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1. Number Systems and Data structuresAmr Elmasryelmasry@mpi-inf.mpg.deA. Elmasry, C. Jensen and J. Katajainen, The strictly-regular number system and worst-case efficient data structures.A. Elmasry, C. Jensen and J. Katajainen, The magic of a number system. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA

2. Outline • Some existing number systems • Properties • Data-structural applications • The strictly-regular system • Properties • Application • The quinary-skew system • Properties • Application Elmasry, Amr

3. What is a Number System? x = <d0,d1,d2,… dk-1> b-ary wj = bj wj = bj+1 – 1 (skew) Examples: • Binary: dj є {0,1}, wj = 2j • Redundant binary: dj є {0,1,2}, wj = 2j • (Canonical) skew binary: The first non-zero digit may be a 2. 0* (ε | 2) (0|1)*, wj = 2j+1 – 1 [Myers 83] • Redundant-regular (RR) binary: Every 2 is preceded by at least one 0. (0 | 1 | 01*2)* [Clancy & Knuth 77] Elmasry, Amr

4. Operations • Extended-RRsystem: dj є {0,1,2,3}.Every 3 is preceded by at least one {0,1}, and every 0 is preceded by at least one {2,3}. [Clancy & Knuth 77 - Kaplan et al. 02] • Zeroless systems: dj ≠ 0 • Others: dk-1 = 2 Operations: increment(x,j): s  val(x) + wj decrement(x,j): s  val(x) – wj add(x,y): s val(x) + val(y) cut(x,j): cut x into two valid sequences. concatenate(x,y): concatenate x and z and return a valid sequence. Elmasry, Amr

5. Properties Properties • Increments involve amortized constant digit flips in binary system. • But, worst-case constant digit flips for the RR system. • Increments and decrements only at d0 involve constant digit flips in the skew-binary system. • Increments and decrements involve worst-case constant digit flips when using two back-to back RR system. • Increments and decrements involve worst-case constant digit flips for the extended-RR system. • The sum of digits of a k-digit number in the RR binary system is at most k. • The sum of digits of a k-digit number in the zeroless-RR binary system is at most 2k. • The sum of digits of a k-digit number in the extended-RR binary system is at most 2k. • The value of a k-digit number in the zeroless RR binary system is at least 2k-1 (the exponentiality property). Elmasry, Amr

6. How does it Work? Example: RR-binary system (least-significant digit first) Increment least significant bit + fix least significant 2 fix-carry: 2 x  0 (x+1) 01200001112012020011 11010001112012020011 02010001112012020011 10110001112012020011 01110001112012020011 Keep track of a list of 2’s from least to most significant. For general increments use Brodal’s guides. Elmasry, Amr

7. From Number Systems to Data Structures • Define the notion of the rank. • There would be dj objects of rank j in the data structure. • The size of an object of rank j is sj sj =wj  perfect components In general, sj ≤wj (for skew binomial queues 2j ≤ sj ≤2j+1 – 1) • fix-carry resembles a join of b objects of rank j to one of rank j+1. • fix-borrow resembles a split of an object of rank j to b objects of rank j-1. Elmasry, Amr

8. Worst-Case Efficient Data Structures • Finger trees. [Guibas et al. 77] • Using the RR-binary system: binomial queues with constant worst-case insertion cost. [Carlsson et al. 88] • Using the skew-binary system: skew binomial queues with constant worst-case insertion cost. [Brodal & Okasaky 96] • Using the zeroless-RR system: a priority queue that supports meld in constant worst-case cost. [Brodal 95] • Using two RR-systems back-to-back with dj ≥ 2: both meld and decrease in constant worst-case cost. [Brodal 96] • Using two RR-systems back-to-back: purely functional catenable deques in constant worst-case cost per operation. [Kaplan & Tarjan 95] • Using the extended-RR system: Fat heaps. [Kaplan et al. 02] • Using the extended-RR system: worst-case efficient priority queue with the working-set property. [Elmasry 06] Elmasry, Amr

9. The Strictly-Regular System Every 2 is preceded by a 0, and every 0 is preceded by a 2, except for the last block that starts with a 0. (1+ | 01*2)* (ε | 01*) Primitives: fix-carry: 2 x  0 (x+1) fix-borrow: 0 x  2 (x-1) Deciding which digit to fix depends on the value of the current digit (digit under operation) and the next extreme digit (0 or 2). Properties: • Increments, decrements, catenations and cuts involve worst-case constant digit flips. All can be implemented in worst-case constant time once you know the closest extreme digit. • Addition of k-digit numbers involve k+O(1) carry propagations. • The sum of digits of a k-digit number is either k or k-1 (compactness). • The value of a k-digit number is at least Φk (exponentiality). Elmasry, Amr

10. Strictly-Regular Trees The rank of a node equals the number of its children. The ranks of the children of a node form a strictly-regular sequence. Properties: • A strictly-regular tree of rank r has at least 2r nodes. • Adding a subtree with a non-larger rank or detaching a subtree is done in worst-case constant time if one can access the trees whose ranks resemble the adjacent extreme digits. • Splitting and concatenating sequences of children can be done worst-case efficiently. Application: Improving the constant factors for the number of comparisons of data structures. For meldable heaps (no decrease) delete-min requires at most 2 lg n +O(1) comparisons instead of 7 lg n + O(1). Elmasry, Amr

11. The Skew Five-Symbol Number System dj є {0,1,2,3,4}, wj = 2j+1 – 1 increment by 1: • Increase d0 by 1. • Find the smallest j where dj є {3,4}. • If j exists, perform a fix for dj . fix dj : • Decrease dj by 3. • Increase dj+1 by 1. • If j≠0, Increase dj -1 by 2. A fix does not change the value of a number. Elmasry, Amr

12. Properties Numbers from one to thirty: 1, 2, 01, 11, 21, 02, 12, 22, 03, 301, 111, 211, 021, 121, 221, 031, 302, 112, 212, 022, 122, 222, 032, 303,113, 2301, 0401, 3111,1211, 2211 A decrement is implemented as an inverse of an increment, by using an undo stack to remember where the fixes were performed. Properties: (Any number is composed of a sequence of blocks each ends with a 3 or 4 and the last subsequence is the tail) • The body of a block ending with 4 comprises either 0 or 12*1. • The body of a block ending with 3 comprises either 0, 12*1, or 2*. • Each 4, 23 and 33 is followed by either 0 or 1. • There can be at most one 0 in the tail, which must be its first digit. • The sum of digits of a k-digit number is at most 2k. Elmasry, Amr

13. Application The fix is done in constant worst-case time: The minimum root is detached from its tree and attached as the root of the other two trees forming a larger tree. Application: A priority queue can be implemented as a forest of complete binary trees having worst-case costs: O(1) insert and O(lg n) deletemin. Elmasry, Amr

14. Conclusions and Future Work: • Conclusions • Number systems are interesting. • The connection between number systems and worst-case-efficient data structures is efficacious. • Two new number systems were introduced. • Future work • New number systems with applications. • More applications for the strictly-regular system. • Extending the strictly-regular system to be ternary (b-ary). • Is it possible to implement general increments and decrements in worst-case constant cost for the strictly-regular binary system? • Supporting decrements for the skew five-symbol system without the usage of the undo stack. Elmasry, Amr