Comparison Based Dictionaries: Fault Tolerance versus I/O Efficiency Gerth Stølting Brodal Allan Grønlund Jørgensen Thomas Mølhave University of Aarhus ADS 2007, 3rd Bertinoro Workshop on Algorithms and Data Structures University Residential Centre of Bertinoro, Italy, September 30-October 5, 2007

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 2 Binary Searching Fault tolerance I/O Efficiency This talk Future work

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 3 Search(17) O(log N) comparisons

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 4 Search(17) ? 9 soft memory error

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 5 Faulty-Memory RAM Model Finocchi and Italiano, STOC’04  Content of memory cells can get corrupted  Corrupted and uncorrupted content cannot be distinguished  O(1) safe registers  Assumption: At most δ corruptions  Example: Sorting requires time Θ(N·log N+δ 2 ) Finocchi, Grandoni, Italiano, ICALP‘06

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 6 Faulty-Memory RAM: Searching  Lower bound  Upper bound Θ(log N + δ) comparisons Finocchi, Grandoni, Italiano, ICALP’06 Brodal, Fagerberg, Finocchi, Grandoni, Italiano, Jørgensen, Moruz, Mølhave, ESA’07

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 7 Faulty-Memory RAM: Searching 17? Problem? High confidenceLow confidence Requirement: If there exists an uncorrupted element equal to the search key, we should find such an element

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 8 Faulty-Memory RAM: Searching When are we done ( δ=3 )? Contradiction, i.e. at least one fault If range contains at least δ+1 and δ+1 then there is at least one uncorrupted and, i.e. x must be contained in the range

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 9 If verification fails → contradiction, i.e. ≥1 memory-fault → ignore 4 last comparisons → backtrack one level of search Faulty-Memory RAM: Θ(log N + δ) Searching Brodal, Fagerberg, Finocchi, Grandoni, Italiano, Jørgensen, Moruz, Mølhave, ESA’07

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 10 Faulty-Memory RAM: Θ(log N + δ) Searching  Standard binary search + verification steps  At most δ verification steps can fail/backtracking  Detail: Avoid repeated comparison with the same (wrong) element by grouping elements into blocks of size O(δ) Brodal, Fagerberg, Finocchi, Grandoni, Italiano, Jørgensen, Moruz, Mølhave, ESA’07

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 11 Faulty-Memory RAM: Reliable Values  Store 2δ+1 copies of value x - at most δ copies uncorrupted  x = majority  Time O(δ) using two safe registers (candidate and count) δ=5 y y y x x y x x x y x Candidate y y y y y y y – x – x Count Boyer and Moore ‘91

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 12 Faulty-Memory RAM: Dynamic Dictionaries  Packed array  Reliable pointers and keys  Updates O(δ ·log 2 N)  Searches = fault tolerant O(log N+δ)  2-level buckets of size O(δ·log N)  Root: Reliable pointers and keys  Bucket search/update amortized O(log N+δ)... Θ(δ·log N) elements Itai, Konheim, Rodeh, 1981  Search and update amortized O(log N+δ)... Θ(δ) elements Brodal, Fagerberg, Finocchi, Grandoni, Italiano, Jørgensen, Moruz, Mølhave, ESA’07

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 13 I/O Model

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 14 I/O Model  N = problem size  M = memory size  B = I/O block size  One I/O moves B consecutive records from to disk  Complexity = number of I/Os CPU External I/O Memory y r o m e M Aggarwal and Vitter 1988  Example: Sorting requires I/Os

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 15 B-trees O(log B N).... Ω(B)Ω(B) Search path  Search and update O(log B N)

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 16 Fault-Tolerance versus I/O Efficiency

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 17 Lower Bound for Fault-Tolerant External Searching  Adversary argument  If B ε slabs per I/O → factor B ε reduction and B 1-ε faults  After k I/Os N/(B ε ) k –k· B 1-ε elements remain Possible values  I/Os required [minimized wrt ε ]

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 18 Randomized Upper Bound for Fault-Tolerant External Searching  Sorted array + 2δ identical B-trees (over N/(2δ) elements, stored in BFS layout)  Search: Select random tree for each node on search path + verification  Probability no faults on path: where Σ β i ≤δ  Search O(log B N+δ/B) expected....

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 19  Sorted array + 2δ/B 1-ε identical B-trees of degree B ε + B 1-ε copies of each key + min/max  Search: Verify against min/max in each step – if fail, backtrack one level and advance to next copy  Search I/Os Deterministic Upper Bound for Fault-Tolerant External Searching

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 20 Dynamic Fault-Tolerant External Dictionaries Static structure + Packed arrays + Buckets of size O(δ ·log 3 N)  Deterministic I/Os search and updates  Randomized Expected O(log B N+δ/B) I/Os search and updates ... Static

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 21  Fault-tolerant external memory searching I/Os worst-case [minized wrt ε ]  Randomized O(log B N+δ/B) I/Os Conclusion

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 22 Future Work Fault Tolerance versus I/O Efficiency  Randomized algorithms: Memory faults in internal memory?  Sorting: ? ...

Dictionaries: Fault Tolerance versus I/O Efficiency Brodal, Jørgensen, Mølhave 23 THANKS