O(log n) bits or atomic elements

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O(log n) bits or atomic elements Implicit model atomic elements CPU, O(1) registers O(log n) bits or atomic elements 1 x1 2 x2 3 x3 4 x4 5 x5 xn-1 n xn - XOR OR write shift-left + * ≤ shift-right Memory, n words read ... NOT AND only allowed operation on elements # reads Complexity = + # writes + # instructions performed Inaccessible

Sorting Search trees O(n∙log n) Yes O(n2) O(n) No O(log n) Comparisons Data moves/writes Implicit HeapSort O(n∙log n) Yes SelectionSort O(n2) O(n) SearchTree No [FG05] Implicit priority queue [G. Franceschini, V. Geffert, An in-place sorting with O(n log n) comparisons and O(n) moves, J.ACM, 52(4), 515-537, 2005] Search trees Search & updates Range searching Implicit Cache-oblivious Red-black, ... O(log n) O(log n + t) No Sorted array Yes [FG02] [FG03] - [BFJ02], ... (no updates) Open problems Worst-case range searching + cache oblivious Range searching+ implicit + cache oblivious am. [G. Franceschini, R. Grossi, Optimal Cache-Oblivious Implicit Dictionaries, Proc. 30th International Colloquium on Automata, Languages, and Programming, volume 2719 of Lecture Notes in Computer Science, 316-331, Springer-Verlag, 2003.] [G. Franceschini, R. Grossi, J.I. Munro, L. Pagli. Implicit B-trees: New results for the dictionary problem. IEEE Symposium on Foundations of Computer Science, 145-154, 2002] [G.S. Brodal, R. Fagerberg, R. Jacob, Cache-Oblivious Search Trees via Binary Trees of Small Height, 13th Annual ACM-SIAM Symposium on Discrete Algorithms, 39-48, 2002]

The foundamental implicit trick The relative two elements x < y, can encode a bit = 0 = 1 2log n elements can encode integer {0,...,n-1} x y y x

Search trees O(log2 n) O(log2 n + t) Yes Search & updates Range searching Implicit O(log2 n) O(log2 n + t) Yes Search trees [J. Ian Munro, An Implicit Data Structure Supporting Insertion, Deletion, and Search in O(log² n) Time, Journal of Computer and System Sciences, 33(1), 66-74, 1986] Each nodes stores ..2-1 elements encoding the below fields (=8∙log n+2) Red-black search tree x x x x x x field values encoded by #elements left (address) 0,1,..., n-1 2∙log n right (address) parent(address) color (red/black) 0,1 2 node size s 0,1,...,n-1 x x x x x x x x x x x x x x x x x ps mod s = 0 root ps qs ps-1 qs-1 ps-2 qs-2 ps-4 qs-4 ps-3 =qs-3 p2-1 =q2-1 s s-1 s-2 s-4 gap xxxxxx xxxxx xxxx xx root p2-1,q2-1,... ,p,q (2+1)∙log n s ∙ (1 + # size-s-nodes) Total gap: 2+(2-1)+∙ ∙ ∙+ = arbitrary elements

Implicit merging O(n + m) can be used in an implicit O(n∙log n) MergeSort 2 4 5 7 9 11 3 6 8 14 n m 2 3 4 5 6 7 8 9 11 14 (Honours project 2)