Trading Determinism for Time: The k-Reach Problem

TL;DR

This paper explores a trade-off between determinism and time in space-bounded computations by introducing a new reachability problem involving paths with limited switches, demonstrating algorithms with different guarantees.

Contribution

It presents a novel reachability problem and shows how to solve it with algorithms that trade off between determinism, space, and time, expanding understanding of computational trade-offs.

Findings

An algorithm solving the problem in O(klog f+log n) space without guaranteed polynomial runtime.

An unambiguous nondeterministic algorithm solving the same problem in polynomial time and similar space.

The example illustrates a new perspective on the trade-off between determinism and time in space-bounded computation.

Abstract

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(klog 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(klog 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.