Randomized Catalytic Algorithm for s-t Connectivity

• The paper introduces a randomized catalytic algorithm that leverages a push-based approach and catalytic tape to efficiently test s-t connectivity.
• It employs random shifts, modular arithmetic, and reversible register updates to achieve space efficiency in Õ(n) catalytic space and Õ(nm) runtime.
• The work reveals that randomized and deterministic catalytic logspace classes collapse, offering significant implications for space-bounded graph algorithms.

A randomized catalytic algorithm for $s\to t$ connectivity is an algorithm that tests whether a path exists between two designated vertices $s$ and $t$ in a network, leveraging both randomization and the "catalytic" computation model. Catalytic computation augment the standard space-bounded Turing machine model with a large read/write tape (“catalytic tape”) that must retain its initial contents throughout the computation. Recent research has clarified the power and practical algorithmics of this model, showing that randomization within catalytic logspace yields efficient and implementable algorithms for fundamental graph connectivity problems (Cook et al., 7 Sep 2025), with complexity-theoretic consequences for the collapse of randomized/nondeterministic catalytic classes (Koucký et al., 11 Apr 2025).

1. Catalytic Computation and Complexity

Catalytic machines are space-bounded Turing machines equipped with a large auxiliary tape whose initial contents must be restored at the end. The main complexity-theoretic result is that deterministic, nondeterministic, and randomized catalytic logspace classes collapse: $\text{CL} = \text{CNL} = \text{CPrL}$ where $\text{CL}$ denotes deterministic catalytic logspace, $\text{CNL}$ non-deterministic catalytic logspace, and $\text{CPrL}$ the randomized variant with acceptance probability arbitrarily close to $1/2$ (Koucký et al., 11 Apr 2025). This indicates that randomness (and non-determinism) do not increase computational power in the presence of catalytic space.

Algorithmically, the compress-or-compute framework (Koucký et al., 11 Apr 2025) modularizes catalytic computations as follows:

• Traverse the configuration graph induced by the computation.
• If the graph is small, compress and solve directly.
• Otherwise, compress portions of the catalytic tape, freeing space iteratively until a brute-force or recursive computation becomes feasible.

The implication is that randomized catalytic algorithms for $s\to t$ connectivity—such as those running in logspace with randomness—can be deterministically simulated with only modest overhead.

2. Randomized Catalytic Algorithm for $s\to t$ Connectivity: Push-Based Approach

The randomized catalytic connectivity algorithm exploits the catalytic tape to maintain a large set of registers. The algorithm "lifts" the $n$-vertex graph $G$ into $n$ layers and defines, for each $(i, v)$ (layer/time, vertex), a register $R_{i,v}$ (Cook et al., 7 Sep 2025). The algorithm proceeds as follows:

• Initialization: For each vertex $v$ and time $i$, allocate a register $R_{i,v}$; the tape begins in an arbitrary state, and all registers are shifted by a random $\beta$ modulo a randomly chosen modulus $q$ (to ensure validity).
• Push Step: For consecutive layers $i=0, ..., n-1$ and every edge $(u, v)$, update

$$R_{i+1, v} \leftarrow R_{i+1, v} + R_{i,u} \pmod{q}$$

• Perturbation: Increment the register for the start vertex $s$ at layer 0 by 1.
• Termination and Detection: At layer $n$, the value at register $R_{n, t}$ differs from its initial value if and only if there is an $s \to t$ path (difference counts the number of paths modulo $q$).

Register sizes are $O(\log n)$ bits (versus $O(n)$ bits for the deterministic version), and the entire procedure uses only $O(n)$ catalytic space.

Pseudocode:

for i in range(n): for u in V: for (u, v) in E: R[i + 1, v] = (R[i + 1, v] + R[i, u]) % q

Detection: after all pushes,

if (R[n, t]_perturbed - R[n, t]_unperturbed) % q != 0: output "Path exists" else: output "No path"

Random choice of $q$ (polylogarithmic range) ensures that wraparound does not affect correctness with high probability.

3. Runtime, Space, and Comparison

Algorithm Type	Time Complexity	Catalytic Space Complexity	Register Size
Deterministic CL	$\widetilde{O}(n^3m)$	$\widetilde{O}(n^2)$	$\Omega(n)$ bits
Randomized Catalytic	$\widetilde{O}(nm)$	$\widetilde{O}(n)$	$O(\log n)$ bits

The randomized algorithm is the first to make explicit algorithmic use of randomization in the $\mathsf{CL}$ model (Cook et al., 7 Sep 2025). Compared to classical logspace approaches (e.g., polynomial time and logspace random walks (Li et al., 2022), or compress-or-compute (Koucký et al., 11 Apr 2025)), the randomized catalytic algorithm is a factor of $n$ slower than BFS with linear workspace but much faster than previous deterministic catalytic implementations.

4. Implementation Aspects and Tape Reversibility

The scheme achieves rapid local reversibility: registers can be restored to their initial values (arbitrary) by subtracting contributions from previous layers, since arithmetic is done modulo a small $q$. Organizing registers per layer supports reversibility at the cost of additional tape; grouping (e.g., odd/even layers) can cut space but complicates tape restoration.

Random shifting of initial values is essential—registers start in an arbitrary (unknown) state, and randomization ensures that the "push" procedure does not inadvertently nullify the connectivity signal.

5. Connection to Complexity Theory and Lower Bounds

Randomized catalytic algorithms do not exceed the power of deterministic ones ($\text{CL} = \text{CPrL}$) (Koucký et al., 11 Apr 2025). Thus, the randomized push-based algorithm is essentially optimal for the catalytic setting; further speedups would require fundamentally new breakthroughs, likely violating known circuit lower bounds for small-depth connectivity (Chen et al., 2015).

From a complexity perspective, s-t connectivity is NL-complete, and the catalytic technique simulates nondeterminism and randomness using a large pre-filled tape, aligning with compress-or-compute and derandomization frameworks (Koucký et al., 11 Apr 2025, Li et al., 2022).

6. Applications and Extensions

Potential uses of the randomized catalytic algorithm include:

• Memory-constrained or streaming scenarios where workspace is severely limited and only a large, arbitrarily-filled tape is available.
• Space-efficient verification of connectivity, with minimal arithmetic per vertex.
• Supporting efficient simulation of random walks and more general combinatorial processes.

The push-based algorithmic principle—updating state with local arithmetic and leveraging randomness for arithmetic efficiency—extends naturally to random walk simulation and may inform future developments in space-bounded algorithmics, derandomization, and graph computation over noisy memory (Cook et al., 7 Sep 2025).

7. Limitations and Practical Considerations

The main limitation is model non-standardness: the algorithm assumes the existence of a fully-filled, restorable catalytic tape. While theoretically compelling, practical implementation demands careful management of reversibility and register mapping. The randomized approach's correctness is probabilistic (one-sided error); amplification by repetition is possible but incurs further runtime.

In summary, randomized catalytic algorithms for $s\to t$ connectivity combine efficient modular computation, reversible tape manipulation, and explicit randomization, achieving theoretical optimality with $\widetilde{O}(nm)$ runtime and $\widetilde{O}(n)$ catalytic space (Cook et al., 7 Sep 2025, Koucký et al., 11 Apr 2025). These results clarify the power of randomization in the catalytic space model and provide a practical framework for efficient connectivity algorithms in space-limited settings.