Average Sensitivity of Dynamic Programming

Abstract

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 −1 log 3 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.

Creating Autonomous Multi-Object Safe Control via Different Forms of Neural Constraints of Dynamic Programming

“…An important test for evaluating dynamic programming algorithms is the examination of their sensitivity to changes in various parameters of the control process [35][36][37].…”

“…Figure 9a shows the safe and optimal trajectory of object 0 in conditions of good visibility, and Figure 9b in conditions of limited visibility. An important test for evaluating dynamic programming algorithms is the examination of their sensitivity to changes in various parameters of the control process [35][36][37]. The measure of sensitivity s was the relative change in the optimal time t opt of the safe and optimal trajectory of object 0 caused by a change in the selected parameter of the control process p:…”