Token-based Function Computation with Memory Algorithm Analysis

Table of Contents

1. Introduction

Distributed function computation serves as a fundamental building block in numerous network applications where computing a function of initial node values in a distributed manner is required. Traditional approaches based on spanning trees, while efficient in terms of message and time complexities, suffer from robustness issues in the presence of node failures or dynamic network topologies.

The Token-based function Computation with Memory (TCM) algorithm presents a novel approach that addresses these limitations through a token-based mechanism where node values attached to tokens travel across the network and coalesce when they meet, forming new token values through function application.

2. TCM Algorithm Design

The TCM algorithm introduces an innovative approach to distributed function computation that improves upon traditional Coalescing Random Walk (CRW) methods through strategic token movement and memory utilization.

2.1 Token Movement Mechanism

In TCM, each token carries both a value and memory of its computation history. Unlike random walk approaches, token movement is directed toward optimizing meeting opportunities. The algorithm ensures that when two tokens meet, they coalesce into a single token with a new value computed as $g(v_i, v_j)$, where $g$ is the rule function specific to the target computation.

2.2 Chasing Mechanism

The core innovation of TCM is its chasing mechanism, where tokens actively seek each other rather than moving randomly. This strategic movement pattern significantly reduces the expected meeting time compared to conventional random walk approaches, particularly in structured networks.

3. Mathematical Framework

The TCM algorithm operates within a rigorous mathematical framework that ensures correctness and enables complexity analysis.

3.1 Rule Function Definition

The rule function $g(.,.)$ must satisfy specific properties to ensure correct distributed computation. For a target function $f_n(v_1^0, \cdots, v_n^0)$, the rule function must be:

• Commutative: $g(v_i, v_j) = g(v_j, v_i)$
• Associative: $g(g(v_i, v_j), v_k) = g(v_i, g(v_j, v_k))$
• Identity element existence: $\exists e$ such that $g(v, e) = g(e, v) = v$

3.2 Complexity Analysis

The time complexity improvement of TCM over CRW is substantial across different network topologies:

• Erdős-Rényi and complete graphs: $O(\frac{\sqrt{n}}{\log n})$ improvement factor
• Torus networks: $O(\frac{\log n}{\log \log n})$ improvement factor

The message complexity shows at least constant factor improvement across all tested topologies, making TCM more efficient in both time and communication overhead.

4. Experimental Results

Extensive simulations demonstrate the performance advantages of TCM across various network configurations and scales.

4.1 Time Complexity Comparison

Experimental results show that TCM achieves significant reduction in convergence time compared to CRW. In Erdős-Rényi graphs with 1000 nodes, TCM reduces convergence time by approximately 40% while maintaining the same accuracy guarantees.

4.2 Message Complexity Analysis

The message complexity of TCM shows consistent improvement over CRW, with reductions ranging from 15% to 30% depending on network density and topology. This improvement stems from the reduced number of token movements required due to the chasing mechanism.

5. Implementation Details

The practical implementation of TCM requires careful consideration of token management and failure handling mechanisms.

5.1 Pseudocode Implementation

class TCMNode:
    def __init__(self, node_id, initial_value):
        self.id = node_id
        self.value = initial_value
        self.tokens = []
        self.neighbors = []
    
    def process_token(self, token):
        # Check for coalescing opportunities
        for local_token in self.tokens:
            if should_coalesce(token, local_token):
                new_value = rule_function(token.value, local_token.value)
                new_token = Token(new_value, merge_memory(token, local_token))
                self.tokens.remove(local_token)
                self.tokens.append(new_token)
                return
        
        # No coalescing, add token to collection
        self.tokens.append(token)
        
    def token_movement_decision(self):
        # Implement chasing mechanism
        target = find_chasing_target(self.tokens, self.neighbors)
        if target:
            move_token(self.tokens[0], target)

5.2 Node Failure Handling

The robustness of TCM in the presence of node failures is enhanced through parallel execution of multiple algorithm instances. This approach ensures that temporary node failures don't compromise the overall computation, with recovery mechanisms that reintegrate recovered nodes seamlessly.

6. Future Applications

The TCM algorithm has promising applications in several emerging domains:

• Edge Computing Networks: Efficient aggregation of sensor data in IoT deployments
• Federated Learning Systems: Distributed model parameter aggregation while preserving privacy
• Blockchain Networks: Consensus mechanism optimization through efficient value propagation
• Autonomous Vehicle Networks: Collaborative decision making through distributed computation

Future research directions include extending TCM to dynamic networks, investigating energy-efficient variants for battery-constrained devices, and developing security-enhanced versions resistant to malicious nodes.

7. References

1. Salehkaleybar, S., & Golestani, S. J. (2017). Token-based Function Computation with Memory. arXiv:1703.08831
2. Boyd, S., Ghosh, A., Prabhakar, B., & Shah, D. (2006). Randomized gossip algorithms. IEEE Transactions on Information Theory
3. Kempe, D., Dobra, A., & Gehrke, J. (2003). Gossip-based computation of aggregate information. FOCS
4. Dimakis, A. G., Kar, S., Moura, J. M., Rabbat, M. G., & Scaglione, A. (2010). Gossip algorithms for distributed signal processing. Proceedings of the IEEE
5. Shi, E., Chu, C., & Zhang, B. (2011). Distributed consensus and optimization in multi-agent networks. Foundations and Trends in Systems and Control