Consequences of APSP, triangle detection, and 3SUM hardness for separation between determinism and non-determinism

Let NDTIME(f(n),g(n)) denote the class of problems solvable in O(g(n)) time by a multi-tape Turing machine using an f(n)-bit non-deterministic oracle, and let DTIME(g(n)) = NDTIME(0, g(n)). We show that if all-pairs shortest paths problem (APSP) for directed graphs with N vertices integer edge weights within super-exponential range { −2Nk+o(1),....,2Nk+o(1) }, k≥1 does not admit truly subcubic algorithm then any ∈>0, NDTIME([ 1/2 log2 n ], n)⊆DTIME(n1+12+k−∈). If APSP already when are moderate size we obtain even stronger implication, namely n)⊆DTIME(n1.5−∈). Similarly, triangle detection (DT) graph on sub-Nω -time n)⊆DTIME(nw/2−∈), where ω stands exponent fast matrix multiplication. For more general detecting minimum weight ℓ-clique (MWCℓ) size, non-existence sub−Nℓ−time yields NDTIME((ℓ−2)[ 12 log2n ],n)⊆DTIME(n1+ℓ−22−∈). Next, 3SUM integers −2Nk+o(1),....2Nk+o(1) } some k≥0, subquadratic ],n)⊆DTIME(n1+11+k−∈). Finally, observe Exponential Time Hypothesis (ETH) implies k ],n)⊆DTIME(n) k>0, while strong ETH (SETH) ],n)⊆DTIME(n2−ε). comparison, strongest known result separation between deterministic only asserts NDTIME(O(n),n)⊆DTIME(n).