We present a deterministic algorithm for 3-SAT running in $1.32736^n$ time. The algorithm is built on a branching-and-chain framework augmented with a polynomial-time expansion procedure that joins short chains through additional clauses to form bounded-size blocks, which impose stronger constraints on the assignment space. After branching, the algorithm chooses between derandomized local search guided by the block structure and enumeration of block assignments together with selected port variables, reducing the remaining formula to 2-SAT. A weighted analysis of branching, partial-assignment, and local-search costs, together with inductive bounds for chains and block interactions, establishes the running-time guarantee. The framework is self-contained and also offers guidance for practical SAT solver design by identifying useful clause-overlap structures and suggesting when to switch between local-search and reduction-based strategies.
We give a randomized ARBITRARY CRCW PRAM algorithm for computing a spanning forest of an \(n\)-vertex, \(m\)-edge graph. Let \(d\) be the maximum component diameter. With good probability, the algorithm runs in \(O(\log d+\log\log n)\) time using \(O(m+n)\) processors and outputs a rooted spanning forest of original edges with maximum depth \(2^{O(\log d+\log\log n)}\). For finding bridges, our algorithm identifies every bridge edge in \(O(\log d+\log\log n)\) time. For biconnectivity, it outputs all blocks in \(O(\log d+\log d'+\log\log n)\) time, where \(d'\) is the bi-diameter. All these problems are fundamental in graph algorithms with numerous applications. Our results improve the best previous time bounds parameterized by the number of vertices and diameter. Finally, we prove deterministic \(Ω(\log n/\log\log n)\) lower bounds for outputting a DFS Sequence, a weighted spanning forest, and an ear decomposition.