Atomic snapshots in O (log³ n) steps using randomized helping

Atomic snapshots in O(log³ n) steps using randomized helping James Aspnes, Yale Keren Censor-Hillel, Technion

Snapshot Objects … update(v) scan read all locations update your location p1 p2 pn …

Model R1 R2 R … read v write(v) ok p1 p2 pn … System of nprocesses, multi-writer registers Asynchronous schedule controlled by an adversary Crash failures – require wait-free implementations Linearizable implementations

Snapshots - Step Complexity Using multi-writer registers: can be done in O(n)steps [Inoue and Chen, WDAG 1994] and requires Ω(n) steps [Jayanti, Tan, and Toueg, SICOMP 1996] Goal: a faster snapshot implementation (polylog) Limited-use: O(log3(n)) steps per operation, for polynomially many update operations [Aspnes, Attiya, Censor-Hillel, and Ellen, PODC 2012] This Talk: O(log3(n)) steps per operation, with high probability(without a usage bound)

Tree structure, Updates help Scans [Aspnes, Attiya, Censor-Hillel, and Ellen, PODC 2012] O(log n) steps? Array of views Pointer to array location X 5 2 0 1 4 3 5 s1+...+s4 1 2 0 0 2 3 1 s1+s2 s3+s4 s1 s2 s3 s4

Two Challenges 1. Coping with slow operations. Max-register: returns largest value previously written [Aspnes, Attiya, and Censor-Hillel, JACM 2012] Consecutive values differ by at most n 5 s1+...+s4 s1+s2 s3+s4 s1 s2 s3 s4

Two Challenges 2. Guaranteeing consistent views. Max-array: returns comparable pairs of max-register values [Aspnes, Attiya, Censor-Hillel, and Ellen, PODC 2012] 5 s1+...+s4 s1+s2 s3+s4 s1 s2 s3 s4

Our Results Randomized max-register inO(logn) steps with high probability Randomized 2-component max-array O(log2n) steps whp Randomized snapshot in O(log3n) steps whp Main technique: randomized helping

Max-Register switch = 0 switch = 1 read switch = 0 switch = 0 … … value 0 value 1 value v value v’ … write(v)

Max-Register switch = 0 switch = 1 read switch = 0 switch = 1 switch = 0 … … value 0 value 1 value v value v’ … write(v) write(v’) write(v’’)

Randomized Max-Register switch = 0 switch = 1 switch = 0 switch = 1 … m-valued max register switch = 0 … write(v)

Randomized Max-Register switch = 0 switch = 1 k-bounded increments:value of write ≤ k + value of largest write switch = 0 switch = 1 … m-valued max register switch = 0 … O(log m + kn/m) = O(log n) steps per write write(v)

Writing to the Max-Register pj(cyclic) 1 TS: read 1 v’ = max(returned value, v) read write(v) pi … HELP: 0 v', ts[j]TS[j] (random) … size n3 POINTER: i write(v)

Reading the Max-Register pj read 1 TS: +1 1 read (logarithmic no. of steps) pi … HELP: 0 read (returns v, ts[j]) Main argument: many read stepsmany fresh values in pointer array, whp (random) … size n3 POINTER: read (returns i) if ts[j]==TS[j] return v

2-Component Max Array write1(v) write0(v) update second max-register update first max-register read read both max-registers

MaxArray Implementation Max1 Max1 switch = 0 write1(v) read switch = 0 Max1 Max1 switch = 0 Max1 … … value 0 Max1 value 1 Max1 value v value v’ Max1 Max1 Max1 … write0(v)

Randomized 2-Component Max Array Problem: readers do not travel all the way from root to value switch = 0 switch = 1 Max1 Max1 Max1 switch = 1 switch = 0 Max1 Max1 Max1 Max1 Max1 … Max1 switch = 0 Max1 Max1 Solution: read Max at root instead of Max at previous location … Max1 write(v)

Summary Randomized snapshot in O(log3n) steps whp Main Technique: Randomized helping Open problems: • Snapshot implementations using single-writer registers • Additional randomized implementations • Randomized lower bounds Thank you! Questions?

Atomic snapshots in O (log³ n) steps using randomized helping