From Independence to Expansion and Back Again Tobias Christiani, - PowerPoint PPT Presentation

From Independence to Expansion and Back Again Tobias Christiani, Rasmus Pagh Mikkel Thorup IT University of Copenhagen University of Copenhagen From Independence to Expansion and Back Again 1

Introduction Topic of this talk: • Upper bounds on the space-time tradeoff of k -independent functions in the word RAM model From Independence to Expansion and Back Again 2

Introduction Topic of this talk: • Upper bounds on the space-time tradeoff of k -independent functions in the word RAM model Definition A family of functions F from [ u ] to [ r ] is k -independent if for every set of k distinct keys x 1 , x 2 , . . . , x k ∈ [ u ] and k values y 1 , y 2 , . . . , y k ∈ [ r ] we have that f 2 F [ f ( x 1 ) = y 1 , f ( x 2 ) = y 2 , . . . , f ( x k ) = y k ] = r � k Pr From Independence to Expansion and Back Again 2

Introduction Topic of this talk: • Upper bounds on the space-time tradeoff of k -independent functions in the word RAM model Definition A family of functions F from [ u ] to [ r ] is k -independent if for every set of k distinct keys x 1 , x 2 , . . . , x k ∈ [ u ] and k values y 1 , y 2 , . . . , y k ∈ [ r ] we have that f 2 F [ f ( x 1 ) = y 1 , f ( x 2 ) = y 2 , . . . , f ( x k ) = y k ] = r � k Pr • Example of a k -independent function: k � 1 a i x i mod p X f ( x ) = i =0 From Independence to Expansion and Back Again 2

Introduction Topic of this talk: • Upper bounds on the space-time tradeoff of k -independent functions in the word RAM model Definition A family of functions F from [ u ] to [ r ] is k -independent if for every set of k distinct keys x 1 , x 2 , . . . , x k ∈ [ u ] and k values y 1 , y 2 , . . . , y k ∈ [ r ] we have that f 2 F [ f ( x 1 ) = y 1 , f ( x 2 ) = y 2 , . . . , f ( x k ) = y k ] = r � k Pr • Example of a k -independent function: k � 1 a i x i mod p X f ( x ) = i =0 • Tradeoff: – Space used to represent f ∈ F – Time used to evaluate f ∈ F From Independence to Expansion and Back Again 2

Space-time tradeoff: Lower and upper bounds Lower bound Theorem [Siegel’89] A data structure for representing a k -independent function f : [ u ] → [ r ] with evaluation time t < k must use at least ku 1 /t words of space Domain Memory Probes f ( x ) x . . . t . . . . . . From Independence to Expansion and Back Again 3

Space-time tradeoff: Lower and upper bounds Lower bound Theorem [Siegel’89] A data structure for representing a k -independent function f : [ u ] → [ r ] with evaluation time t < k must use at least ku 1 /t words of space Main result (vanilla version) Randomized data structure for representing a k -independent function f : [ u ] → [ r ] with a space usage of O ( ku 1 /t t ) and evaluation time O ( t log t ) From Independence to Expansion and Back Again 3

Space-time tradeoff: Lower and upper bounds Lower bound Theorem [Siegel’89] A data structure for representing a k -independent function f : [ u ] → [ r ] with evaluation time t < k must use at least ku 1 /t words of space Main result (vanilla version) Randomized data structure for representing a k -independent function f : [ u ] → [ r ] with a space usage of O ( ku 1 /t t ) and evaluation time O ( t log t ) Previous results Reference Space Time O ( k ) O ( k ) Polynomials [Joffe’74] O ( k t u 1 /t ) O ( t ) t Graph powering [Siegel’89] O (poly k + u 1 /t ) O ( t log t ) Recursive tabulation [Thorup’13] From Independence to Expansion and Back Again 3

From expansion to independence • Constructions of k -independent families of functions based on bipartite expander graphs U • Neighbor function Γ : U → V d V . . . d From Independence to Expansion and Back Again 4

From expansion to independence • Constructions of k -independent families of functions based on bipartite expander graphs U • Neighbor function Γ : U → V d V . Definition A bipartite graph Γ is k -unique if . . d for every S ⊆ U with | S | ≤ k there exists v ∈ V with exactly one neighbor in S From Independence to Expansion and Back Again 4

From expansion to independence • Constructions of k -independent families of functions based on bipartite expander graphs U • Neighbor function Γ : U → V d x 1 V Definition A bipartite graph Γ is k -unique if for every S ⊆ U with | S | ≤ k there exists x 2 v ∈ V with exactly one neighbor in S x 3 From Independence to Expansion and Back Again 4

From expansion to independence • Constructions of k -independent families of functions based on bipartite expander graphs U • Neighbor function Γ : U → V d x 1 V v 1 Definition A bipartite graph Γ is k -unique if for every S ⊆ U with | S | ≤ k there exists x 2 v 2 v ∈ V with exactly one neighbor in S x 3 v 3 v 4 v 5 From Independence to Expansion and Back Again 4

From expansion to independence • Constructions of k -independent families of functions based on bipartite expander graphs U • Neighbor function Γ : U → V d x 1 V v 1 Definition A bipartite graph Γ is k -unique if for every S ⊆ U with | S | ≤ k there exists x 2 v 2 v ∈ V with exactly one neighbor in S x 3 v 3 v 4 v 5 From Independence to Expansion and Back Again 4

From expansion to independence • Constructions of k -independent families of functions based on bipartite expander graphs U • Neighbor function Γ : U → V d x 1 V v 1 Definition A bipartite graph Γ is k -unique if for every S ⊆ U with | S | ≤ k there exists x 2 v 2 v ∈ V with exactly one neighbor in S x 3 v 3 v 4 v 5 Lemma [Siegel’89] Let Γ : U → V d be k -unique and h : V → [ r ] be a random function. Then f ( x ) = P i h ( Γ ( x ) i ) mod r defines a k -independent family of functions From Independence to Expansion and Back Again 4

From expansion to independence • Constructions of k -independent families of functions based on bipartite expander graphs U • Neighbor function Γ : U → V d x 1 V v 1 Definition A bipartite graph Γ is k -unique if for every S ⊆ U with | S | ≤ k there exists x 2 v 2 v ∈ V with exactly one neighbor in S x 3 v 3 v 4 • Peeling argument example: v 5 f ( x 3 ) = h ( v 2 ) + h ( v 4 ) + h ( v 5 ) mod r Lemma [Siegel’89] Let Γ : U → V d be k -unique and h : V → [ r ] be a random function. Then f ( x ) = P i h ( Γ ( x ) i ) mod r defines a k -independent family of functions From Independence to Expansion and Back Again 4

From expansion to independence • Constructions of k -independent families of functions based on bipartite expander graphs U • Neighbor function Γ : U → V d x 1 V v 1 Definition A bipartite graph Γ is k -unique if for every S ⊆ U with | S | ≤ k there exists x 2 v 2 v ∈ V with exactly one neighbor in S v 3 v 4 • Peeling argument example: v 5 f ( x 3 ) = h ( v 2 ) + h ( v 4 ) + h ( v 5 ) mod r Lemma [Siegel’89] Let Γ : U → V d be k -unique and h : V → [ r ] be a random function. Then f ( x ) = P i h ( Γ ( x ) i ) mod r defines a k -independent family of functions From Independence to Expansion and Back Again 4

From expansion to independence • Constructions of k -independent families of functions based on bipartite expander graphs U • Neighbor function Γ : U → V d x 1 V v 1 Definition A bipartite graph Γ is k -unique if for every S ⊆ U with | S | ≤ k there exists x 2 v 2 v ∈ V with exactly one neighbor in S v 3 v 4 • Peeling argument example: v 5 f ( x 3 ) = h ( v 2 ) + h ( v 4 ) + h ( v 5 ) mod r Lemma [Siegel’89] Let Γ : U → V d be k -unique and h : V → [ r ] be a random function. Then f ( x ) = P i h ( Γ ( x ) i ) mod r defines a k -independent family of functions From Independence to Expansion and Back Again 4

From expansion to independence • Constructions of k -independent families of functions based on bipartite expander graphs U • Neighbor function Γ : U → V d V v 1 Definition A bipartite graph Γ is k -unique if for every S ⊆ U with | S | ≤ k there exists x 2 v 2 v ∈ V with exactly one neighbor in S v 3 v 4 • Peeling argument example: v 5 f ( x 3 ) = h ( v 2 ) + h ( v 4 ) + h ( v 5 ) mod r Lemma [Siegel’89] Let Γ : U → V d be k -unique and h : V → [ r ] be a random function. Then f ( x ) = P i h ( Γ ( x ) i ) mod r defines a k -independent family of functions From Independence to Expansion and Back Again 4

From expansion to independence • Constructions of k -independent families of functions based on bipartite expander graphs U • Neighbor function Γ : U → V d V v 1 Definition A bipartite graph Γ is k -unique if for every S ⊆ U with | S | ≤ k there exists x 2 v 2 v ∈ V with exactly one neighbor in S v 3 v 4 • Peeling argument example: v 5 f ( x 3 ) = h ( v 2 ) + h ( v 4 ) + h ( v 5 ) mod r Lemma [Siegel’89] Let Γ : U → V d be k -unique and h : V → [ r ] be a random function. Then f ( x ) = P i h ( Γ ( x ) i ) mod r defines a k -independent family of functions From Independence to Expansion and Back Again 4

From expansion to independence • Constructions of k -independent families of functions based on bipartite expander graphs U • Neighbor function Γ : U → V d V v 1 Definition A bipartite graph Γ is k -unique if for every S ⊆ U with | S | ≤ k there exists v 2 v ∈ V with exactly one neighbor in S v 3 v 4 • Peeling argument example: v 5 f ( x 3 ) = h ( v 2 ) + h ( v 4 ) + h ( v 5 ) mod r Lemma [Siegel’89] Let Γ : U → V d be k -unique and h : V → [ r ] be a random function. Then f ( x ) = P i h ( Γ ( x ) i ) mod r defines a k -independent family of functions From Independence to Expansion and Back Again 4