A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs

We consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. For an $n\times n$ 0-1 matrix $C,$ let $K_{C}$ be the complete weighted graph on the rows of $C$ where the weight of an edge between two rows is equal to their Hamming distance. Let $MWT(C)$ be the weight of a minimum weight spanning tree of $K_{C}.$ We show that the all-pairs shortest path problem for a directed graph $G$ on $n$ vertices with nonnegative real weights and adjacency matrix $A_G$ can be solved by a combinatorial randomized algorithm in time $$\widetilde{O}(n^{2}\sqrt {n + \min\{MWT(A_G), MWT(A_G^t)\}})$$ As a corollary, we conclude that the transitive closure of a directed graph $G$ can be computed by a combinatorial randomized algorithm in the aforementioned time. $\widetilde{O}(n^{2}\sqrt {n + \min\{MWT(A_G), MWT(A_G^t)\}})$ We also conclude that the all-pairs shortest path problem for uniform disk graphs, with nonnegative real vertex weights, induced by point sets of bounded density within a unit square can be solved in time $\widetilde{O}(n^{2.75})$

Computing the Boolean product of two n\times n Boolean matrices using O(n^2) mechanical operation

We study the problem of determining the Boolean product of two n\times n Boolean matrices in an unconventional computational model allowing for mechanical operations. We show that O(n^2) operations are sufficient to compute the product in this model.Comment: 11 pages, 7 figure

Improved Lower Bounds for Testing Triangle-freeness in Boolean Functions via Fast Matrix Multiplication

Understanding the query complexity for testing linear-invariant properties has been a central open problem in the study of algebraic property testing. Triangle-freeness in Boolean functions is a simple property whose testing complexity is unknown. Three Boolean functions $f_1$, $f_2$ and $f_3: \mathbb{F}_2^k \to \{0, 1\}$ are said to be triangle free if there is no $x, y \in \mathbb{F}_2^k$ such that $f_1(x) = f_2(y) = f_3(x + y) = 1$. This property is known to be strongly testable (Green 2005), but the number of queries needed is upper-bounded only by a tower of twos whose height is polynomial in 1 / \epsislon, where \epsislon is the distance between the tested function triple and triangle-freeness, i.e., the minimum fraction of function values that need to be modified to make the triple triangle free. A lower bound of $(1 / \epsilon)^{2.423}$ for any one-sided tester was given by Bhattacharyya and Xie (2010). In this work we improve this bound to $(1 / \epsilon)^{6.619}$. Interestingly, we prove this by way of a combinatorial construction called \emph{uniquely solvable puzzles} that was at the heart of Coppersmith and Winograd's renowned matrix multiplication algorithm

Quantum Algorithms for Matrix Products over Semirings

In this paper we construct quantum algorithms for matrix products over several algebraic structures called semirings, including the (max,min)-matrix product, the distance matrix product and the Boolean matrix product. In particular, we obtain the following results. We construct a quantum algorithm computing the product of two n x n matrices over the (max,min) semiring with time complexity O(n^{2.473}). In comparison, the best known classical algorithm for the same problem, by Duan and Pettie, has complexity O(n^{2.687}). As an application, we obtain a O(n^{2.473})-time quantum algorithm for computing the all-pairs bottleneck paths of a graph with n vertices, while classically the best upper bound for this task is O(n^{2.687}), again by Duan and Pettie. We construct a quantum algorithm computing the L most significant bits of each entry of the distance product of two n x n matrices in time O(2^{0.64L} n^{2.46}). In comparison, prior to the present work, the best known classical algorithm for the same problem, by Vassilevska and Williams and Yuster, had complexity O(2^{L}n^{2.69}). Our techniques lead to further improvements for classical algorithms as well, reducing the classical complexity to O(2^{0.96L}n^{2.69}), which gives a sublinear dependency on 2^L. The above two algorithms are the first quantum algorithms that perform better than the $\tilde O(n^{5/2})$-time straightforward quantum algorithm based on quantum search for matrix multiplication over these semirings. We also consider the Boolean semiring, and construct a quantum algorithm computing the product of two n x n Boolean matrices that outperforms the best known classical algorithms for sparse matrices. For instance, if the input matrices have O(n^{1.686...}) non-zero entries, then our algorithm has time complexity O(n^{2.277}), while the best classical algorithm has complexity O(n^{2.373}).Comment: 19 page

Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments

In this paper we present a quantum algorithm solving the triangle finding problem in unweighted graphs with query complexity $\tilde O(n^{5/4})$, where $n$ denotes the number of vertices in the graph. This improves the previous upper bound $O(n^{9/7})=O(n^{1.285...})$ recently obtained by Lee, Magniez and Santha. Our result shows, for the first time, that in the quantum query complexity setting unweighted triangle finding is easier than its edge-weighted version, since for finding an edge-weighted triangle Belovs and Rosmanis proved that any quantum algorithm requires $\Omega(n^{9/7}/\sqrt{\log n})$ queries. Our result also illustrates some limitations of the non-adaptive learning graph approach used to obtain the previous $O(n^{9/7})$ upper bound since, even over unweighted graphs, any quantum algorithm for triangle finding obtained using this approach requires $\Omega(n^{9/7}/\sqrt{\log n})$ queries as well. To bypass the obstacles characterized by these lower bounds, our quantum algorithm uses combinatorial ideas exploiting the graph-theoretic properties of triangle finding, which cannot be used when considering edge-weighted graphs or the non-adaptive learning graph approach.Comment: 17 pages, to appear in FOCS'14; v2: minor correction

Finding Even Cycles Faster via Capped k-Walks

In this paper, we consider the problem of finding a cycle of length $2k$ (a $C_{2k}$) in an undirected graph $G$ with $n$ nodes and $m$ edges for constant $k\ge2$. A classic result by Bondy and Simonovits [J.Comb.Th.'74] implies that if $m \ge100k n^{1+1/k}$, then $G$ contains a $C_{2k}$, further implying that one needs to consider only graphs with $m = O(n^{1+1/k})$. Previously the best known algorithms were an $O(n^2)$ algorithm due to Yuster and Zwick [J.Disc.Math'97] as well as a $O(m^{2-(1+\lceil k/2\rceil^{-1})/(k+1)})$ algorithm by Alon et al. [Algorithmica'97]. We present an algorithm that uses $O(m^{2k/(k+1)})$ time and finds a $C_{2k}$ if one exists. This bound is $O(n^2)$ exactly when $m=\Theta(n^{1+1/k})$. For $4$-cycles our new bound coincides with Alon et al., while for every $k>2$ our bound yields a polynomial improvement in $m$. Yuster and Zwick noted that it is "plausible to conjecture that $O(n^2)$ is the best possible bound in terms of $n$". We show "conditional optimality": if this hypothesis holds then our $O(m^{2k/(k+1)})$ algorithm is tight as well. Furthermore, a folklore reduction implies that no combinatorial algorithm can determine if a graph contains a $6$-cycle in time $O(m^{3/2-\epsilon})$ for any $\epsilon>0$ under the widely believed combinatorial BMM conjecture. Coupled with our main result, this gives tight bounds for finding $6$-cycles combinatorially and also separates the complexity of finding $4$- and $6$-cycles giving evidence that the exponent of $m$ in the running time should indeed increase with $k$. The key ingredient in our algorithm is a new notion of capped $k$-walks, which are walks of length $k$ that visit only nodes according to a fixed ordering. Our main technical contribution is an involved analysis proving several properties of such walks which may be of independent interest.Comment: To appear at STOC'1