Given a graph G and two vertices s and t, the Long Path problem asks whether there exists a path of length at least k from s to t. For general graphs, this problem is NP-hard when k is part of the input. For directed acyclic graphs (DAGs), however, it is solvable in polynomial time and is NL-complete. A nondeterministic logspace computation is said to be unambiguous if, on every input, there is at most one accepting computation path. The class UL consists of all problems solvable by such machines, and whether NL = UL is an open question. In this work, we study the unambiguous complexity of the Long Path problem on DAGs under parameterization. Specifically, we consider the problem of deciding whether there exists a path of length at least n-k between two given vertices in a DAG. Bhadra and Tewari [Ronak Bhadra and Raghunath Tewari, 2025] showed that this problem can be solved in unambiguous and co-unambiguous O(klog n) space. We improve this result by giving an algorithm that runs in unambiguous O(k+log n) space. Additionally, we obtain an algorithm that achieves unambiguous and co-unambiguous O(klog n) space while running in time polynomial in both n and k, improving the previous O^*(n^k) time bound.

Kallampally and Tewari showed in 2016 that there can be a trade-off between determinism and time in space-bounded computations. This they did by describing an unambiguous non-deterministic algorithm to solve Directed Graph Reachability that requires O (log 2 n ) space and simultaneously runs in polynomial time. Savitch’s 1970 algorithm that solves the same problem deterministically also requires O (log 2 n ) space but doesn’t guarantee polynomial running time and hence the trade-off. We describe a new problem for which we can show a similar trade-off between determinism and time. We consider a collection P of f directed paths. We show that the problem of finding reachability from one vertex to another in the union G of these path graphs via a path that switches amongst the paths in P at most k times can be solved in O ( k log f + log n ) space but the algorithm doesn’t guarantee polynomial runtime. On the other hand, we also show that the same problem can be solved by an unambiguous non-deterministic algorithm that simultaneously runs in O ( k log f + log n ) space and polynomial time. Since these two algorithms are not dependent on Savitch, therefore this example sheds new light on how such a trade-off between determinism and time happens in space-bounded computations and makes the phenomenon less elusive.