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Trading Determinism for Time: The k-Reach Problem

Ronak Bhadra, Raghunath Tewari

TL;DR

This paper investigates the trade-off between determinism and time in space-bounded computations by introducing the kReach problem, where a path from s to t in the union of f directed paths may switch among those paths at most k times. It demonstrates a deterministic algorithm that uses O(k log f + log n) space (not necessarily polytime) and an unambiguous nondeterministic algorithm with the same space bound that runs in polynomial time, thereby showing a time–determinism trade-off beyond Savitch’s framework. By reducing kReach to layered-digraph reachability, the work derives bounds in terms of both k and f, including a deterministic O(log n log k) space and an unambiguous O(log n sqrt(log k)) space, while also providing a combined O(k log f + log n) bound that is efficient for certain parameter regimes. The results illuminate when the joint consideration of k and f yields superior space-time performance, offering concrete parameter settings where the new bounds outperform existing ones and highlighting the role of unambiguous computation in achieving polytime performance with tight space use.

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.

Trading Determinism for Time: The k-Reach Problem

TL;DR

This paper investigates the trade-off between determinism and time in space-bounded computations by introducing the kReach problem, where a path from s to t in the union of f directed paths may switch among those paths at most k times. It demonstrates a deterministic algorithm that uses O(k log f + log n) space (not necessarily polytime) and an unambiguous nondeterministic algorithm with the same space bound that runs in polynomial time, thereby showing a time–determinism trade-off beyond Savitch’s framework. By reducing kReach to layered-digraph reachability, the work derives bounds in terms of both k and f, including a deterministic O(log n log k) space and an unambiguous O(log n sqrt(log k)) space, while also providing a combined O(k log f + log n) bound that is efficient for certain parameter regimes. The results illuminate when the joint consideration of k and f yields superior space-time performance, offering concrete parameter settings where the new bounds outperform existing ones and highlighting the role of unambiguous computation in achieving polytime performance with tight space use.

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.
Paper Structure (8 sections, 6 theorems, 3 algorithms)

This paper contains 8 sections, 6 theorems, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Upper Bound in terms of $f$
  4. Upper Bounds on in terms of $k$
  5. Upper Bounds on in terms of both $f$ and $k$ together
  6. Deterministic Algorithm for k-Reach
  7. Unambiguous Nondeterministic Algorithm for k-Reach
  8. Comparison among the Upper Bounds for $\kReach$

Key Result

Theorem 2.1

Given a collection of paths $P$ consisting of $f$ directed paths, $\kReach$ in $P$ can be decided in $O(f\log n)$ space and polynomial time.

Theorems & Definitions (11)

  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1
  • proof
  • Remark 1
  • Theorem 4.2
  • ...and 1 more