Label updating to avoid point-shaped obstacles in fixed model

Abstract

In this paper, we present efficient algorithms for updating the labeling of a set of n points after the presence of a random obstacle that appears on the map repeatedly. We update the labeling so that the given obstacle does not appear in any of the labels, the new labeling is valid, and the labels are as large as possible (called the optimal labeling). Each point is assumed to have an axis-parallel, square-shaped label of unit size, attached exclusively to that point in the middle of one of its edges. We consider two models: (1) the 2PM model, where each label is attached to its feature only on the middle of one of its horizontal edges, and (2) the r4PM model, where each label is attached to its feature on the middle of either one of its horizontal or vertical edges (known in advance). We assume that a sequence of point-shaped obstacles appear on the map on random locations. Three settings are considered for the behavior of the obstacle: (1) the obstacle is removed afterwards, (2) it remains on the map, and (3) it receives a similar label and remains on the map. Only two operations are permitted on the labels: flipping one or more labels, and/or resizing all labels. In the first setting, we suggest a data structure of O(n) space and O(n lg n) time in the 2PM model, and of O(n 2) time in the r4PM model, so that the updated labeling can be constructed for any obstacle position in O(lg n + k) time, where k is the minimum number of operations needed. For the second and third problems, we suggest an O(n) space and O(n lg n) time data structure that can place each obstacle (possibly with a label) on the map in O(lg n + k) time, if k label flips are sufficient to make room to place the new obstacle. Otherwise, two O(n) time algorithms are suggested when a relabeling of all points is required.

Incremental labeling in closed-2PM model

“…Since each vertex of G has no more than 4 outgoing edges, there there are O(n) vertices. Moreover, there are O(n) edges in G that can be constructed in O(n lg n) time [16].…”

“…We present another special weighed and directed graph called conflict graph to represent all possible flip and resize operations of a given labeling [14][15][16]. For clarity, we define the conflict graph precisely and briefly in the following.…”

“…4PM labeling is NP-Hard and is first considered in [9,5] to model and solve a problem in meta-font. To convert 4PM problem to a non NP-Hard problem, a labeling direction for each point, which is known in advance, is introduced [16]. Label direction of each point restricts its label to be attached to it, via just either horizontal or vertical edges.…”

“…In r4PM, there are two label candidates per each point, hence, n points can be labeled optimally, in r4PM model, in O(n lg n) time using a reduction to 2SAT problem (like the reduction given in [11]). In [14,16] we introduced the r4PM model and provided a new approach to the this relabeling problem: How a labeling can be maintained when an obstacle appears in the map? We considered the case of a moving obstacle in a labeled map where at each given time, an optimal obstacle-avoiding labeling is constructed.…”

“…With an O(n 2 ) preprocessing time, we presented an algorithm that obtains such a labeling in O(lg n+k) response-time, where k is the number of changes needed to avoid the obstacle. Later in [15,16],we considered the same problem in 2PM model and reduced the preprocessing time from O(n 2 ) to O(n lg n). The latter algorithm was much easier to implement and required simpler data structures than the former one.…”