Dynamic Maintenance of Majority Information in Constant Time per Update? Gudmund S. Frandsen and Sven Skyum BRICS 1 Department of Computer Science, Un

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1 Dynamic Maintenance of Majority Information in Constant Time per Update? Gdmnd S. Frandsen and Sven Skym BRICS 1 Department of Compter Science, University of arhs, Ny Mnkegade, DK-8000 arhs C, Denmark We show how to maintain information abot the existence of a majority color in a set of elements nder insertion and deletion of single elements sing O(1) time and at most 4 eqality tests on colors per pdate. No ordering information is sed. Key words: Data strctres, nalysis of algorithms. 1 Introdction We consider the problem of maintaining information abot the existence of a majority in a set of elements nder insertion and deletion of single elements. The notion of majority is formalised by considering each element to have a color. If strictly more than half of the elements have the same color, this color is a majority color. O-line, the existence of a majority color may be decided in time O(n log n) by sorting, and several people [1,] independently fond a linear pper bond and determined that precisely b 3 (n? 1)c eqality tests on colors are needed in the worst case to determine the existence of a majority color (withot the se of any ordering information). We are not aware of any similar reslts for the dynamic problem, prior to this paper. For the dynamic problem one may obtain a soltion sing O(log n)? This research was spported by the ESPRIT II BR Programme of the EC nder contract # 7141 (LCOM II) and by CCI-Erope. 1 Basic Research in Compter Science, Centre of the Danish National Research Fondation. 15 pril 1997

2 comparisons per pdate and O(1) time per qery by sing ordering information. In this paper we describe a data strctre for the optimal bond of (1) per pdate and qery. We se eqality tests on colors, bt no ordering information. 1.1 Problem Denition We let the dynamic majority maintenance problem consist in implementing the following data type. { memory: S: the set of elements. Initially, S = ;. { operations: Init : S ;; Insert(e) : S S [ feg; Delete(e) : S S? feg; Qery? : retrn (yes; e), if there are at least d jsj+1 e elements in S of some single color c, and e is some element of this color c; retrn (no) otherwise. s or model of comptation, we se an ordinary nit cost RM with O(log n) word size. Colors cannot be sed as addresses. In fact the only allowed way to extract color information is by making an eqality test on two colors. 1. Reslts We present a soltion for the dynamic majority problem that ses time (1). This soltion ses at most 4 eqality tests on colors per delete, at most 3 eqality tests per insert and no eqality tests on a qery. The lower bond of b 3 (n? 1)c eqality tests for the o-line majority problem [1,] implies that a single insert reqires at least eqality tests in the worst case no matter how mch time we allow. simple constant time constrction.1 Tri-ladder data strctre Or soltion ses a special pointer strctre, which we have called a triladder for storing the elements of the set (see Fig. 1). The tri-ladder stores

3 X Y Z and B have distinct colors B and B have identical colors B It is known, whether and B have identical colors B It is nknown, whether and B have identical colors? B Fig. 1. General tri-ladder and symbol explanation No majority color: Majority color exists: Fig.. Typical sets with and withot a majority color information abot identity and distinctness of colors. The tri-ladder organises the elements in two opposing lists, sch that an element Y has for adjacent elements, two opposing elements U; V in the other list, and two neighbors X; Z in the same list. n element may have fewer adjacent elements when placed at the end of one list or beyond the end of the other list. We maintain the following invariant: { For any pair of adjacent elements it is known whether colors are eqal or distinct. { ll pairs of adjacent opposing elements have distinct colors. { The left ends of the two lists are adjacent (and opposed) to each other. { If there is no majority color, then the length of the two lists dier by at most 1 (see typical tri-ladder in Fig. ). { If there is a majority color, then the two list are neqal in length and all the elements in the longer list have the majority color (see typical tri-ladder in Fig. ). ny maximal length rn of identically colored elements in one of the two lists is called a block (it is possible that a block consists of a single element). block has pointers between the two endpoints, allowing s to go from one endpoint of a block to the other endpoint in constant time. 3

4 ? X Y W Z? (i) If Z then X Y W Z U Z W? (ii) If Z and U V then X Y V? (iii) If Z and U V then?. The qery operation Fig. 3. Insertion of a new element Z. X Z W Y ccording to the invariant, there exists a majority color precisely when { one list is longer than the other, and { the elements of the longer list forms a single block. These criteria can be checked in time (1) sing the tri-ladder representation..3 The insert operation We shall show how to maintain the tri-ladder strctre when inserting a new element. We will assme that there is no majority color in the present set. The reader shold nd no diclty in modifying or constrction to the other case, when there is a majority color. Fig. 3 illstrates the insertion of a new element into a typical tri-ladder. Initially, we compare the color of the new element Z with the color of the rightmost element in the crrent tri-ladder. Depending on the otcome of this comparison and internal color relations, there are three cases: (i) If and Z have distinct colors, we can simply insert Z at the right end of the tri-ladder opposing. One additional comparison is needed to 4

5 maintain invariants. (ii) If and Z have identical colors and U and V have distinct colors then we split the tri-ladder to the right of U and Y and trn the right part of the tri-ladder pside down. Z can then be inserted between U and W. This reqires one additional comparison to maintain invariants. In case (ii) the reqirement that U and V have distinct colors garantees that we do not split the tri-ladder in the middle of any block, and therefore the insert is handled in constant time in this case. (iii) If and Z have identical colors and U and V also have identical colors (i.e. U and V cold be in the middle of a large block for all we know) then we may replace the element Y next to 's block with Z and reinsert Y at the right end of the tri-ladder. This reqires two additional comparisons to maintain invariants. It shold be clear that a single insertion can be done in time O(1) and ses at most 3 color comparisons to maintain the tri-ladder strctre..4 The delete operation We shall show how to maintain the tri-ladder strctre when deleting an element. Fig. 4 illstrates the deletion of an element Z from a typical triladder. There are several cases depending on whether the elements opposing Z named U and V in Fig. 4 have distinct colors or not, and depending on whether Z is at one end of its block. (i) If U and V have distinct colors and Z is at one end of its block (i.e. B and Z have distinct colors or Z and D have distinct colors) then we split the tri-ladder to the right of B and U and trn the right part of the tri-ladder pside down. The two parts of the tri-ladder are re-connected after removal of Z. Two color comparisons sce to maintain invariants. (ii) If U and V have identical colors then we remove both Z and V from the tri-ladder, and reinsert V sing the method of the previos section. Maintenance of invariants reqires one color comparison in addition to those needed for the reinsertion of V. (iii) If B, Z and D all have identical colors, we act similarly to (ii). It shold be clear that a single deletion can be done in time O(1) and ses at most 4 color comparisons (4 = maxf; 1 + 3g) to maintain the tri-ladder strctre. 5

6 B Z D C (i) If U V and ( B Z or Z D ) B V C? then? U D (ii) If U B D? V then and reinsert V U C B D (iii) If B Z D then? U C and reinsert V References Fig. 4. Deletion of an element Z. [1] Fischer, M. J. and Salzbrg S. L., Soltion to problem Jornal of lgorithms 3 (198) 376{379. [] Matla, D. W., n Optimal lgorithm for the Majority Problem. Manscript, Sothern Methodist University, Texas,