Fast Algorithms for D Dominance Rep orting and Counting Qingmin Shi and Joseph JaJa Institute for Advanced Computer Studies Department of Electrical and Computer Engineering University of Maryland Col lege Park MD fqshijosephumiacsumdedug January Abstract We present in this pap er fast algorithms for the D dominance rep orting and counting problems and generalize the results to the ddimensional case Our D dominance rep orting algorithm achieves O log n log log n f query time using O n log n space where f is the numb er of p oints satisfying the query and is an arbitrary small constant For the D dominance counting problem whichisequivalent to the D range counting problem our algorithm runs in O log n log log n time using O nl og n log log n space Intro duction d Let S be a set of n points in Thedominancereporting problem is to store S in a data structure so that the subset Q of p oints in S that dominate an arbitrarily given p oint q can b e rep orted ecientlyGiven two p oints p p p p andq q q q we d d say p dominates q if and only if p q for all i d Another related problem is the i i dominancecounting problem In this problem the size k of Q only needs to b e rep orted Many computational geometry problems such as the rectangle intersection problems can b e reduced to the dominance search problem In this pap er weprovide faster algorithms than previously known for b oth the D dominance rep orting and counting problems Our mo del of computation is the RAM mo del as mo died byFredman and Willard In this mo del it is assumed that eachword is of w size w and that the numb er of data elements n never exceeds that is w log nIn addition arithmetic and bitwise logical op erations take constanttime Before outlining our results we give a brief overview of related literature The D dom inance rep orting is a sp ecial case of the D orthogonal range search problem Hence results Supp orted in part by the National Science Foundation through the National Partnership for Advanced Computational Infrastructure NPACI DoDMD Pro curement under contract MDAC and NASA under the ESIP Program NCC for orthogonal range search apply immediately to our problem In particular a class of algorithms for orthogonal range rep orting uses constant time for each rep orted p ointand p olylogarithmic search time see for example The b est known result for D orthogonal range search is given by Alstrup and Bro dal whichachieves O n log n space and O log n f query time Several other data structures use less space but require larger query time In Chazelle gives two data structures one with query time O log n f log log n f and using space O n log n log log n and the other with query time O log n f log nf and using O n log n space The b est known dominance rep ort ing algorithm is due to Makris and Tsakalidis whichfollows the approach of Chazelle and Edelsbrunner and achieves linear space and O log n f query time Unlike dominance rep orting which seems to b e inherently simpler than general orthogo nal range rep orting the dominance counting problem is equivalent to the orthogonal range counting problem if the dimension d is assumed to b e constant Indeed it is easy to show that if a data structure of size O sn exists for answering any ddimensional dominance counting query in O tn time then any ddimensional range counting query can b e an d d swered with O sn space and using O tn query time Not many results are known for the multidimensional dominance counting problem In Chazelle uses a compressed form of the range tree to achieve O log n query time and O n space for the D dominance counting problem This result easily leads to a solution to the D dominance counting with O n log n space and O log n query time On the other hand we can reduce the counting query problem to the aggregation range query problem in the same dimension by assign ingaweight of to eachpoint Willard in shows howtocombine the fusion tree and its variant called the q heaps and the fractional cascading technique to achieve O n log n space and O log n log log n query time to handle D aggregation range queries where is arbitrarily small constant In this pap er we establish the following results that provide faster algorithms for D dominance rep orting and counting An algorithm for threesided D range rep orting that achieves O n space and O log n log log n f query time This result is similar to Willards mo dication of priority tree but the algorithm is much simpler This algorithm plays an imp ortant role in our D dominance rep orting algorithm An algorithm for D dominance rep orting that uses O n log n space and O log n log log n f query time where f is the number of points satisfying the query An algorithm for D dominance counting problem whichusesO n log n space and runs in O log n log log n query time This algorithm can b e seen as an improvement on the query time over Chazelles algorithm at the exp ense of O log n additional space An extension of the D dominance counting algorithm and D dominance rep orting d algorithm to multidimensional dominance search which leads to an O log n log log n d query time and O n log nlog n log log n space algorithm for the counting case d d and an O log n log log n f query time and O n log nlog n log log n space algorithm for the rep orting case where d is the numb er of dimensions and d for the counting case and d for the rep orting case In Section we briey discuss several known techniques used heavily in this pap er The threesided D range rep orting algorithm is describ ed in Section while the description of our D dominance rep orting algorithm is given in Section Our D dominance count ing algorithm is describ ed in Section Results in Sections and are extended to the multidimensional case in Section Preliminaries Toavoid tedious details in describing the algorithms we assume for the rest of this pap er that no two p oints have the same co ordinate in any dimension For simplicitywe call the p oint with the largest xco ordinate smaller than or equal to a real number r the xpredecessor of r and the one with the smallest xco ordinate larger than or equal to r the xsuccessor of r The y and zpredecessorssuccessors are dened similarly Cartesian Trees Cartesian trees dened over a nite set of D p oints were rst intro duced byVuillemin Let p p p be a set of n D p oints sorted by their xco ordinates The corresp onding n Cartesian tree C is dened recursively as follows Let p b e the p oint with the largest y i co ordinate Then p is asso ciated with the ro ot r of C The left child of r is the ro ot of the i Cartesian tree built on p p and the rightchild of r is the ro ot of the Cartesian tree i built on p p The leftright child do es exist if i n We call such a Cartesian i n tree an x y Cartesian tree An imp ortant prop erty of the Cartesian tree is given by the following observation Observation Consider a set S of D points and the corresponding x y Cartesian tree C Let x x be the xcoordinates of two points in S and let and be their respective vertices Then the point with the largest ycoordinate among those whose xcoordinates are between x and x is stored in the nearest common ancestor nca of and Fractional Cascading Supp ose wehave a tree T of b ounded degree c and ro oted at no de r suchthateachnode v contains a list Lv of sorted elements Let n b e the total size of all the lists stored in T and let F b e an arbitrary forest with p no des consisting of subtrees of T determined by some of the children of r F may b e sp ecied on line ie not necessarily known during the prepro cessing step The following lemma is a direct derivation from the one given by Chazelle and Guibas for identifying all the successors of a value g in the lists stored in F Lemma There exits a linear size fractional cascading data structure that can beusedto determine the successors of a given item g in the lists storedin F in O p log c tn time where tn is the time it takes to identify the successor of g in Lr A simple way to implement the fractional cascading strategy is to use augmented lists For eachnode v in T we store in addition to Lv another sorted list L v called the augmented list which consists of elements in the lists asso ciated with b oth v and its children We asso ciate with each element e in L v c pointers One p ointer called the intranode pointerpoints to the successor of e in Lv while the other c internode pointers each p oints to the successor of e in the augmented lists asso ciated with a dierentchild of v Let v be an internal no de and u b e its j th child starting from the left If weknow the successor e v of g in L v then by following the j th interno de p ointer asso ciated with e we can obtain v the successor e of g in L u The successor of g in Lu is then obtained by following the u intrano de p ointer asso ciated with e Note that this works correctly since Lu is a subset u of L u Since each element is stored at most twice and there are c pointers asso ciated with each element the storage cost of this scheme is O cn where n is the total size of all the lists Wethus haveavariation of the fractional cascading structure which allows constanttime lo okup in nonro ot no des Wecallitfast fractional cascading and the linearspace fractional cascading structure compact fractional cascading Lemma The fast fractional cascading structure al lows the identication of the successors of a given item g in the lists storedin F in O p tn time where tn is the