[Paper Review] Dijkstras algorithm with predictions to solve the single-source many-targets shortest-path problem

Willem Feijen, Guido Schäfer|arXiv (Cornell University)|Dec 22, 2021

Reliability and Maintenance Optimization4 citations

TL;DR

This paper proposes a learning-augmented Dijkstra's algorithm that uses machine learning predictions to prune the search space in the single-source many-targets shortest-path problem (SSMTSP). By predicting shortest path distances early, the algorithm reduces priority queue operations while guaranteeing optimality, achieving up to 35% faster performance than Dijkstra’s algorithm on favorable graph instances with binomial heaps.

ABSTRACT

We study the use of machine learning techniques to solve a fundamental shortest path problem, known as the single-source many-targets shortest path problem (SSMTSP). Given a directed graph with non-negative edge weights, our goal is to compute a shortest path from a given source node to any of several target nodes. Basically, our idea is to equip an adapted version of Dijkstras algorithm with machine learning predictions to solve this problem: Based on the trace of the algorithm, we design a neural network that predicts the shortest path distance after a few iterations. The prediction is then used to prune the search space explored by Dijkstras algorithm, which may significantly reduce the number of operations on the underlying priority queue. We note that our algorithm works independently of the specific method that is used to arrive at such predictions. Crucially, we require that our algorithm always computes an optimal solution (independently of the accuracy of the prediction) and provides a certificate of optimality. As we show, in the worst-case this might force our algorithm to use the same number of queue operations as Dijkstras algorithm, even if the prediction is correct. In general, however, our algorithm may save a significant fraction of the priority queue operations. We derive structural insights that allow us to lower bound these savings on partial random instances. In these instances, an adversary can fix the instance arbitrarily except for the weights of a subset of relevant edges, which are chosen randomly. Our bound shows that the number of relevant edges which are pruned increases as the prediction error decreases. We then use these insights to derive closed-form expressions of the expected number of saved queue operations on random instances.

Motivation & Objective

• To improve the efficiency of solving the single-source many-targets shortest-path problem (SSMTSP) using machine learning predictions.
• To design a modified Dijkstra’s algorithm that leverages predictions to prune the search space without sacrificing optimality.
• To ensure the algorithm remains robust even when predictions are inaccurate, maintaining worst-case performance guarantees.
• To derive theoretical bounds on the expected savings in priority queue operations based on prediction accuracy.
• To empirically validate the performance gains on random and structured graph instances.

Proposed method

• An adapted version of Dijkstra’s algorithm is used, where predictions guide the pruning of nodes not on the shortest path.
• A neural network is trained on algorithm traces to predict the shortest path distance after a few iterations.
• The algorithm maintains a reserve set for nodes not yet processed, updating their tentative distances via 'Decrease-Prio' operations.
• The method supports any prediction model, as long as predictions are used to avoid inserting nodes into the priority queue.
• The algorithm guarantees optimality by only pruning nodes that are provably not on the shortest path, based on the prediction and current distances.
• Theoretical analysis uses a 'fortunate graph' model with random edge weights on a subset of edges to derive lower bounds on pruning gains.

Experimental results

Research questions

• RQ1 Can machine learning predictions be used to reduce the number of priority queue operations in Dijkstra’s algorithm for SSMTSP without compromising optimality?
• RQ2 How do prediction errors affect the number of pruned edges and the resulting performance gain?
• RQ3 What is the theoretical lower bound on the expected number of saved queue operations under partial random edge weights?
• RQ4 How do different heap data structures (binomial vs. Fibonacci) influence the performance gains of the prediction-based algorithm?
• RQ5 To what extent do the actual runtime improvements match the theoretical expectations on random graph instances?

Key findings

• The proposed algorithm achieves up to 35% faster execution than Dijkstra’s algorithm on binomial heap implementations for graphs with 25% of nodes on the shortest path.
• On favorable instances with $ r = 0.35 $, the prediction-based algorithm reduced runtime to 4.82 seconds compared to 6.54 seconds for Dijkstra’s algorithm using a binomial heap.
• Theoretical analysis shows that the number of pruned relevant edges increases as prediction error decreases, with a closed-form expression derived for expected savings on random instances.
• Even with perfect predictions, the algorithm may still perform as many queue operations as Dijkstra’s in the worst case, but in practice, savings are substantial.
• The algorithm maintains optimality and provides a certificate of correctness regardless of prediction quality, ensuring robustness.
• Experimental results show that actual runtime improvements often exceed theoretical expectations, especially with binomial heaps and larger values of $ r $.