Papers › Faster all-pairs shortest paths via circuit complexity

Faster all-pairs shortest paths via circuit complexity

23 Dec 2013arXiv:1312.6680links table onlyarchive 2025-07-28

Ryan Williams

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We present a new randomized method for computing the min-plus product (a.k.a., tropical product) of two n ×n matrices, yielding a faster algorithm for solving the all-pairs shortest path problem (APSP) in dense n-node directed graphs with arbitrary edge weights. On the real RAM, where additions and comparisons of reals are unit cost (but all other operations have typical logarithmic cost), the algorithm runs in time n³/(2^(Ω(logn)^(1/2))) and is correct with high probability. On the word RAM, the algorithm runs in n³/2^(Ω(logn)^(1/2)) + n²⁺ᵒ⁽¹⁾logM time for edge weights in ([0,M] ∩ℤ)∪{∞}. Prior algorithms used either n³/(logᶜ n) time for various c ≤2, or O(M^αn^β) time for various α> 0 and β> 2. The new algorithm applies a tool from circuit complexity, namely the Razborov-Smolensky polynomials for approximately representing AC⁰[p] circuits, to efficiently reduce a matrix product over the (min,+) algebra to a relatively small number of rectangular matrix products over 𝔽₂, each of which are computable using a particularly efficient method due to Coppersmith. We also give a deterministic version of the algorithm running in n³/2^(log^δ n) time for some δ> 0, which utilizes the Yao-Beigel-Tarui translation of AC⁰[m] circuits into "nice" depth-two circuits.

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