Information Processing Letters 24 (1987) 95-96 30 January 1987 North-Holland

A N O T E O N A NEW DATA S T R U C T U R E FOR IN-THE-PAST Q U E R I E S

Dan F I E L D

Computer Graphics Laboratory, Department of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Communicated by W.L. Van der Poel Received 20 February 1986

This note describes a new data structure which records information and supports queries about elements that have been previously inserted and deleted. Both time and space parameters match those of Overmars (1981); the advantage to the structure described here is its simplicity.

Keywords: Data structure, in-the-past query

I. Introduction

Dobkin and Munro [1] invested a data struc- ture that allows queries to be answered about previously held states. Subsequent papers by Overmars [2] and Aviad and Shamir [3] improved and generali7ed their data structure. This note, while not improving the asymptotic time or space bounds of Overmars, describes a data structure that is simpler to understand and implement .

Let xx, x 2 , . . . , x n be a totally ordered set of elements. An update operat ion consists of the insertion or deletion of an element x i, and is assigned a t ime-stamp t indicating that t - 1 up- dates have preceded. The s ta te of the data struc- ture at t ime t is the set of elements present after update t and before update t + 1. The data struc- ture must be able to answer two types of query:

Q1 " W h a t rank does x have in state t?" Q2 " W h a t element has rank r in state t?"

These so-called ' in-the-past ' queries are useful for solving geometric searching and intersection prob- lems.

Let N t denote the number of elements present in state t. Initially, N o = 0. Overmars demonstrates a data structure and algorithm that answers a query of state t in O(log Na) time. An update of

state t - 1 takes O(log Nt) time and space for a total of 0020 ,~ i ,~ t log N i) over t updates.

2. The data structure

The data structure used here is a forest; ele- ments present at state t are stored in rank order at the leaves of balanced search trees To, T1 , . . . , Tt, one tree for each state of the data structure. Each node v contains gv, the number of leaves descend- ing from the left child of v. It is not hard to see how queries Q1 and Q2 may be answered for a particular state t by performing the appropriate search of T t. Since there are N t leaves in Tt, the query can be answered in O0og N t ) time.

As things now stand, space requirements are O~0 ~i*g tNi) • This is reduced by embedding sub- trees of T 0, Tx , . . . , Tt_ 1 in T t.

Suppose update t removes element x. Let S be the set of nodes in T t_ 1 that would be modif ied by the deletion of x in the tree. Define

S' = S t3 (w I w is an ancestor of v e S }.

T t is created by copying nodes in S' and linking them to subtrees in Tt_ 1 that would not be af- fected by the deletion of x. T t is now an exact copy of Tt_ 1 with total space O(ITt_xl+lS'l).

0020-0190/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland) 95

Volume 24, Number 2 INFORMATION PROCESSING LETTERS 30 January 1987

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(b)

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The delet ion of x is now pe r fo rmed on T t in the usual m a n n e r wi thout affecting shared por t ions of the tree (see Fig. 1). The same basic steps above are fol lowed when the upda te is an insertion.

For m a n y balanced trees including 2-3 trees [4], A V L trees [5], and BB[~] trees [6], [S'[ = O(log Nt). Therefore, by sharing c o m m o n sub- trees, the size of the data s tructure has been reduced to O(E 0 ¢ i ~ t log N i).

References

[1] G.M. Adelson-Velskii and Ye.M. Landis, An algorithm for the organiTation of information, Soviet Math. Dokl. 3 (1962) 1259-1263.

[2] A.V. Aho, J.E. Hopcroft and J.D. Ullman, Design and Analysis of Computer Algorithms (Addison-Wesley, Read- ing, MA, 1975).

[3] Z. Aviad and E. Shamir, A direct dynamic solution to range search and related problems for product regions, Manuscript.

[4] D.P. Dobkin and J.I. Munro, Efficient uses of the past, Proc. 21st I1~1~1~ Syrup. on Foundations of Computer Sci- ence (1980) 200-206.

[5] J. Nievergelt and E.M. Reingold, Binary search trees of bounded balance, SIAM J. Comput. 2 (1) (1973) 33-43.

[6] M.H. Overmars, Searching in the past I, Tech. Rept. RUU-CS-81-7, Univ. of Utrecht, April 1981.

(d)

Fig. 1. The sequence of operations to insert a leaf containing the value 12 into an AVL tree T. Part (a) depicts T before the insertion. Part (b) shows the tree T ' resulting from the inser- tion of a leaf containing 12. Note that a new interior node e ' was created and a rotation performed. Part (c) depicts the nodes S' of T ' that are different from those in T. Part (d) shows the result of embedding the subtrees of T into T'.

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