Binhai Zhu

Jacob Sriraman, Eli Barton, Brittany Terese Fasy et al.

We consider the parametric shortest paths problem in a linearly interpolated graph. Given two positively-weighted directed graphs $G_0=(V,E,ω_0)$ and $G_1=(V,E,ω_1),$ the linearly interpolated graph is the family of graphs $(1-λ)G_0+λG_1$, parameterized by $λ\in [0,1]$. The problem is to compute all distinct parametric shortest paths. We compute a data structure in $Θ(k|E|\log |V|)$ time, where~$k$ is the number of distinct parametric shortest paths over all~$λ\in [0,1]$ that exist for a nontrivial interval of parameters, each corresponding to a linear function in a maximal sub-interval of $[0,1]$. Using this data structure, a shortest path query takes~$Θ(\log k)$ time.

Braeden Sopp, Adiesha Liyanage, Mingyang Gong et al.

Given \(k\) strings each of length at most $n$, computing the shortest common supersequence of them is a well-known NP-hard problem (when \(k\) is unbounded). On the other hand, when \(k=2\), such a shortest common supersequence can be computed in \(O(n^2)\) time using dynamic programming as a textbook example. In this paper, we consider the problem of computing a \emph{minimal} common supersequence and enumerating all minimal common supersequences for \(k=2\) input strings. Our results are summarized as follows. A minimal common supersequence of \(k=2\) input strings can be computed in $O(n)$ time. (The method also works when \(k\) is a constant). All minimal common supersequences between two input strings can be enumerated with a data structure of $O(n^2)$ space and an $O(n)$ time delay, and the data structure can be constructed in $O(n^3)$ time.