Abstract
A major factor affecting the clarity of graphical displays that include text labels is the degree to which labels obscure display features (including other labels) as a result of spatial overlap. Point-feature label placement (PFLP) is the problem of placing text labels adjacent to point features on a map or diagram so as to maximize legibility. This problem occurs frequently in the production of many types of informational graphics, though it arises most often in automated cartography. In this paper we present a comprehensive treatment of the PFLP problem, viewed as a type of combinatorial optimization problem. Complexity analysis reveals that the basic PFLP problem and most interesting variants of it are NP-hard. These negative results help inform a survey of previously reported algorithms for PFLP; not surprisingly, all such algorithms either have exponential time complexity or are incomplete. To solve the PFLP problem in practice, then, we must rely on good heuristic methods. We propose two new methods, one based on a discrete form of gradient descent, the other on simulated annealing, and report on a series of empirical tests comparing these and the other known algorithms for the problem. Based on this study, the first to be conducted, we identify the best approaches as a function of available computation time.
Highlights
Tagging graphical objects with text labels is a fundamental task in the design of many types of informational graphics
We concentrate on point-feature label placement (PFLP) without loss of generality; in Section 5 of the paper we describe how our results generalize to labeling tasks involving line and area features
The restricted set of problem instances in which no potential label position overlaps more than one other potential label position can be solved efficiently.6. These polynomially solvable subcases notwithstanding, the previous complexity results imply that PFLP problems likely to be of practical interest are NP-hard
Summary
Tagging graphical objects with text labels is a fundamental task in the design of many types of informational graphics. The set of potential label positions for each point feature characterizes the PFLP search space. The function to be optimized, the objective function, should assign to each element of the search space (a potential labeling of the points) a value that corresponds to the relative quality of that labeling. Imhof’s analysis is descriptive, not prescriptive; coming up with an appropriate definition of the objective function for a general label-placement problem (that is, one that includes point, line, and area features) is a difficult task. The PFLP problem is a combinatorial optimization problem defined by its search space and objective function; we wish to identify a general algorithm that is able to find a relatively good element of the search space.
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