EFFICIENT MULTIPLE SEARCH TREE STRUCTURE

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1 EFFICIENT MULTIPLE SEARCH TREE STRUCTURE Mohammad Reza Ghaeii 1 ad Mohammad Reza Mirzababaei 1 Departmet of Computer Egieerig ad Iformatio Techology, Amirkabir Uiversity of Techology, Tehra, Ira mr.ghaeii@aut.ac.ir Departmet of Computer Egieerig ad Iformatio Techology, Amirkabir Uiversity of Techology, Tehra, Ira mirzababaei@aut.ac.ir ABSTRACT This paper describes a efficiet multiple search tree structure istead of biary search tree. The search tree is more robust agaist differet sequetial of umbers ad more balace tree. Our tree height is less tha BST (ear to half); therefore, our search tree is faster i searchig. I deletio, our search tree is more stable ad does t eed recostructio, so the algorithm doe i O(log()). KEYWORDS Tree, Biary search tree, Search Structure, Sigmoid fuctio 1. INTRODUCTION A good data structure searches quickly, uses less memory, ad reasoable costructio time. We have may either liear or o-liear data structures used i computer sciece. The tree is oe of the o-liear fudametal. We use trees i compiler desig, text processig ad searchig algorithms [1]. Trees made up from some odes ad ay ode has some child odes. Tree has a mai ode as root its. Trees have may types ad usages; oe of those is the biary search tree []. I biary search tree, ay ode has zero, oe or two child odes. We have a policy i odes, all of the left subtree keys are less tha the paret s key ad all of the right subtree keys are bigger tha the paret s key. I best case, the height of the biary search tree is ad the search algorithm has O(log()) time complexity. Tree makig time complexity accordig to Kuth proof equals to O(log()) [3]: log( k) log(!) log( ) log( e) (log( )) log( ) (1) k 1 I worst case, the height of biary search tree is ad O() time is the complexity for searchig. So the tree makig time complexity is equal to O( ): ( 1) k k 1 Despite the wide popularity of biary search tree, ufortuately it has few serious problems. First, its shape depeds o the ature of the iput. Secod, its shape depeds o the deletes ad iserts too [4]. Accordig to the importace ad usage of biary search tree we cocluded that fidig a efficiet search tree is so valuable. ()

2 . RELATED WORKS The shape of the biary search tree depeds o the ature of the iput but some researches have bee doe to warraty the shape regular tree for ay iput, e.g.: -AVL Trees with Relaxed Balace [5]. - The Suffix Biary Search Tree ad Suffix AVL Tree [6]. - A Algorithm for the Orgaizatio of Iformatio, (AVL Tree) [7]. - Optimally Grow Biary Trees with a Sortig Algorithm [8]. - Self-Adjustable Biary Search Trees [9]. Aother type of those trees is the chromatic trees, which studied by Nurmi ad Soisalo- Soiie [10]. - Chromatic biary search trees: A structure for cocurret rebalacig [10]. - A dichromatic framework for balaced trees [11]. - Amortizatio results for chromatic search trees, with a applicatio to priority queues [1]. - Left-leaig Red-Black Trees [13]. - Symmetric Biary B-Trees: Data Structure ad Maiteace Algorithms [14]. Below papers describe o-blockig biary search tree i a asychroous shared-memory system usig sigle-word compare-ad-swap operatios. - No-blockig Biary Search Tree [15]. - No-blockig k-ary Search Tree [16]. Oe of tree problems is its traversal, for biary search tree we have three commo traversal methods: Iorder, Preorder, ad Postorder. These methods commoly implemet recursively. But i some situatios we wat to implemet these methods o-recursively, below paper presets a o-recursive algorithm for recostructig a biary tree. -Modified No-Recursive Algorithm for Recostructig a Biary Tree [17]. The other papers try to preset ew tree or optimize biary search tree. - O Dyamic Optimality for Biary Search Trees [18]. - A Forest of Hashed Biary Search Trees with Reduced Iteral Path Legth ad better Compatibility with the Cocurret Eviromet [4]. - A isertio algorithm for a miimal iteral path legth biary search tree [19]. - A Empirical Study of Isertio ad Deletio i Biary Search Trees [0]. Below paper is about oe of the biary search tree usages. - The research of quadtree search algorithms for ati-collisio i radio frequecy idetificatio systems [1]. This paper maily is about RFID (Radio Frequecy Idetificatio); the collisio resolutio is the most importat issues i RFID system that affects the data itegrity [1]. Oe of the aticollisio algorithms is biary search tree that is used i this paper. This area is so wide, but we would t elaborate ay further.

3 3. MULTIPLE SEARCH TREE STRUCTURE Figure 1, Data Structure Diagram I this structure every ode store three keys ad three Booleas. I every ode, we have a sigmoid fuctio that classifies umbers to blocks ad buckets. This fuctio is: size y( x) (3) 1 e ( x middle) w Figure, Sigmoid diagram with differet parameters Node fuctio determies bucket that searched key ca be there. The tree policy is the ode fuctio, each key that classify i a bucket, stores i that bucket or i bucket subtree Algorithm Specificatio First middle ad size equal to zero ad w equals to oe. I first umber additio, we store it i first positio; size variable sets to oe ad its fillig flag sets. I secod umber additio, we sort these data ad store them i icreasigly sorted order ad set size variable to two, ow we make middle equal to average of them. I third umber additio, after sortig, storig ad settig size, we make middle equal to average of maximum ad miimum, ext, 0 ) /1. 37 w ( ode ode (4) We divide sigmoid fuctio to three area, first area less tha 1/3, secod area betwee 1/3 ad /3, ad third area bigger tha /3. Calculatio shows that legth of middle area is ear to 1.38 (Equatio 5) ad for avoidig margial positio, we use 1.37 i our calculatio. By w, we map middle data betwee maximum ad miimum i middle bucket ad middle area.

4 y ( x) x e & x y) l y y 1 ( x & x (5) 3 3 Whe umber additio i filled bucket occurred, we call subtree to do umber additio request. For searchig, i ay ode we fid the bucket that our data ca be i there, if our data ot equals to ode bucket data, we cotiue searchig i bucket subtree. For deletio, first we fid that umber, after fidig, we delete the data ad cotiue traversal of that subtree arbitrarily to reach a leaf (a ode that do t have ay child), ad exchage a arbitrary leaf bucket umber with the deleted bucket umber ad reset its flag, ad if the deleted bucket have t subtree, we oly delete that umber ad reset its flag. 3.. Pseudo Code middle 0 w 1 size 0 child[3] {ull, ull, ull} ode[3] isfill[3] {false, false, false} add(umber): if (size == 0) the ode[0] umber isfill[0] true size 1 if (size == 1) the if (umber ode[0]) the ode[1] umber ode[1] ode [0] ode[0] umber middle (ode[1]+ode[0])/ isfill[1] true size if (size == ode[1]) the ode[] umber (if umber ode[0]) the ode[] ode[1] ode[1] umber ode[] ode[1] ode[1] ode[0] ode[0] umber middle (ode[] + ode[0])/ w (ode[] ode[0])/1.37 isfill[3] true size 3 b getbucketnumber(umber) if (isfill[b]) the child[b].add(umber) ode[b] umber isfill[b] true

5 getbucketnumber(umber): result 1 umber middle ( ) 1 e w retur result fidnumber(umber): b getbucketnumber(umber) if (ode[b] == umber) the retur true if (child[b] ull) the retur child[b].fidnumber(umber) retur false delete(umber): isdeleted false b getbucketnumber(umber) if (ode[b] umber) the if (child[b] ull) the retur child[b].delete(umber) force b temp child[b] if (temp ull) the while (temp.child[0] ull temp.child[1] ull temp.child[] ull) do if (temp.child[force] ull) the temp temp.child[force] if (temp.child[0] ull) the temp temp.child[0] force 0 if (temp.child[1] ull) the temp temp.child[1] force 1 if (temp.child[] ull) the temp temp.child[] force temp.size temp.size -1 ode[b] temp.ode[temp.size] isfill false retur true retur false 4. IMPLEMENTATION AND SIMULATION We implemet our data structure by JAVA. The system properties are: Processor: Itel Core Duo E4600,.40GHz / Memory: *1GB DDR 800. I this implemetatio we study two feathers, Tree makig time ad Tree Height. To decrease system errors ad iput situatios, we tested each dataset test 10 times, ad accepted the

6 miimum, we examied 10 differet dataset i every step ad declared average of them as result of that step. (Datasets make absolutely radomly) I implemetatio, we ca use some optimizatio, e.g., we ca determie umber of bucket for data without sigmoid fuctio calculatio, we ca determie bucket with e power amout. Figure 3, Tree testig ad compariso result Time of makig tree is importat too, fidig ad allocatig memory affect to this time, i biary search tree we have more extra ad useless allocated memory, therefore, makig that is slower (Figure 3). I ay ode, probable bucket of umbers ad keys determie i oe actio ad while each ode does t fill, we stay i that ode ad do t make a ew ode. We have local ad flexible policy i each ode; so, our tree is more robust agaist differet atural of iput ad sequece of umbers. 5. COMPARISON I best case, our tree height is log 3 ad O(log()) time complexity for searchig. Makig time complexity accordig to Kuth proof equals to O((log())) [3]: log( k) log(!) log( ) log( e) (log( )) log( ) (6) k 1 I worst case, our search tree height is /3 ad O() time complexity for searchig. Makig time complexity equals to O( ): 3 k ( ) k 1 3 ( 3 1) I average case, geerally, we ca say it is similar to best case ad worst-case occurrece possibility is too low. Because of similarity betwee this search tree ad biary search tree, we ca desig ad implemet some algorithms that belog to biary search tree for our search tree, E.g. traversal algorithms. Accordig to lower height of our tree, certaily, we ca expect lower time ad fewer cycle for searchig i this search tree. I this search tree we have some advatages icludig: lower searchig time, lower deletig time, stable shape after deletig, lower tree height, lower tree makig time, more robust i differet atural of iput (by local policy i ay ode), ad efficiet usig of memory (lower useless allocated memory). So, our search tree is efficiet substitute for biary search tree. (7)

7 6. CONCLUSIONS Accordig to the poits which metioed i last of compariso sectio, we ca seriously say that our search tree has may advatages i each usages ad we ca substitute that with biary search tree. 7. FUTURE WORKS The most importat issues ad future works are usig this search tree i problems ad applicatios to achieve its optimizatio ad its advatages. REFERENCES [1] V.V. Das, A ew No-Recursive algorithm for recostructig a biary tree from its traversals, IEEE Commuicatio ad Computig, pp.61-63, 010. [] T.N. Hibbard, Some combiatorial properties of certai trees with applicatios to searchig ad sortig, Joural of the ACM, Vol.9, No.1, pp.13 8, 196. [3] D.E. Kuth. The Art of Programmig-Sortig ad Searchig, Addiso Wesley, Readig, Mass, [4] V. Prasad. A Forest of Hashed Biary Search Trees with Reduced Iteral Path Legth ad better Compatibility with the Cocurret Eviromet, Iteratioal Joural of Computer Applicatios, Vol.33, No.10, Nov [5] K.S. Larse, AVL Trees With Relaxed Balace, Joural of Computer ad System Scieces. Vol.61, No.3, pp.508-5, 000. [6] R.W. Irvig, ad L. Love. The suffix biary search tree ad suffix AVL tree. Joural of Discrete Algorithms, Vol.1, No.5-6, pp , 003. [7] G.M. Adel so-vel skii, ad E.M. Ladis A Algorithm for the Orgaizatio of Iformatio, Soviet Mathematics Doklady, Vol.3, No.5, pp , 196. [8] W.A. Marti, ad D.N. Ness. Optimal Biary Trees Grow with a Sortig Algorithm, Commuicatio of the ACM, Vol.15, No., pp.88-93, 197. [9] D.D. Sleator, ad R.E. Tarjo. Self-Adjustig Biary Search Trees, Joural of The ACM, Vol.3, No.3, pp , [10] O. Nurmi, ad E. Soisalo-Soiie, Chromatic biary search trees: A structure for cocurret rebalacig, Acta Iformatica, Vol.33, No.6, pp , [11] L.J. Guibas, R. Sedgewick, A dichromatic framework for balaced trees. I Proc. 19th IEEE Symposium o Foudatios of Computer Sciece, pp.8 1, [1] J. Boyar, R. Fagerberg, ad K.S. Larse, Amortizatio results for chromatic search trees, with a applicatio to priority queues, Joural of Computer ad System Scieces, Vol.55, No.3, pp , [13] R. Sedgewick, Left-leaig red-black trees, Workshop o Aalysis of Algorithms, Maresias, Brazil, 008. (Available olie at [14] R. Bayer, Symmetric Biary B-Trees: Data Structure ad Maiteace Algorithms, Acta Iformatica, Vol.1, No.4, pp , 197. [15] F. Elle, E. Ruppert, P. Fatourou, ad F. va Breugel, No-blockig Biary Search Trees, Proceedigs of the 9th Aual ACM SIGACT-SIGOPS Symposium o Priciples of Distributed Computig (PODC'10), Zurich, Switzerlad, July 010. [16] T. Brow, ad J. Helga. No-blockig k-ary Search Trees, Departmet of Computer Sciece ad Egieerig York Uiversity, 011. [17] N. Arora, V.K. Tamta, ad S. Kumar. Modified No-Recursive Algorithm for Recostructig a Biary Tree, Iteratioal Joural of Computer Applicatios, V.43, No.10, April 01.

8 [18] N. Goyal, ad M. Gupta. O Dyamic Optimality for Biary Search Trees, 011. (Available olie at [19] T.E. Gerasch. A isertio algorithm for a miimal iteral path legth biary search tree, Commuicatios of the ACM, Vol.31, No.5, pp , [0] J.L. Eppiger, A Empirical Study of Isertio ad Deletio i Biary Search Trees, Commuicatio of the ACM, Vol.6, No.9, pp , [1] B.Y. Shih, T.W. Lo, C.Y.ad Che, The research of quadtree search algorithms for aticollisio i radio frequecy idetificatio systems, Scietific Research ad Essays, Vol.6, No.5, pp , 30 Oct.011.