dynamic data structures Shachar Lovett IAS Ely Porat Bar Ilan University Synergies in lower bounds June 2011 Information theoretic lower bounds Information theory is a powerful tool to prove lower bounds eg in data structures ID: 1003042
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1. Lower bounds for approximate membershipdynamic data structuresShachar LovettIASEly PoratBar-Ilan UniversitySynergies in lower bounds, June 2011
2. Information theoretic lower boundsInformation theory is a powerful tool to prove lower bounds, e.g. in data structuresStudy size of data structure (unlimited access)Static d.s.: pure information theoryDynamic d.s.: communication game
3. Talk overviewApproximate set membership problemBloom filters (simple near-optimal solution)Lower bounds – static caseNew dynamic lower bounds
4. Talk overviewApproximate set membership problemBloom filters (simple near-optimal solution)Lower bounds – static caseNew dynamic lower bounds
5. Approximate set membershipLarge universe URepresent subset S UQuery: is x S?Data structure representing S approximately:If x S: answer YES alwaysIf x S: answer NO with high probabilityWhy approximately? To save spaceUS~S
6. ApplicationsStorage (or communication) is costly, but a small false positive error can be toleratedOriginal applications (70’s): dictionaries, databases – Bloom filtersNowadays: mainly network applications
7. Talk overviewApproximate set membership problemBloom filters (simple near-optimal solution)Lower bounds – static caseNew dynamic lower bounds
8. Bloom filtersS={x1,x2,…,xn}0000000000Bit array of length mHash function h:U {1,…,m}
9. Bloom filtersS={x1,x2,…,xn}0001000000Bit array of length mHash function h:U {1,…,m}h(x1)=4
10. Bloom filtersS={x1,x2,…,xn}1001000000Bit array of length mHash function h:U {1,…,m}h(x2)=1
11. Bloom filtersS={x1,x2,…,xn}1001000000Bit array of length mHash function h:U {1,…,m}h(x3)=4
12. Bloom filtersS={x1,x2,…,xn}1001000000Bit array of length mHash function h:U {1,…,m}Query: y S?
13. Bloom filtersS={x1,x2,…,xn}1001000000Bit array of length mHash function h:U {1,…,m}Query: y S?h(y)=3
14. Bloom filtersS={x1,x2,…,xn}1001000000Bit array of length mHash function h:U {1,…,m}Query: y S? NOh(y)=3
15. Bloom filters: analysisS={x1,x2,…,xn}Query: y S?If y S: returns YES alwaysIf y S: returns NO with probability Error ½: Error : (repetition)1001000000Bit array of length mhash
16. Known boundsUpper bounds (e.g. algorithms)Bloom filter:Improvements: [Porat-Matthias’03, Arbitman-Naor-Segev’10] Lower bounds:information theoretic:Can be matched by static data structures [Charles-Chellapilla’08,Dietzfelbinger-Pagh’08,Porat’08]This work: dynamic d.s.
17. Talk overviewApproximate set membership problemBloom filters (simple near-optimal solution)Lower bounds – static caseNew dynamic lower bounds
18. Static lower boundsStatic settings: insert + queryYao’s min-max principle: prove lower bound for deterministic data structure, randomized inputsInsert: x1,…,xnm bitsQuery: y
19. Static lower boundsDeterministic data structure: compression maps all sets to a small family of setsInput: random set Accept set: Properties:Small memory: No false negatives:Few false positives:Optimal setting:Insert: x1,…,xnm bitsQuery: y
20. Static lower boundsInsert: x1,…,xnm bitsQuery: yUSA(S) Set S, Represented byGoal: show #A(S) large
21. Static lower boundsProperties:Assume thatIf then General case: convexityInsert: x1,…,xnm bitsQuery: y
22. Talk overviewApproximate set membership problemBloom filters (simple near-optimal solution)Lower bounds – static caseNew dynamic lower bounds
23. Dynamic lower boundsBasic dynamic settings: two inserts + queryBreak inputs to k, n-k chunksInsert: x1,…,xkm bitsInsert: xk+1,…,xnm bitsQuery: y
24. Dynamiclower boundsAccepting sets:Properties: General approach: analyze size of accepting setsSets A(x1,…,xk) can’t be too small (covering)Sets A(A(x1,…,xk),xk+1,…,xn) can’t be too large (error)These yield the trivial lower bound again… Insert: x1,…,xkm bitsInsert: xk+1,…,xnm bitsQuery: y
25. Dynamiclower boundsMethod of typical inputsOn a typical input:A(x1,…,xk) not too smallA(A(x1,…,xk),xk+1,…,xn) not too largeInputs uncorrelated with data structure: Yields an improved lower bound (note: “typical” can be 1% of inputs)Insert: x1,…,xkm bitsInsert: xk+1,…,xnm bitsQuery: y
26. Dynamic lower boundsInsert: x1,…,xkm bitsInsert: xk+1,…,xnm bitsQuery: yFunctional inequality:Free parameter: k – how to break inputOptimal choice:Extension: break input into more partsDoesn’t seem to help much
27. SummaryApproximate membership problemStatic algorithms match static information theoretic lower bound: This work: new dynamic information theoretic lower boundTHANK YOU!