A Novel Skip Orthogonal List for Dynamic Optimal Transport Problem

TL;DR

The paper tackles the problem of updating discrete optimal transport plans under dynamic changes to weights and point positions. It introduces a novel 2D Skip Orthogonal List coupled with Euler Tour techniques to support efficient, localized cost updates and global minimum queries, enabling the dynamic OT problem to be solved via a sequence of simplex pivots in $O(|V|)$ time per iteration and $O(s|V|)$ overall for $s$ pivots. The key contributions are (i) the Skip Orthogonal List data structure with lazy propagation for range updates, (ii) a dynamic network simplex framework that reduces dynamic updates to pivot operations, and (iii) substantial empirical speedups over static competent methods like Network Simplex and Sinkhorn in large-scale dynamic OT settings. This work enables real-time or streaming OT updates in large datasets, with potential impact on dataset similarity, time-series analysis, and neuroimaging applications where rapid re-computation of transport costs is valuable.

Abstract

Optimal transport is a fundamental topic that has attracted a great amount of attention from the optimization community in the past decades. In this paper, we consider an interesting discrete dynamic optimal transport problem: can we efficiently update the optimal transport plan when the weights or the locations of the data points change? This problem is naturally motivated by several applications in machine learning. For example, we often need to compute the optimal transport cost between two different data sets; if some changes happen to a few data points, should we re-compute the high complexity cost function or update the cost by some efficient dynamic data structure? We are aware that several dynamic maximum flow algorithms have been proposed before, however, the research on dynamic minimum cost flow problem is still quite limited, to the best of our knowledge. We propose a novel 2D Skip Orthogonal List together with some dynamic tree techniques. Although our algorithm is based on the conventional simplex method, it can efficiently find the variable to pivot within expected $O(1)$ time, and complete each pivoting operation within expected $O(|V|)$ time where $V$ is the set of all supply and demand nodes. Since dynamic modifications typically do not introduce significant changes, our algorithm requires only a few simplex iterations in practice. So our algorithm is more efficient than re-computing the optimal transport cost that needs at least one traversal over all $|E| = O(|V|^2)$ variables, where $|E|$ denotes the number of edges in the network. Our experiments demonstrate that our algorithm significantly outperforms existing algorithms in the dynamic scenarios.

Figures (8)

• Figure 1: Overview of Euler Tour Tree with Skip Lists
• Figure 2: An Orthogonal List Example.
• Figure 3: An illustration for Skip Orthogonal List.
• Figure 4: Illustrations of Domination in Skip Lists
• Figure 5: Illustrations of a "cut" operation

Theorems & Definitions (17)

• definition thmcounterdefinition: Euler Tour Representation
• definition thmcounterdefinition
• definition thmcounterdefinition
• theorem thmcountertheorem
• theorem thmcountertheorem
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof