Sorting and searching in the presence of memory faults (without redundancy) Irene Finocchi Giuseppe F. Italiano DISP, University of Rome “Tor Vergata”
The problem Large, inexpensive and error-prone memories Classical algorithms may not be correct in the presence of (even very few) memory faults A B Out An example: merging two ordered lists (n) (n 2 ) inversions
Faulty- memory model Memory fault = the correct value stored in a memory location gets altered (destructive faults) Fault appearance At any time At any memory location Simultaneously Faulty Random Access Machine: –O(1) words of reliable memory –Corrupted values indistinguishable from correct ones Fault-tolerant algorithms = able to get a correct output (at least) on the set of uncorrupted values
Related work LiesTransient failures Algorithms can exploit query replication strategies The liar model: comparison questions answered by a possibly lying adversary [Ulam 77, Renyi 76] At most k lies Linearly bounded model (n log n + k n) [Lakshmanan et al., IEEE TOC 91] O(n log n) for k = O (log n / log log n) [Ravikumar, COCOON 02] (n log (n/q)), correct with probability (1-q) [Feige et al., SICOMP 94] Exponential lower bound [Borgstrom & Kosaraju, STOC 93] Probabilistic model
Why not data replication? Data replication can be quite inefficient in certain highly dynamic scenarios, especially if objects to be replicated are large and complex What can we do without data replication? Q1. Can we sort the correct values in the presence of, e.g., polynomially many memory faults? Q2. How many faults can we tolerate in the worst case if we wish to maintain optimal time and space? E.g., with respect to sorting:
A fault tolerant algorithm We show an algorithm resilient up to O ( (n log n) 1/3 ) memory faults Based on mergesort Main difficulty: merging step Can we sort (at least) the correct values on O(n log n) time and optimal space in the presence of, e.g., polynomially many memory faults? Q1.
A hierarchy of disorder k-unordered faithfully ordered ordered k-weakly fault tolerant strongly fault tolerant faithfully ordered = ordered except for the corrupted keys k-unordered = ordered except for k (correct or corrupted) keys 3-unordered
Solving a relaxation solve the k-weakly FT merging problem (for k not too large) and use it to reconstruct a faithful order Idea of our merging algorithm: k-unordered faithfully ordered ordered
The merging algorithm: a big picture AB k-weaklyFT-merge C E naïf-mergesort stronglyFT-merge F purify SD Slow, but D is short... Slow in general, fast on unbalanced sequences Faithfully ordered, long Disordered, short Very fast Faithfully ordered, short Faithfully ordered k-unordered, but k is not so large Very fast
The merging algorithm: a big picture AB k-weaklyFT-merge C E naïf-mergesort stronglyFT-merge F purify SD O( )-unordered, ≤ O(n) O(n+ ) Faithfully ordered, long |D| = O( ) O (n+ ) Faithfully ordered, short Faithfully ordered O ()O () Running time O (n+ ) Strongly fault tolerant
Summing up We obtain an O(n log n) strongly fault tolerant sorting algorithm that is resilient up to O(n log n) 1/3 memory faults and uses O(n) space By plugging in the merging algorithm into mergesort, we can sort in time O (n log n+ ) and thus:
A (polynomial) lower bound No more than O ( (n log n) 1/2 ) To prove this, we first prove a lower bound on fault tolerant merging: If n 2/(3-2 , for some [0,1/2], then (n+ 2- ) comparisons are necessary for merging two faithfully ordered n-length lists We use an adversary based argument How many faults can we tolerate in the worst case maintaining space and running time optimal? Q2.
Adversary-based argument: big picture If Paul asks less than 2- /2 questions, than he cannot determine the correct faithful order univocally Carole’s power: doesn’t need to choose the sequences in advance, can play with memory faults Carole’s limits: if challenged at any time, she must exhibit two input sequences and at most memory faults that prove that her answers were consistent with the memory image Paul (the merging algorithm) asks comparison questions of the form “x<y?” Carole (the adversary) must answer consistently Delphi’s Oracle
Carole’s strategy A and B: n-length faithfully ordered sequences A B A1A1 A2A2 A 1 AA... B1B1 B2B2 B 1 BB... n/ Carole answers as if the sorted sequence were: A 1 B 1 A 2 B 2 … B -1 A B
Sparse sets If n 2/(3-2 , for some [0,1/2], and Paul asks less than 2- /2 questions, then sparse set S containing two elements a A i S and b B j S that have not been directly compared by Paul We prove that both the order a < b and the order b < a can be consistent with Carole’s answers A set S of consecutive subsequences is sparse if the number of comparisons between elements in S is at most /2 A 1 B 1 A 2 B 2 A 3 B 3 … B -1 A B S
Paul’s dilemma: a < b or b < a ? For each such question asked by Paul, Carole asserts that she answered after corrupting x (if x ≠ a,b) and y (if y ≠ a,b) How many memory faults introduced by Carole? –1 or 2 faults only if x and y are both in S –S sparse (at most /2 comparisons) at most /2) = faults All possibly useful elements are corrupted! a and b not directly compared: but why Paul can’t deduce their order by transitivity? “x<y?” is a possibly “useful” comparison if both x and y S a x b y ≤ ≤ ≤ x y
Implications If n 2/(3-2 , for some [0,1/2], then Paul must ask at least 2- /2 questions to merge A and B (n+ 2- ) comparisons for merging (n log n+ 2- ) comparisons for sorting If n 2/(3-2 ?
n 0 What if is large? If n 2/(3-2 , (n log n+ 2- ) comparisons for sorting (n log n) 1/2 n 2/3 (n log n+ 2 ) (n log n) 2/3 n (n log n+ 3/2 ) An O(n log n) strongly fault tolerant sorting algorithm cannot be resilient to (n log n) 1/2 memory faults (n log n) 6/11 n 3/4 (n log n+ 11/6 )
Open questions Can randomization help (e.g., to tolerate more than (n log n) 1/2 memory faults) ? Closing the gap: –our algorithm is resilient to O(n log n) 1/3 memory faults –no optimal algorithm can be resilient to (n log n) 1/2 memory faults External fault-tolerant sorting