Analysis of a Greedy Heuristic for the Labeling of a Map with a Time-Window Interface

TL;DR

This paper evaluates a greedy heuristic for map labeling with time windows, analyzing its approximation quality and providing bounds under various label shapes and weights, relevant for interactive map applications.

Contribution

It offers the first approximation guarantees for a greedy heuristic in time-window map labeling, including bounds for different label geometries and weight distributions.

Findings

Greedy heuristic guarantees at least 1/8 of optimal quality for square labels.

For disk-shaped labels, the heuristic achieves at least 1/10 of optimal quality.

Approximation ratio is Theta(log b) when label weights are bounded by a ratio b.

Abstract

In this paper, we analyze the approximation quality of a greedy heuristic for automatic map labeling. As input, we have a set of events, each associated with a label at a fixed position, a timestamp, and a weight. Let a time-window labeling be a selection of these labels such that all corresponding timestamps lie in a queried time window and no two labels overlap. A solution to the time-window labeling problem consists of a data structure that encodes a time-window labeling for each possible time window; when a user specifies a time window of interest using a slider interface, we query the data structure for the corresponding labeling. We define the quality of a time-window labeling solution as the sum of the weights of the labels in each time-window labeling, integrated over all time windows. We aim at maximizing the quality under the condition that a label may never disappear when the user shrinks the time window. In this paper, we analyze how well a greedy heuristic approximates the maximum quality that can be realized under this condition. On the one hand, we present an instance with square labels of equal size and equal weight for which the greedy heuristic fails to find a solution of at least 1/4 of the quality of an optimal solution. On the other hand, we prove that the greedy heuristic does guarantee a solution with at least 1/8 of the quality of an optimal solution. In the case of disk-shaped labels of equal size and equal weight, the greedy heuristic gives a solution with at least 1/10 of the quality of an optimal solution. If the labels are squares or disks of equal size and the maximum weight divided by the minimum weight is at most b, then the greedy heuristic has approximation ratio Theta(log b).