Integer Representations and Counting in the Bit Probe Model

M. Zaiur Rahman and J. Ian MunroCheriton School of Computer Science

University of WaterlooWaterloo On

Canada

Integer Representations

• Standard: n bits representing {0..2n}

• # bits … optimal

• Increment: – Θ(n) bit inspections/changes worst case– O(1) amortized, but not if we include

decrement

Redundant Binary Numbers

• Use “digits” 0,1,2 but representation is in binary, 2 is a delayed carrye.g. 01220 = 10100

• Space 2n bits (or reduce to about (lg 3) n)– Increment O(1), as we de-amortize the scan

to release the rightmost carry– Decrement is tricky, easiest to use digit “-1”– Note the extra lg n bits for the scan pointer

Gray Codes .. Due to Gray ..of course

Binary Reflected Gray Code (BRGC)– Of dimension n (i.e. n bits, numbers [0..2n]– A sequence of 2n strings each of length n

G(n) = 0.G(n-1), 1.G(n-1)R

(reversal is on the sequence order, not the individual codes)

Increment on a Gray Code

• Even parity: flip rightmost 1 bit

Increment on a Gray Code

• Even parity: flip rightmost bit

00110000 → 00110001

Increment on a Gray Code

• Even parity: flip rightmost bit

00110000 → 00110001

• Odd parity: flip the bit to left of rightmost 1

01110000 → 01010000

Increment on a Gray Code

• Even parity: flip rightmost bit

00110000 → 00110001

• Odd parity: flip the bit to left of rightmost 1

01110000 → 01010000• Costs:1 bit changed, n inspections (worst case)

• Or: add parity bit, O(1) amortized insert, 2 bits changed

• Integrate parity bit with code (no extra bit [Lucal])

Our First Result: Lower Bound

Theorem: Any representation of [0..2n] using exactly n bits requires Ω(√n) bit inspections in the worst case.

Proof: Model, apply Sunflower Lemma and manipulate

Sunflower Lemma [Ёrdos & Rado]

• Sunflower with p petals– Sets S1, …, Sp so SiSj is same for all i,j

Lemma: S1, …, Sp is a system of sets each of size at most q. If m>(p-1)q+1q!, then it contains a subcollection with p petals

Use the lemma with m as small as possible

Back to Algorithms

• We want a scheme that balances (as well as possible)– Bit changes– Bit inspections– Extra bits

Lower bound says no extra bits lots of inspections

The increment in BRGC

• Key issue … that rightmost 1

Lemma: On increment or decrement, the position of rightmost 1 in any segment of the leftmost bits changes only when that bit is flipped, except for the case of leftmost bit going 1→0 on increment (or 0 → 1 on decrement)

De-amortizing

• The lemma lets us de-amortize: as with redundant binary representation.

With some details:

Theorem: There is a representation using n + lg n + 3 bits (a pointer is in there), requiring 2 lg n +4 bit inspections and 4 bit changes for increment or decrement.

Moving Along

• Complicating issue:Mixing increments and decrements forces us to add flags – (for {inc, dec} and {done yet , not done}

• Idea: use the lower order lg n bits to give the pointer into the leftmost n - lg n bits– This would seem to require a table (or model

issue) to translate

But

• Permute the code of the leading n – lg n bits and use trailing bits directly to walk through them in “code order”

• Indeed, we don’t need n – lg n to be a power of 2

Theorem: There is a representation using n + 3 bits, requiring lg n + 6 bit inspections and 7 bit changes for increment or decrement.

Addition and Subtraction

• Adding or subtracting and m digit (standard notation) integer to update an n digit integer (n > m): – n + O(lg2n) bits, – O(m + lg n) inspections/changes

Open Issues

• Improve the lower bound to n bits inspected in no extra bits

• Lower bound trading off bits changed with space and bits inspected

• Tweaking upper bounds