In [

^{1}], in

Figs. 1,

2,

3, and

4, all occurrences of "-" between words and all occurrences of a space between words are gone; as a result, some words are also not aligned with the boxes and some occurrences of ")" also do not appear in the right places.

Corrections:

Figs. 1,

2,

3, and

4 in [

^{1}] should have appeared as follows (the correct version of [

^{1}] is available from the author's Web site: http://cis.poly.edu/chiang/PTOT-vis09.pdf):

Fig. 1. Persistent structure. (a) The node-copying technique, where each persistent node has three extra fields. (b) Persistent binary search tree (where we use the alphabetical order to compare the keys (letters)) after simulating a sequence of updates. Each node has one extra field. The number associated with a node/pointer denotes its version stamp. The numbers 1 to 7 on the top horizontal line denote the entry-point array . Version 5 is shown in red.

Fig. 2. (a) The time-octree. (b) An example of the time tree for time interval . Each internal node labeled covers the time span , and each leaf labeled corresponds to time step .

Fig. 3. Line-sweep process to insert/delete intervals to the time-octree. There are four time cells whose intervals are respectively . The vertical red line is the sweep line. The interval endpoints subdivide the scalar-value range into ranges , , , , , , , , , where version of the time-octree corresponds to range . Isovalue lies in range , and version 5 of the time-octree contains exactly time cells , which are active for ; and belong to the octree of and belongs to the octree of .

Fig. 4. (a) Standard representation. Inserting/deleting a leaf can create/remove many nodes (shown in red excluding ). (b) The compact representation. (c) With the compact representation, inserting/deleting a leaf (as the event in (a)) can create/remove at most one internal node, the *degree-2 fork* node .

• *The authors are with the Computer Science and Engineering Department, Polytechnic Institute of New York University, Brooklyn, NY 11201.*

*E-mail: cwang05@students.poly.edu, yjc@poly.edu.*

*For information on obtaining reprints of this article, please send e-mail to: tvcg@computer.org.*