Faster all-pairs shortest paths via circuit complexity

We present a new randomized method for computing the min-plus product (a.k.a., tropical product) of two $n \times n$ 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 $$\frac{n^3}{2^{\Omega(\log n)^{1/2}}}$$ and is correct with high probability. On the word RAM, the algorithm runs in $n^3/2^{\Omega(\log n)^{1/2}} + n^{2+o(1)}\log M$ time for edge weights in $([0,M] \cap {\mathbb Z})\cup\{\infty\}$. Prior algorithms used either $n^3/(\log^c n)$ time for various $c \leq 2$, or $O(M^{\alpha}n^{\beta})$ time for various $\alpha > 0$ and $\beta > 2$. The new algorithm applies a tool from circuit complexity, namely the Razborov-Smolensky polynomials for approximately representing ${\sf AC}^0[p]$ circuits, to efficiently reduce a matrix product over the $(\min,+)$ algebra to a relatively small number of rectangular matrix products over ${\mathbb F}_2$, each of which are computable using a particularly efficient method due to Coppersmith. We also give a deterministic version of the algorithm running in $n^3/2^{\log^{\delta} n}$ time for some $\delta > 0$, which utilizes the Yao-Beigel-Tarui translation of ${\sf AC}^0[m]$ circuits into "nice" depth-two circuits.Comment: 24 pages. Updated version now has slightly faster running time. To appear in ACM Symposium on Theory of Computing (STOC), 201

Faster Shortest Paths in Dense Distance Graphs, with Applications

We show how to combine two techniques for efficiently computing shortest paths in directed planar graphs. The first is the linear-time shortest-path algorithm of Henzinger, Klein, Subramanian, and Rao [STOC'94]. The second is Fakcharoenphol and Rao's algorithm [FOCS'01] for emulating Dijkstra's algorithm on the dense distance graph (DDG). A DDG is defined for a decomposition of a planar graph $G$ into regions of at most $r$ vertices each, for some parameter $r < n$. The vertex set of the DDG is the set of $\Theta(n/\sqrt r)$ vertices of $G$ that belong to more than one region (boundary vertices). The DDG has $\Theta(n)$ arcs, such that distances in the DDG are equal to the distances in $G$. Fakcharoenphol and Rao's implementation of Dijkstra's algorithm on the DDG (nicknamed FR-Dijkstra) runs in $O(n\log(n) r^{-1/2} \log r)$ time, and is a key component in many state-of-the-art planar graph algorithms for shortest paths, minimum cuts, and maximum flows. By combining these two techniques we remove the $\log n$ dependency in the running time of the shortest-path algorithm, making it $O(n r^{-1/2} \log^2r)$. This work is part of a research agenda that aims to develop new techniques that would lead to faster, possibly linear-time, algorithms for problems such as minimum-cut, maximum-flow, and shortest paths with negative arc lengths. As immediate applications, we show how to compute maximum flow in directed weighted planar graphs in $O(n \log p)$ time, where $p$ is the minimum number of edges on any path from the source to the sink. We also show how to compute any part of the DDG that corresponds to a region with $r$ vertices and $k$ boundary vertices in $O(r \log k)$ time, which is faster than has been previously known for small values of $k$

A Decentralized Algorithm for Finding the Shortest Paths in Defense Communications Networks

This paper presents a decentralized shortest path algorithm which finds the shortest distances between all pairs of nodes without requiring that any particular node have information about the complete topology of the network. The algorithm requires at most ½N³ additions, ½N³ comparisons, and ½N³ transmissions of simple messages between individual nodes. The computational upper bound of the present algorithm is lower than that of Dijkstra’s centralized shortest path algorithm and is 1/N of the upper bound of Abram and Rhodes’ decentralized shortest path algorithm.Foundation Research Program of the Naval Postgraduate SchoolChief of Naval ResearchApproved for public release; distribution is unlimited

Energy-Efficient Algorithms

We initiate the systematic study of the energy complexity of algorithms (in addition to time and space complexity) based on Landauer's Principle in physics, which gives a lower bound on the amount of energy a system must dissipate if it destroys information. We propose energy-aware variations of three standard models of computation: circuit RAM, word RAM, and transdichotomous RAM. On top of these models, we build familiar high-level primitives such as control logic, memory allocation, and garbage collection with zero energy complexity and only constant-factor overheads in space and time complexity, enabling simple expression of energy-efficient algorithms. We analyze several classic algorithms in our models and develop low-energy variations: comparison sort, insertion sort, counting sort, breadth-first search, Bellman-Ford, Floyd-Warshall, matrix all-pairs shortest paths, AVL trees, binary heaps, and dynamic arrays. We explore the time/space/energy trade-off and develop several general techniques for analyzing algorithms and reducing their energy complexity. These results lay a theoretical foundation for a new field of semi-reversible computing and provide a new framework for the investigation of algorithms.Comment: 40 pages, 8 pdf figures, full version of work published in ITCS 201