# Average Sensitivity of Dynamic Programming

arXiv: https://arxiv.org/abs/2111.02657 · HTML: https://mlsanity.com/paper/2111.02657/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Nov 2021.

## Built on by (most cited first, 2 of 2)

- [A Batch-to-Online Transformation under Random-Order Model](https://mlsanity.com/paper/2306.07163/) (2023)
- [From Average Sensitivity to Small-Loss Regret Bounds under Random-Order Model](https://mlsanity.com/paper/2602.09457/) (2026)

## Abstract (arXiv)

When processing data with uncertainty, it is desirable that the output of the algorithm is stable against small perturbations in the input. Varma and Yoshida [SODA'21] recently formalized this idea and proposed the notion of average sensitivity of algorithms, which is roughly speaking, the average Hamming distance between solutions for the original input and that obtained by deleting one element from the input, where the average is taken over the deleted element. In this work, we consider average sensitivity of algorithms for problems that can be solved by dynamic programming. We first present a (1-δ)-approximation algorithm for finding a maximum weight chain (MWC) in a transitive directed acyclic graph with average sensitivity O(δ-1log3 n), where n is the number of vertices in the graph. We then show algorithms with small average sensitivity for various dynamic programming problems by reducing them to the MWC problem while preserving average sensitivity, including the longest increasing subsequence problem, the interval scheduling problem, the longest common subsequence problem, the longest palindromic subsequence problem, the knapsack problem with integral weight, and the RNA folding problem. For the RNA folding problem, our reduction is highly nontrivial because a naive reduction generates an exponentially large graph, which only provides a trivial average sensitivity bound.