Dynamic Set Intersection

Consider the problem of maintaining a family $F$ of dynamic sets subject to insertions, deletions, and set-intersection reporting queries: given $S,S'\in F$, report every member of $S\cap S'$ in any order. We show that in the word RAM model, where $w$ is the word size, given a cap $d$ on the maximum size of any set, we can support set intersection queries in $O(\frac{d}{w/\log^2 w})$ expected time, and updates in $O(\log w)$ expected time. Using this algorithm we can list all $t$ triangles of a graph $G=(V,E)$ in $O(m+\frac{m\alpha}{w/\log^2 w} +t)$ expected time, where $m=|E|$ and $\alpha$ is the arboricity of $G$. This improves a 30-year old triangle enumeration algorithm of Chiba and Nishizeki running in $O(m \alpha)$ time. We provide an incremental data structure on $F$ that supports intersection {\em witness} queries, where we only need to find {\em one} $e\in S\cap S'$. Both queries and insertions take O\paren{\sqrt \frac{N}{w/\log^2 w}} expected time, where $N=\sum_{S\in F} |S|$. Finally, we provide time/space tradeoffs for the fully dynamic set intersection reporting problem. Using $M$ words of space, each update costs $O(\sqrt {M \log N})$ expected time, each reporting query costs $O(\frac{N\sqrt{\log N}}{\sqrt M}\sqrt{op+1})$ expected time where $op$ is the size of the output, and each witness query costs $O(\frac{N\sqrt{\log N}}{\sqrt M} + \log N)$ expected time.Comment: Accepted to WADS 201

Improving Function Coverage with Munch: A Hybrid Fuzzing and Directed Symbolic Execution Approach

Fuzzing and symbolic execution are popular techniques for finding vulnerabilities and generating test-cases for programs. Fuzzing, a blackbox method that mutates seed input values, is generally incapable of generating diverse inputs that exercise all paths in the program. Due to the path-explosion problem and dependence on SMT solvers, symbolic execution may also not achieve high path coverage. A hybrid technique involving fuzzing and symbolic execution may achieve better function coverage than fuzzing or symbolic execution alone. In this paper, we present Munch, an open source framework implementing two hybrid techniques based on fuzzing and symbolic execution. We empirically show using nine large open-source programs that overall, Munch achieves higher (in-depth) function coverage than symbolic execution or fuzzing alone. Using metrics based on total analyses time and number of queries issued to the SMT solver, we also show that Munch is more efficient at achieving better function coverage.Comment: To appear at 33rd ACM/SIGAPP Symposium On Applied Computing (SAC). To be held from 9th to 13th April, 201

Crystal constructions in Number Theory

Weyl group multiple Dirichlet series and metaplectic Whittaker functions can be described in terms of crystal graphs. We present crystals as parameterized by Littelmann patterns and we give a survey of purely combinatorial constructions of prime power coefficients of Weyl group multiple Dirichlet series and metaplectic Whittaker functions using the language of crystal graphs. We explore how the branching structure of crystals manifests in these constructions, and how it allows access to some intricate objects in number theory and related open questions using tools of algebraic combinatorics

Dynamic Maxflow via Dynamic Interior Point Methods

In this paper we provide an algorithm for maintaining a $(1-\epsilon)$-approximate maximum flow in a dynamic, capacitated graph undergoing edge additions. Over a sequence of $m$-additions to an $n$-node graph where every edge has capacity $O(\mathrm{poly}(m))$ our algorithm runs in time $\widehat{O}(m \sqrt{n} \cdot \epsilon^{-1})$. To obtain this result we design dynamic data structures for the more general problem of detecting when the value of the minimum cost circulation in a dynamic graph undergoing edge additions obtains value at most $F$ (exactly) for a given threshold $F$. Over a sequence $m$-additions to an $n$-node graph where every edge has capacity $O(\mathrm{poly}(m))$ and cost $O(\mathrm{poly}(m))$ we solve this thresholded minimum cost flow problem in $\widehat{O}(m \sqrt{n})$. Both of our algorithms succeed with high probability against an adaptive adversary. We obtain these results by dynamizing the recent interior point method used to obtain an almost linear time algorithm for minimum cost flow (Chen, Kyng, Liu, Peng, Probst Gutenberg, Sachdeva 2022), and introducing a new dynamic data structure for maintaining minimum ratio cycles in an undirected graph that succeeds with high probability against adaptive adversaries.Comment: 30 page

Investigation into Indexing XML Data Techniques

The rapid development of XML technology improves the WWW, since the XML data has many advantages and has become a common technology for transferring data cross the internet. Therefore, the objective of this research is to investigate and study the XML indexing techniques in terms of their structures. The main goal of this investigation is to identify the main limitations of these techniques and any other open issues. Furthermore, this research considers most common XML indexing techniques and performs a comparison between them. Subsequently, this work makes an argument to find out these limitations. To conclude, the main problem of all the XML indexing techniques is the trade-off between the size and the efficiency of the indexes. So, all the indexes become large in order to perform well, and none of them is suitable for all users’ requirements. However, each one of these techniques has some advantages in somehow