On Minimizing Wiggle in Stacked Area Charts

TL;DR

This paper analyzes the computational complexity of minimizing wiggle in stacked area charts, proving NP-hardness, and introduces an exact optimization approach while exploring related number ordering problems.

Contribution

It provides the first formal complexity analysis of wiggle minimization and compares an exact mixed-integer linear programming method with heuristics.

Findings

Wiggle minimization is NP-hard and hard to approximate.

An exact mixed-integer linear programming formulation is proposed.

Complexity results for related number ordering problems are established.

Abstract

Stacked area charts are a widely used visualization technique for numerical time series. The x-axis represents time, and the time series are displayed as horizontal, variable-height layers stacked on top of each other. The height of each layer corresponds to the time series values at each time point. The main aesthetic criterion for optimizing the readability of stacked area charts is the amount of vertical change of the borders between the time series in the visualization, called wiggle. While many heuristic algorithms have been developed to minimize wiggle, the computational complexity of minimizing wiggle has not been formally analyzed. In this paper, we show that different variants of wiggle minimization are NP-hard and even hard to approximate. We also present an exact mixed-integer linear programming formulation and compare its performance with a state-of-the-art heuristic in an experimental evaluation. Lastly, we consider a special case of wiggle minimization that corresponds to the fundamentally interesting and natural problem of ordering a set of numbers as to minimize their sum of absolute prefix sums. We show several complexity results for this problem that imply some of the mentioned hardness results for wiggle minimization.