From Circuit Complexity to Faster All-Pairs Shortest Paths
نویسندگان
چکیده
We present a randomized method for computing the min-plus product (a.k.a. tropical product) of two $n \times n$ matrices, yielding faster algorithm solving all-pairs shortest path problem (APSP) in dense $n$-node directed graphs with arbitrary edge weights. In real random-access machine model, where additions and comparisons reals are unit cost (but all other operations have logarithmic cost), runs time $\frac{n^3}{2^{\Omega(\log n)^{1/2}}}$ is correct high probability. On word which permits constant-time on $\log(n)$-bit words, $n^3/2^{\Omega(\log n)^{1/2}} + n^{2+o(1)}\log(nM)$ weights $([0,M] \cap \mathbb{Z})\cup\{\infty\}$. Prior algorithms needed either $\Theta(n^3/\log^c n)$ various $c \leq 2$, or $\Theta(M^{\alpha}n^{\beta})$ $\alpha > 0$ $\beta 2$. Our applies tool from circuit complexity, namely, Razborov--Smolensky polynomials approximately representing ${AC}^0[p]$ circuits, to efficiently reduce matrix over algebra relatively small number rectangular products $\mathbb{F}_2$. Each can be computed using particularly efficient due Coppersmith. also give deterministic version running $n^3/2^{\log^{\delta} n}$ some $\delta 0$, utilizes Yao--Beigel--Tarui translation ${AC}^0[m]$ circuits into “nice” depth-two circuits.
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ژورنال
عنوان ژورنال: Siam Review
سال: 2021
ISSN: ['1095-7200', '0036-1445']
DOI: https://doi.org/10.1137/21m1418654