents O(n^1.9992) and O(n^2.998), respectively.

Problem setting and principal claims

The paper identifies a new algorithmic regime for sparse outputs of highly rectangular matrix products and uses it to obtain polynomial improvements for several canonical fine-grained problems. Its central result is a deterministic algorithm for computing a prescribed sparse set of entries of a thin product $XY$, where $X$ is $N \times D$, $Y$ is $D \times N$, and $D$ is a small polynomial in $N$. In the principal parameter setting, if $N \ge D^{18}$ and the wanted set $W$ contains at most $N^2/\sqrt D$ positions, then the entries $X$0 for $X$1 can be computed in

$X$2

operations. A stronger data-structure formulation preprocesses $X$3 and $X$4 in $X$5 time and answers any individual entry query in $X$6 time. More generally, the method applies whenever $X$7 for every $X$8, with a polynomial saving depending on the density of the queried positions and the desired query time (2610.06783).

The immediate graph interpretation is a lopsided All-Edges Sparse Triangle algorithm. In a tripartite graph with parts $X$9 and $N \times D$0 of size $N \times D$1, a middle part of size at most $N \times D$2, and a prescribed set $N \times D$3 of query pairs, the algorithm counts or detects common neighbors in the middle part for all pairs in $N \times D$4. In the regime $N \times D$5 and $N \times D$6, this is truly subquadratic. This is significant because the balanced All-Edges Sparse Triangle problem has long served as a central hardness source for $N \times D$7SUM, APSP, Set Disjointness, and related problems, while the lopsided instances produced by several established reductions had not been exploited algorithmically.

Composing the new triangle algorithm with known reductions yields the headline bounds:

Problem	Input restriction	Running time
$N \times D$8SUM	Polynomially bounded integers	$N \times D$9
APSP	Polynomially bounded integer weights	$Y$0
$Y$1-product	Polynomially bounded integer entries	$Y$2
Exact Triangle	Polynomially bounded integer weights	$Y$3
Real-valued $Y$4SUM	Comparisons, additions, subtractions	$Y$5 expected
Real-valued APSP	Comparisons, additions, subtractions	$Y$6 expected

The exponents in the introduction are stated with slightly different rounded versions depending on whether the basic sparse-product theorem or its stronger data-structure refinement is used. The improvements are small in absolute exponent but polynomial rather than subpolynomial, thereby contradicting the standard $Y$7SUM, APSP, and Exact Triangle hypotheses in their stated integer-input models.

The sparse thin-product theorem

The technical core is a selective matrix-multiplication algorithm. It does not assume structure in the entries of $Y$8 and $Y$9, nor in the set $D \times N$0 beyond its cardinality. This distinguishes the result from conventional fast matrix multiplication, which computes every entry of $D \times N$1 and therefore incurs an $D \times N$2 output cost.

The construction adapts the second, tensor-based algorithm of Coppersmith for rectangular matrix multiplication. Its algebraic foundation is a ten-multiplication identity due to Schönhage. The identity simultaneously represents:

1. an outer product of two length-three vectors, requiring nine scalar products; and
2. an inner product of two length-four vectors, requiring four scalar products.

The identity expresses these thirteen products through ten bilinear multiplications, at the cost of additional error terms. Crucially, its variables divide into “outer” variables associated with the nine-term product and “inner” variables associated with the four-term product. The error terms contain an outer output variable together with at least one inner input variable. This separation makes the error terms harmless when the recursion is restricted to the particular output strings used to encode thin matrix products.

Applying the identity recursively for $D \times N$3 levels produces a recursion tree with $D \times N$4 leaves. Each leaf corresponds to a product of two encoded input values. The recursion simultaneously computes many matrix products whose inner dimension is $D \times N$5, where $D \times N$6 is the number of levels at which the inner product is selected. For a fixed set $D \times N$7 of $D \times N$8 inner levels, the recursion represents a product of an $D \times N$9 matrix and a $D$0 matrix, with $D$1. The $D$2 possible choices of $D$3 are packed into one recursive computation.

The authors then tile the target $D$4 product into products of shape $D$5. A tile contains a grid of block products, each assigned to a distinct inner-level set $D$6. This arrangement is important because the input array for a tile depends on one row band of $D$7 and one column band of $D$8. The corresponding encodings can therefore be computed once per band and reused across many tiles.

The decisive modification is to separate encoding, recursive multiplication, and decoding. Encoding computes, for each input array, all leaf values $D$9 or $N$0. Once these values are available, the algorithm can prune the recursive decoding computation: a recursive branch is visited only if at least one wanted output depends on it. The resulting cost is controlled not by the number of wanted entries multiplied by the number of leaves per entry, but by the size of the union of the leaf sets contributing to all wanted entries.

Leaf sharing and the source of the polynomial saving

The leaf-counting argument is the most distinctive combinatorial component of the paper. An output string with exactly $N$1 inner positions depends on $N$2 leaves. Computing each output independently would therefore cost roughly $N$3, while direct inner-product evaluation would cost $N$4. Neither bound yields the desired improvement.

The authors classify a leaf by its order. A leaf has order $N$5 if it chooses a non-special term rather than $N$6 at exactly $N$7 of the $N$8 inner levels. The private leaf of an output chooses $N$9 at all inner levels and has order zero. Leaves of higher order are shared by more output strings because replacing a $N \ge D^{18}$0 by one of the nine other terms makes the leaf compatible with multiple choices of the inner-level set.

For a fixed output, the number of order-$N \ge D^{18}$1 contributing leaves is

$N \ge D^{18}$2

The total number of order-$N \ge D^{18}$3 leaves in the recursion is

$N \ge D^{18}$4

With $N \ge D^{18}$5, the global number of leaves decays geometrically:

$N \ge D^{18}$6

Thus, low-order leaves are few per output but potentially numerous globally, whereas high-order leaves are numerous per output but sparse globally and heavily shared. Splitting at order $N \ge D^{18}$7 gives

$N \ge D^{18}$8

where $N \ge D^{18}$9|U|=M/\sqrt D$W$0O(M/D^{1/18}).</sup>Thealgorithmconsequentlyevaluatesfewerthanoneleafperoutputentryofthefullproduct,despitecomputingarbitraryintegermatrixproducts.</p><p>Thisisthemechanismbehindthecentralnumericalsaving.Itisnotaconsequenceofafasterboundonordinarymatrixmultiplication;rather,itexploitstheoverlappatternbetweentheleafsetsassociatedwithdifferentoutputpositions.Thepaper’sexplicittheoremthereforedemonstratesaformofoutput−sensitivebilinearcomputationthatisunavailablefromstandardrectangularmatrixmultiplicationboundsalone.</p><h2class=′paper−heading′id=′data−structures−and−parameter−trade−offs′>Datastructuresandparametertrade−offs</h2><p>Thepaperextendsthebatchalgorithmtoonlinequeriesforentriesof.</sup> The algorithm consequently evaluates fewer than one leaf per output entry of the full product, despite computing arbitrary integer matrix products.</p> <p>This is the mechanism behind the central numerical saving. It is not a consequence of a faster bound on ordinary matrix multiplication; rather, it exploits the overlap pattern between the leaf sets associated with different output positions. The paper’s explicit theorem therefore demonstrates a form of output-sensitive bilinear computation that is unavailable from standard rectangular matrix multiplication bounds alone.</p> <h2 class='paper-heading' id='data-structures-and-parameter-trade-offs'>Data structures and parameter trade-offs</h2> <p>The paper extends the batch algorithm to online queries for entries of W$1 whose positions are not known during preprocessing. This requires replacing query-dependent pruning with query-independent aggregation.

The preprocessing partitions the leaves into two classes. Leaves below a switching order $W$2 are read individually during a query. Leaves of order at least $W$3 are grouped into “boxes,” each a structured subcube of the recursion tree. The value of a box is the sum of the products at all leaves in that subcube. Boxes are shared among many output entries and are computed by dynamic programming: a box with one free position is obtained by summing ten boxes in which that position is fixed.

For an output entry, the high-order leaves are partitioned into exactly $W$4 boxes, while the number of boxes stored globally is controlled by the decay of $W$5. This yields a preprocessing/query trade-off. If $W$6 and $W$7, then the query exponent and preprocessing saving depend on

$W$8

and on the decay ratio $W$9H) is the binary entropy function. The thinness threshold approaches

$N^2/\sqrt D$0

as $N^2/\sqrt D$$1D\le N^{{0.1204-o(1)}).}</p> <p>One explicit setting,$$N^2/\sqrt D$2 and $N^2/\sqrt D$3, yields preprocessing time $N^2/\sqrt D$4 and query time $N^2/\sqrt D$5. The same setting gives the main lopsided triangle bound

$N^2/\sqrt D$6

For $N^2/\sqrt D$7, the query term is dominated by the preprocessing term. The result is therefore simultaneously faster than evaluating each queried inner product directly and faster than computing the full product by known rectangular matrix multiplication whenever the middle dimension is sufficiently thin.

Reductions to Exact Triangle, $N^2/\sqrt D$8SUM, and APSP

The graph algorithm is applied through a deterministic reduction from Exact Triangle. The reduction hashes edge-weight sums modulo a carefully selected prime $N^2/\sqrt D$9 of size $X$00. The prime is chosen deterministically by counting false positives for every candidate prime and selecting one minimizing that count.

For each residue class of an edge weight and each piece of the third vertex part, the reduction constructs a lopsided triangle instance whose middle vertices are pairs consisting of a graph vertex and a residue label. A query pair has a common middle neighbor exactly when the corresponding triangle has total weight zero modulo $X$01. The number of instances is at most $X$02, each with at most $X$03 query pairs. The false positives are subsequently scanned directly, and their total number is bounded by an averaging argument over the candidate primes.

Balancing the number of oracle instances against the witness-scanning cost gives an Exact Triangle algorithm in $X$04 time, with $X$05O(n^{{3-1/648}\log²} n)$ bound. The deterministic nature of the prime selection and the reduction is essential: it establishes a deterministic refutation of the integer Exact Triangle hypothesis rather than merely a randomized one.

The reductions from $X$06SUM and APSP then transfer this improvement. For $X$07SUM, the known reduction creates approximately $X$08 Exact Triangle instances of size $X$09, yielding an exponent improvement of approximately half the Exact Triangle saving. The paper obtains

$X$10

For APSP and $X$11-product, the standard reduction invokes Exact Triangle at size $X$12, so only one third of the Exact Triangle exponent saving is retained. The resulting bound is

$X$13

reported in the abstract as $X$14. The same chain also gives truly subcubic algorithms for the broader APSP equivalence class, including Negative Triangle, Minimum Weight Cycle, Replacement Paths, Radius, Median, Tree Edit Distance, and several verification and witness-counting problems.

The reduction structure is algorithmically important. Many fine-grained reductions were developed to transfer conditional lower bounds and were not optimized for preserving exponent improvements. Here, the losses are explicit: the Exact Triangle reduction preserves only part of the lopsided-triangle saving, and the reductions from $X$15SUM and APSP preserve additional fractions. The paper therefore demonstrates both the utility and the current inefficiency of the existing reduction network.

Real inputs and related consequences

For real-valued inputs, integer hashing is unavailable. The paper instead composes the thin-product algorithm with the randomized reductions of Chan, Vassilevska Williams, and Xu, which use Fredman’s trick to transform comparisons between sums into comparisons between row-dependent and column-dependent differences. The required comparison counts are reduced to selected entries of a thin integer matrix product using a block decomposition associated with Matoušek’s dominance-product technique.

With $$d=n^{1/40}), the resulting algorithms use only comparisons, additions, and subtractions on real numbers and are Las Vegas:</p> <ul> <li>real$$X$16SUM in $X$17 expected time;

real Exact Triangle, $X$18-product, and APSP in $X$19 expected time.

The algorithms never return an incorrect answer; randomization affects only the running time. The paper also states high-probability bounds obtained by restarting after an excessive running time.

Further consequences include polynomial improvements for weighted zero-, min-, and max-weight $X$20-Clique through the classical Nešetřil–Poljak reduction. For constant $X$21$X$22$X$23\varepsilon_T&gt;0). The same framework gives a truly subquadratic algorithm for 3XOR with logarithmic dimension and refutes the thin-regime versions of three hinted Online Matrix–Vector conjectures. In particular, when the hint dimension is $X$24 with $X$25O(n<sup>{2-0.063\tau})$X$26O(n<sup>{1+0.437\tau})$ time. These bounds are faster than both obvious strategies—precomputing the full product and evaluating a query by a direct matrix-vector product.

The consequences for dynamic data-structure lower bounds are localized. The main hinted-OMv parameter regimes used for dynamic matrix inverse, determinant, matching, and related problems typically have much thicker matrices and are not refuted. What fails are the conjectural lower bounds at the thin end of their trade-offs, where the middle dimension is below the threshold supported by the new matrix algorithm.

Limitations and open questions

The result does not yield a faster algorithm for balanced All-Edges Sparse Triangle, whose conjectured $X$27 complexity remains unaffected. Nor does it directly improve problems known only to be 3SUM-hard or APSP-hard in the reverse direction, such as several geometric problems, dynamic graph problems, and some distance-oracle problems. In particular, the paper does not resolve whether three collinear points among $X$28 planar points can be found in truly subquadratic time.

The method is algebraic, has very large hidden constants, and is not presented as practical. Its applicability depends critically on the specific sparsity and sharing properties of Schönhage’s ten-multiplication identity. The authors do not establish that comparable pruning is possible for Coppersmith–Winograd identities or other tensor decompositions. The threshold $X$29 is therefore a limitation of the construction, not a lower bound on sparse thin matrix products.

Several hypotheses remain untouched, including SETH, Orthogonal Vectors, unhinted OMv, $X$30-SUM and $X$31-XOR for $X$32, and 3SUM-Indexing. The paper emphasizes that these problems lack the combination of reductions to sparse triangles, efficient decision-tree algorithms, and nondeterministic algorithms that characterizes $X$33SUM, Exact Triangle, and APSP. Whether the same algebraic technique can address any of these problems is left open.

The paper also reports that the discovery involved Claude, an AI model developed by Anthropic, while the authors supplied the mathematical analysis, simplification, parameter optimization, reductions, exposition, and formal verification described in the manuscript. The methodology is part of the paper’s provenance, but the mathematical claims stand or fall on the stated algorithms and proofs. The Lean formalization covers the principal integer-input theorems and selected corollaries; it does not eliminate the need to inspect the modeling assumptions, imported prior reductions, or the asymptotic interpretation of the computational model.

Conclusion

The paper’s main contribution is an output-sensitive algorithm for sparse entries of thin matrix products, obtained by pruning a recursive implementation of Schönhage’s identity and exploiting the sharing of high-order leaves across many outputs. This yields truly subquadratic lopsided sparse-triangle algorithms and, through established reductions, the first polynomial improvements over the textbook exponents for integer $X$34SUM and APSP. The resulting refutations are confined to specific fine-grained hypotheses and parameter regimes, but they demonstrate that reductions traditionally used to propagate hardness can also propagate a single algebraic speedup across a broad problem class. The central unresolved issue is whether the same approach can extend beyond lopsided instances and the $X$35 thin-product regime (2610.06783).