Mind the Gap

We examine the complexity of the online Dictionary Matching with One Gap Problem (DMOG) which is the following. Preprocess a dictionary $D$ of $d$ patterns, where each pattern contains a special gap symbol that can match any string, so that given a text that arrives online, a character at a time, we can report all of the patterns from $D$ that are suffixes of the text that has arrived so far, before the next character arrives. In more general versions the gap symbols are associated with bounds determining the possible lengths of matching strings. Finding efficient algorithmic solutions for (online) DMOG has proven to be a difficult algorithmic challenge. We demonstrate that the difficulty in obtaining efficient solutions for the DMOG problem even, in the offline setting, can be traced back to the infamous 3SUM conjecture. Interestingly, our reduction deviates from the known reduction paths that follow from 3SUM. In particular, most reductions from 3SUM go through the set-disjointness problem, which corresponds to the problem of preprocessing a graph to answer edge-triangles queries. We use a new path of reductions by considering the complementary, although structurally very different, vertex-triangles queries. Using this new path we show a conditional lower bound of $\Omega(\delta(G_D)+op)$ time per text character, where $G_D$ is a bipartite graph that captures the structure of $D$, $\delta(G_D)$ is the degeneracy of this graph, and $op$ is the output size. We also provide matching upper-bounds (up to sub-polynomial factors) for the vertex-triangles problem, and then extend these techniques to the online DMOG problem. In particular, we introduce algorithms whose time cost depends linearly on $\delta(G_D)$. Our algorithms make use of graph orientations, together with some additional techniques. Finally, when $\delta(G_D)$ is large we are able to obtain even more efficient solutions.