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Lower Bound Techniques for Data Structures Mihai P ă trașcu … Committee: Erik Demaine (advisor) Piotr Indyk Mikkel Thorup

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Data Structures I don’t study stacks, queues and binary search trees ! I do study data structure problems (a.k.a. Abstract Data Types) predecessor search Preprocess T = { n numbers } pred( q ): max { y є T | y < q } Maintain an array A[ n ] under: update( i, Δ): A[ i ] = Δ sum( i ): return A[0] + … + A[ i ] partial-sums problem

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Motivation? packet forwarding predecessor search Preprocess T = { n numbers } pred( q ): max { y є T | y < q } Maintain an array A[ n ] under: update( i, Δ): A[ i ] = Δ sum( i ): return A[0] + … + A[ i ] partial-sums problem Support both:* list operations – concatenate, split, … * array operations – index

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Binary Search Trees = Upper Bound predecessor search Preprocess T = { n numbers } pred( q ): max { y є T | y < q } Maintain an array A[ n ] under: update( i, Δ): A[ i ] = Δ sum( i ): return A[0] + … + A[ i ] partial-sums problem “Binary search trees solve predecessor search” => Complexity of predecessor ≤ O(lg n)/operation “Augmented binary search trees solve partial sums” =>Complexity of partial sums ≤ O(lg n)/operation my work ≤

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What kind of “lower bound”? Model of computation ≈ real computers: memory words of w > lg n bits(pointers = words) random access to memory any operation on CPU registers (arithmetic, bitwise…) Just prove lower bound on # memory accesses Lower bounds you can trust. TM “Black box” “Array Mem[1.. S ] of w -bit words”

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Why Data Structures? Other settings: streaming L.B. : many not very “computational” mostly storage / info thy space-bounded (P vs L) L.B. : a few, Ω(n √lg n) unnatural questions algebraic L.B. : some cool, but not real computing… depth 3 circuits with mod-6 gates ?? The gospel: data structures L.B. : some understand some nontrivial computational phenomena efficient algorithms circuit L.B. not forthcoming hard optimization NP-completeness L.B. : one per STOC/FOCS I want to understand computation.

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Why Data Structures? Other settings: streaming L.B. : many not very “computational” mostly storage / info thy space-bounded (P vs L) L.B. : a few, Ω(n √lg n) unnatural questions algebraic L.B. : some cool, but not real computing… depth 3 circuits with mod-6 gates ?? The gospel: data structures L.B. : some understand some nontrivial computational phenomena efficient algorithms circuit L.B. not forthcoming hard optimization NP-completeness L.B. : one per STOC/FOCS I want to understand computation. Weak as some of the lower bounds may be, it’s the area that has gotten farthest towards “understanding computation”

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History * [Yao, FOCS’78] [Ajtai’88] -- predecessor (static) [Fredman, Saks’89] -- partial sums, union find (dynamic) *Omitted: bounds for succinct data structures. Observations: huge influence 2 nd papers result wrong (better upper bound known) no journal version; many claims without proof

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History * [Yao, FOCS’78] [Ajtai’88] -- predecessor (static) [Bing Xiao, Stanford’92] ** [Miltersen STOC’94] [Miltersen, Nisan, Safra, Wigderson STOC’95] [Beame, Fich STOC’99] [Sen ICALP’01] (1+ε)-nearest neighbor: [Chakrabarti, Chazelle, Gum, Lvov STOC’99] [Chakrabarti, Regev FOCS’04] [Fredman, Saks’89] -- partial sums, union find (dynamic) [Ben-Amram, Galil FOCS’91] [Miltersen, Subramanian, Vitter, Tamassia’93] [Husfeldt, Rauhe, Skyum’96] [Fredman, Henzinger’98] planar connectivity [Husfeldt, Rauhe ICALP’98] nondeterminism [Alstrup, Husfeldt, Rauhe FOCS’98] marked ancestor [Alstrup, Husfeldt, Rauhe SODA’01] dynamic 2D NN [Alstrup, Ben-Amram, Rauhe STOC’99] union-find === richness lower bounds === [Borodin, Ostrovsky, Rabani STOC’99] p.m. [Barkol, Rabani STOC’00] rand. NN [Jayram,Khot,Kumar,Rabani STOC’03] p.m. [Liu’04] det. ANN *Omitted: bounds for succinct data structures.

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[Yao, FOCS’78] [Ajtai’88] -- predecessor (static) [Bing Xiao, Stanford’92] ** [Miltersen STOC’94] [Miltersen, Nisan, Safra, Wigderson STOC’95] [Beame, Fich STOC’99] [Sen ICALP’01] (1+ε)-nearest neighbor: [Chakrabarti, Chazelle, Gum, Lvov STOC’99] [Chakrabarti, Regev FOCS’04] [Fredman, Saks’89] -- partial sums, union find (dynamic) [Ben-Amram, Galil FOCS’91] [Miltersen, Subramanian, Vitter, Tamassia’93] [Husfeldt, Rauhe, Skyum’96] [Fredman, Henzinger’98] planar connectivity [Husfeldt, Rauhe ICALP’98] nondeterminism [Alstrup, Husfeldt, Rauhe FOCS’98] marked ancestor [Alstrup, Husfeldt, Rauhe SODA’01] dynamic 2D NN [Alstrup, Ben-Amram, Rauhe STOC’99] union-find Three Main Ideas === richness lower bounds === [Borodin, Ostrovsky, Rabani STOC’99] p.m. [Barkol, Rabani STOC’00] rand. NN [Jayram,Khot,Kumar,Rabani STOC’03] p.m. [Liu’04] det. ANN 1. Epochs 3. Round Elimination 2. Asym. Communication, Rectangles

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[Yao, FOCS’78] [Ajtai’88] -- predecessor (static) [Bing Xiao, Stanford’92] ** [Miltersen STOC’94] [Miltersen, Nisan, Safra, Wigderson STOC’95] [Beame, Fich STOC’99] [Sen ICALP’01] (1+ε)-nearest neighbor: [Chakrabarti, Chazelle, Gum, Lvov STOC’99] [Chakrabarti, Regev FOCS’04] [Fredman, Saks’89] -- partial sums, union find (dynamic) [Ben-Amram, Galil FOCS’91] [Miltersen, Subramanian, Vitter, Tamassia’93] [Husfeldt, Rauhe, Skyum’96] [Fredman, Henzinger’98] planar connectivity [Husfeldt, Rauhe ICALP’98] nondeterminism [Alstrup, Husfeldt, Rauhe FOCS’98] marked ancestor [Alstrup, Husfeldt, Rauhe SODA’01] dynamic 2D NN [Alstrup, Ben-Amram, Rauhe STOC’99] union-find Three Main Ideas === richness lower bounds === [Borodin, Ostrovsky, Rabani STOC’99] p.m. [Barkol, Rabani STOC’00] rand. NN [Jayram,Khot,Kumar,Rabani STOC’03] p.m. [Liu’04] det. ANN 1. Epochs 3. Round Elimination 2. Asym. Communication, Rectangles

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Review: Epoch Lower Bounds #updates:r3r3 r2r2 r1r1 r0r0 bits written:t u w∙r 3 t u w∙r 2 t u w∙rtuwtuw update: mark/unmark node [t u ] query: # marked ancestors? [t q ] time epoch j: r j updates epochs {0,.., j-1} write O(t u w∙r j-1 ) bits pick r >> t u w most updates from epoch j not known outside epoch j random query needs to read a cell from epoch j t q = Ω(lg n / lg r) = Ω(lg n / lg(t u w)) max {t q, t u } = Ω(lg n / lglg n)

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Review: Epoch Lower Bounds “Big Challenges” [Miltersen’99] prove some ω(lg n /lglg n ) bound Candidate: Ω(lg n ) for the partial sums problem prove ω(lg n ) in the bit-probe model Maintain an array A[ n ] under: update( i, Δ): A[ i ] += Δ sum( i ): return A[0] + … + A[ i ] See also: [Fredman JACM ’81] [Fredman JACM ’82] [Yao SICOMP ’85] [Fredman, Saks STOC ’89] [Ben-Amram, Galil FOCS ’91] [Hampapuram, Fredman FOCS ’93] [Chazelle STOC ’95] [Husfeldt, Rauhe, Skyum SWAT ’96] [Husfeldt, Rauhe ICALP ’98] [Alstrup, Husfeldt, Rauhe FOCS ’98]

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Our contribution [P., Demaine SODA’04]Ω(lg n ) for partial sums [P., Demaine STOC’04]Ω(lg n ) for dynamic trees, etc. * very simple proof * not based on epochs [P., Tarniţ ă ICALP’05] Ω(lg n ) via epoch argument!! => Ω(lg 2 n /lg 2 lg n ) in the bit-probe model Best Student Paper

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Ω(lg n ) via Epoch Arguments? Old:information about epoch j outside j ≤ #cells written by epochs {0,.., j-1} ≤ O(t u ∙r j-1 ) j

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Ω(lg n ) via Epoch Arguments? New:information about epoch j outside j ≤ #cells read by epochs { 0,.., j -1} from epoch j still≤ O(t u ∙r j-1 ) in the worst case Foil worst-case by randomizing epoch construction! j

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Ω(lg n ) via Epoch Arguments? Foil worst-case by randomizing epoch construction! #cells read by epochs { 0,.., j -1} from epoch j ≤ O ( (t u / #epochs) ∙ r j-1 ) on average => max { t u, t q } = Ω(lg n)

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The “Very Simple Ω(lg n) Proof”

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The hard instance: π = random permutation for t = 1 to n : query: sum(π( t )) Δ t = rand() update(π( t ), Δ t ) π time Δ1Δ1 Δ2Δ2 Δ3Δ3 Δ4Δ4 Δ5Δ5 Δ6Δ6 Δ7Δ7 Δ8Δ8 Δ9Δ9 Δ 10 Δ 11 Δ 12 Δ 13 Δ 14 Δ 15 Δ 16 Maintain an array A[ n ] under: update( i, Δ): A[ i ] = Δ sum( i ): return A[0] + … + A[ i ]

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time Δ1Δ1 Δ2Δ2 Δ3Δ3 Δ4Δ4 Δ5Δ5 Δ6Δ6 Δ7Δ7 Δ8Δ8 Δ9Δ9 Δ 10 Δ 11 Δ 13 Δ 14 Δ 16 Δ 17 Δ 12 t = 5, …, 8 t = 9,…,12 Communication ≈ # memory locations * read during * written during t = 5, …, 8 t = 9,…,12 How can Mac help PC run ?

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time How much information needs to be transferred? Δ7Δ7 Δ8Δ8 Δ9Δ9 Δ1Δ1 Δ1Δ1 Δ2Δ2 Δ3Δ3 Δ4Δ4 Δ5Δ5 Δ 13 Δ 14 Δ 16 Δ 17 Δ1+Δ5+Δ3Δ1+Δ5+Δ3 Δ1+Δ5+Δ3+Δ7+Δ2Δ1+Δ5+Δ3+Δ7+Δ2 Δ1+Δ5+Δ3+Δ7+Δ2 +Δ8 +Δ4Δ1+Δ5+Δ3+Δ7+Δ2 +Δ8 +Δ4 At least Δ 5, Δ 5 +Δ 7, Δ 5 +Δ 7 +Δ 8 => i.e. at least 3 words (random values incompressible)

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The general principle Lower bound = # down arrows How many down arrows? (in expectation) (2 k -1) ∙ Pr[ ] ∙ Pr[ ] = (2 k -1) ∙ ½ ∙ ½ = Ω( k ) k operations

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Recap yellow period pink period Communication = # memory locations * read during * written during Communication between periods of k items = Ω( k ) = Ω( k ) yellow period pink period * read during * written during # memory locations

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Putting it all together time Ω( n /2) Ω( n /4) Ω( n /8) aaaa total Ω( n lg n ) Every load instruction counted lowest_common_ancestor(, ) write timeread time

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Q.E.D. Augmented binary search trees are optimal. First “Ω(lg n)” for any dynamic data structure.

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[Yao, FOCS’78] [Ajtai’88] -- predecessor (static) [Bing Xiao, Stanford’92] ** [Miltersen STOC’94] [Miltersen, Nisan, Safra, Wigderson STOC’95] [Beame, Fich STOC’99] [Sen ICALP’01] (1+ε)-nearest neighbor: [Chakrabarti, Chazelle, Gum, Lvov STOC’99] [Chakrabarti, Regev FOCS’04] [Fredman, Saks’89] -- partial sums, union find (dynamic) [Ben-Amram, Galil FOCS’91] [Miltersen, Subramanian, Vitter, Tamassia’93] [Husfeldt, Rauhe, Skyum’96] [Fredman, Henzinger’98] planar connectivity [Husfeldt, Rauhe ICALP’98] nondeterminism [Alstrup, Husfeldt, Rauhe FOCS’98] marked ancestor [Alstrup, Husfeldt, Rauhe SODA’01] dynamic 2D NN [Alstrup, Ben-Amram, Rauhe STOC’99] union-find Three Main Ideas === richness lower bounds === [Borodin, Ostrovsky, Rabani STOC’99] p.m. [Barkol, Rabani STOC’00] rand. NN [Jayram,Khot,Kumar,Rabani STOC’03] p.m. [Liu’04] det. ANN 1. Epochs 3. Round Elimination 2. Asym. Communication, Rectangles 2. Asym. Communication, Rectangles

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Review: Communication Complexity

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lg S bits w bits lg S bits w bits database => space S Traditional communication complexity: “total #bits communicated ≥ X” => t q ∙(lg S + w) ≥ X => t q = Ω(X/w) But wait ! X ≤ CPU input ≤ O(w) query(a,b,c)

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Review: Communication Complexity lg S bits w bits lg S bits w bits database => space S Asymmetric communication complexity: “either Alice sends A bits or Bob sends B bits” => either t q ∙lg S ≥ A or t q ∙w ≥ B => t q ≥ min { A/lg S, B/w} query(a,b,c)

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Prove: “either Alice sends A bits or Bob sends B bits” Assume Alice sends o(A), Bob sends o(B) => big monochromatic rectangle Show any big rectangle is bichromatic (standard idea in comm. complex.) Richness Lower Bounds Bob Alice 1/2 o(A) 1/2 o(B) output=1 Example:Alice --> q є {0,1} d Bob --> S=n points in {0,1} d Goal: find argmin xєS || x-q || 2 [Barkol, Rabani] A=Ω(d), B=Ω(n 1-ε ) => t q ≥ min { d/lg S, n 1-ε /w }

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Example:Alice --> q є {0,1} d Bob --> S=n points in {0,1} d Goal: find argmin xєS || x-q || 2 [Barkol, Rabani] A=Ω(d), B=Ω(n 1-ε ) => t q ≥ min { d/lg S, n 1-ε /w } Richness Lower Bounds What does this really mean? “optimal space lower bound for constant query time” 1 n 1-o(1) Θ(d/lg n) lower bound S = 2 Ω(d/t q ) upper bound ≈ either: exponential space near-linear query time S tqtq Θ(n) 2 Θ(d) Also: optimal lower bound for decision trees

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Results Partial match -- database of n strings in {0,1} d, query є {0,1,*} d [Borodin, Ostrovsky, Rabani STOC’99] [Jayram,Khot,Kumar,Rabani STOC’03] A = Ω(d/lg n) [P. FOCS’08]A = Ω(d) Nearest Neighbor on hypercube (ℓ 1, ℓ 2 ): deterministic γ-approximate: [Liu’04]A = Ω(d/ γ 2 ) randomized exact: [Barkol, Rabani STOC’00]A = Ω(d) rand. (1+ε)-approx: [Andoni, Indyk, P. FOCS’06]A = Ω(ε -2 lg n) “Johnson-Lindenstrauss space is optimal!” Approximate Nearest Neighbor in ℓ ∞ : [Andoni, Croitoru, P. FOCS’08] “[Indyk FOCS’98] is optimal!” simplify

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Limits of Communication Approach No separation between S =O( n ) and S = n O(1) ! lg S bits w bits branching programs “ Alice must send Ω(A) bits” => t q = Ω(A / lg S) => t q = Ω ( A / lg(Sd/n) ) 1 n 1-o(1) Θ(d/lg n) S tqtq Θ(n) 2 Θ(d) Θ(d/lg d) Separation of Ω(lg n / lglg n ) between S =O( n ) and S = n O(1) ! Implication of richness lower bound undervalued !

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CPU(s) -- > memory communication: one query: lg S k queries: lg ( ) =Θ ( k lg ) SkSk SkSk Richness Gets You More

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SkSk SkSk Prob.1 Prob.2 Prob.3 Prob.k CPU(s) -- > memory communication: one query: lg S k queries: lg ( ) =Θ ( k lg )

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Any richness lower bound “Alice must send A or Bob must send B” ===> k∙Alice must send k∙A or k∙Bob must send k∙B SkSk SkSk Prob.1 Prob.2 Prob.3 Prob.k Richness Gets You More CPU(s) -- > memory communication: one query: lg S k queries: lg ( ) =Θ ( k lg ) Direct Sum

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Any richness lower bound “Alice must send A or Bob must send B” ===> k∙Alice must send k∙A or k∙Bob must send k∙B SkSk SkSk Richness Gets You More CPU(s) -- > memory communication: one query: lg S k queries: lg ( ) =Θ ( k lg ) Direct Sum t q = Ω(A / lg(S/k))

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Three Main Ideas [Yao, FOCS’78] [Ajtai’88] -- predecessor [Bing Xiao, Stanford’92] [Miltersen STOC’94] [Miltersen, Nisan, Safra, Wigderson STOC’95] [Beame, Fich STOC’99] [Sen ICALP’01] (1+ε)-nearest neighbor: [Chakrabarti, Chazelle, Gum, Lvov STOC’99] [Chakrabarti, Regev FOCS’04] [Fredman, Saks’89] - partial sums, union find [Ben-Amram, Galil FOCS’91] [Miltersen, Subramanian, Vitter, Tamassia’93] [Husfeldt, Rauhe, Skyum’96] [Fredman, Henzinger’98] planar connectivity [Husfeldt, Rauhe ICALP’98] nondeterminism [Alstrup, Husfeldt, Rauhe FOCS’98] marked ancestor [Alstrup, Husfeldt, Rauhe SODA’01]dynamic 2D NN [Alstrup, Ben-Amram, Rauhe STOC’99] union-find === richness lower bounds === [Borodin, Ostrovsky, Rabani STOC’99] p.m. [Barkol, Rabani STOC’00] rand. NN [Jayram,Khot,Kumar,Rabani STOC’03] p.m. [Liu’04] det. ANN 1. Epochs 3. Round Elimination 4. Range Queries 2. Asym. Communication, Rectangles

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Open Hunting Season Nice trick, but “Ω(lg n / lglg n ) with O( n polylg n ) space” not impressive argument for “curse of dimensionality” But space n 1+o(1) is hugely important in data structures => open hunting season for range queries etc. SELECT count(*) FROM employees WHERE salary <= AND startdate <= D range counting

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Open Hunting Season SELECT count(*) FROM employees WHERE salary <= AND startdate <= D range counting [P. STOC’07] Ω(lg n / lglg n ) with O( n polylg n ) space N.B. : tight! 1 st bound beyond the semigroup model question from [Fredman JACM’82 ] [Chazelle FOCS’86 ]

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The Power of Reductions SELECT count(*) FROM employees WHERE salary <= AND startdate <= D range counting Preprocess S={ n rectangles} stab(x,y): is (x,y) inside some RєS? 2D stabbing routing ACLs dispatching in some OO languages

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The Power of Reductions SELECT count(*) FROM employees WHERE salary <= AND startdate <= D range counting Preprocess S={ n rectangles} stab(x,y): is (x,y) inside some RєS? 2D stabbing +1

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The Power of Reductions Preprocess S={ n rectangles} stab(x,y): is (x,y) inside some RєS? 2D stabbing reachability oracles in butterfly graph Preprocess G = subgraph of butterfly reachable(x,y): is there a path x->y ?

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The Power of Reductions Preprocess S={ n rectangles} stab(x,y): is (x,y) inside some RєS? 2D stabbing reachability oracles in butterfly graph Preprocess G = subgraph of butterfly reachable(x,y): is there a path x->y ?

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The Power of Reductions Alice: set SBob: set T “are S and T disjoint?” Lopsided Set Disjointness reachability oracles in butterfly graph Preprocess G = subgraph of butterfly reachable(x,y): is there a path x->y ? Hint: S = {one edge out of every node} => n queries from 1 st to last level T = {deleted edges} S disjoint from T => all queries “yes”

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Reachability in Butterfly?? update (node): (un)mark node query (leaf): any marked ancestor? marked ancestor problem

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lopsided set disjointness (LSD) dyn. marked ancestor reachability oracles in the butterfly partial match(1+ε)-ANN ℓ 1, ℓ 2 NN in ℓ 1, ℓ 2 3-ANN in ℓ ∞ worst-case union-find dyn. trees, graphs dyn. 1D stabbing partial sums 2D stabbing 4D reporting2D counting dyn. NN in 2D dyn. 2D reporting [P. FOCS’08]

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Three Main Ideas [Yao, FOCS’78] [Ajtai’88] -- predecessor [Bing Xiao, Stanford’92] [Miltersen STOC’94] [Miltersen, Nisan, Safra, Wigderson STOC’95] [Beame, Fich STOC’99] [Sen ICALP’01] (1+ε)-nearest neighbor: [Chakrabarti, Chazelle, Gum, Lvov STOC’99] [Chakrabarti, Regev FOCS’04] [Fredman, Saks’89] - partial sums, union find [Ben-Amram, Galil FOCS’91] [Miltersen, Subramanian, Vitter, Tamassia’93] [Husfeldt, Rauhe, Skyum’96] [Fredman, Henzinger’98] planar connectivity [Husfeldt, Rauhe ICALP’98] nondeterminism [Alstrup, Husfeldt, Rauhe FOCS’98] marked ancestor [Alstrup, Husfeldt, Rauhe SODA’01]dynamic 2D NN [Alstrup, Ben-Amram, Rauhe STOC’99] union-find === richness lower bounds === [Borodin, Ostrovsky, Rabani STOC’99] p.m. [Barkol, Rabani STOC’00] rand. NN [Jayram,Khot,Kumar,Rabani STOC’03] p.m. [Liu’04] det. ANN 1. Epochs 3. Round Elimination 4. Range Queries 2. Asym. Communication, Rectangles

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Packet Forwarding/ Predecessor Search Preprocess n prefixes of ≤ w bits: make a hash-table H with all prefixes of prefixes | H |=O( n∙w ), can be reduced to O( n ) Given w -bit IP, find longest matching prefix: binary search for longest ℓ such that IP[0: ℓ ] є H [van Emde Boas FOCS’75] [Waldvogel, Varghese, Turener, Plattner SIGCOMM’97] [Degermark, Brodnik, Carlsson, Pink SIGCOMM’97] [Afek, Bremler-Barr, Har-Peled SIGCOMM’99] O(lg w )

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Review: Round Elimination hi lo hash(hi) 0/1 † † 0: continue searching for pred(hi) 1: continue searching for pred(lo) 1 k 2 o(k) bits I want to talk to Alice i Message has negligible info about the typical i => can be eliminated for fixed i

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The Lemma Observe: can’t work worst-case! Traditional fix: introduce 2-sided error Think outside the : easy proof with a different error model [P.-Thorup, STOC’06]

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The Model Alice, Bob receive inputs they may reject inputs if they accept, they start communicating and must produce a correct output The point: error probability ½ is trivial reject probability is still hard “We regret to inform you that your input has not been accepted for communication. We receive a large number of inputs, many of them of high quality, and scheduling constraints unfortunately make it impossible to accept all of them.”

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The Proof Trie for Alice’s input (x 1, …, x k ) leaves = message sent node = set of msgs in subtree Say msg size is m=k/2: |leaf|=1, |root|=2 k/2 => ( ∀ )root-to-leaf path, ½ of nodes have |node|≥ ½|parent| averaging over node-child pairs => ( ∃ ) node: ½ its children have |child|> ½|node| thus ( ∃ ) msg M: ¼ of children have M є child m1m1 m3m3 m2m2 m1m1 m3m3 m1m1 m1m1 m2m2 m2m2 x3x3 x2x2 x1x1 {m 1,m 2 } {m 1,m 2, m 3 } fix i, x 1, …, x i-1 fixed message (eliminate) reject ¾ of inputs (x i )

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Predecessor Search: Timeline after [van Emde Boas FOCS’75] … O(lg w ) has to be tight! [Beame, Fich STOC’99] slightly better bound with O( n 2 ) space … must improve the algorithm for O( n ) space! [P., Thorup STOC’06] tight Ω(lg w ) for space O( n polylg n ) ! Idea: * consider multiple queries * prove round elimination under direct sum

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Predecessor Search: Timeline Idea: * consider multiple queries * prove round elimination under direct sum 1k k 1k I want to talk to Alice 2 I want to talk to Alice 1 I want to talk to Alice 2

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Round Eliminated !

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The End …or Champagne? Questions?

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The Partial Sums Problem Textbook solution: “augmented” binary search trees Running time: O(lg n ) / operation Maintain an array A[ n ] under: update( i, Δ): A[ i ] += Δ sum( i ): return A[0] + … + A[ i ]

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