quantum walks daniel j bernstein university of illinois
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1 Quantum walks Daniel J. Bernstein University of Illinois at Chicago Focusing on quantum walks as an algorithm-design tool: e.g. Grovers algorithm. e.g. Ambainiss algorithm. Can also study quantum walks on much more general graphs.

  1. 7 Normalized graph of q �→ a q for an example with n = 12 after 60 × (Step 1 + Step 2): 1.0 0.5 0.0 −0.5 −1.0

  2. 7 Normalized graph of q �→ a q for an example with n = 12 after 70 × (Step 1 + Step 2): 1.0 0.5 0.0 −0.5 −1.0

  3. 7 Normalized graph of q �→ a q for an example with n = 12 after 80 × (Step 1 + Step 2): 1.0 0.5 0.0 −0.5 −1.0

  4. 7 Normalized graph of q �→ a q for an example with n = 12 after 90 × (Step 1 + Step 2): 1.0 0.5 0.0 −0.5 −1.0

  5. 7 Normalized graph of q �→ a q for an example with n = 12 after 100 × (Step 1 + Step 2): 1.0 0.5 0.0 −0.5 −1.0 Very bad stopping point.

  6. 8 q �→ a q is completely described by a vector of two numbers (with fixed multiplicities): (1) a q for roots q ; (2) a q for non-roots q .

  7. 8 q �→ a q is completely described by a vector of two numbers (with fixed multiplicities): (1) a q for roots q ; (2) a q for non-roots q . Step 1 + Step 2 act linearly on this vector.

  8. 8 q �→ a q is completely described by a vector of two numbers (with fixed multiplicities): (1) a q for roots q ; (2) a q for non-roots q . Step 1 + Step 2 act linearly on this vector. Easily compute eigenvalues and powers of this linear map to understand evolution of state of Grover’s algorithm. ⇒ Probability is ≈ 1 after ≈ ( ı= 4)2 n= 2 iterations.

  9. 9 Ambainis’s algorithm Unique-collision-finding problem: Say f has n -bit inputs, exactly one collision { p; q } : i.e., p � = q , f ( p ) = f ( q ). Problem: find this collision.

  10. 9 Ambainis’s algorithm Unique-collision-finding problem: Say f has n -bit inputs, exactly one collision { p; q } : i.e., p � = q , f ( p ) = f ( q ). Problem: find this collision. Cost 2 n : Define S as the set of n -bit strings. Compute f ( S ), sort.

  11. 9 Ambainis’s algorithm Unique-collision-finding problem: Say f has n -bit inputs, exactly one collision { p; q } : i.e., p � = q , f ( p ) = f ( q ). Problem: find this collision. Cost 2 n : Define S as the set of n -bit strings. Compute f ( S ), sort. Generalize to cost r , success probability ≈ ( r= 2 n ) 2 : Choose a set S of size r . Compute f ( S ), sort.

  12. 10 Data structure D ( S ) capturing the generalized computation: the set S ; the multiset f ( S ); the number of collisions in S .

  13. 10 Data structure D ( S ) capturing the generalized computation: the set S ; the multiset f ( S ); the number of collisions in S . Very efficient to move from D ( S ) to D ( T ) if T is an adjacent set: # S = # T = r , #( S ∩ T ) = r − 1.

  14. 10 Data structure D ( S ) capturing the generalized computation: the set S ; the multiset f ( S ); the number of collisions in S . Very efficient to move from D ( S ) to D ( T ) if T is an adjacent set: # S = # T = r , #( S ∩ T ) = r − 1. 2003 Ambainis, simplified 2007 Magniez–Nayak–Roland–Santha: Create superposition of states ( D ( S ) ; D ( T )) with adjacent S; T . By a quantum walk find S containing a collision.

  15. 11 How the quantum walk works: Start from uniform superposition. Repeat ≈ 0 : 6 · 2 n =r times: Negate a S;T if S contains collision. Repeat ≈ 0 : 7 · √ r times: For each T : Diffuse a S;T across all S . For each S : Diffuse a S;T across all T . Now high probability that T contains collision. Cost r + 2 n = √ r . Optimize: 2 2 n= 3 .

  16. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 0 negations and 0 diffusions: Pr[class (0 ; 0)] ≈ 0 : 938; + Pr[class (0 ; 1)] ≈ 0 : 000; + Pr[class (1 ; 0)] ≈ 0 : 000; + Pr[class (1 ; 1)] ≈ 0 : 060; + Pr[class (1 ; 2)] ≈ 0 : 000; + Pr[class (2 ; 1)] ≈ 0 : 000; + Pr[class (2 ; 2)] ≈ 0 : 001; + Right column is sign of a S;T .

  17. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 1 negation and 46 diffusions: Pr[class (0 ; 0)] ≈ 0 : 935; + Pr[class (0 ; 1)] ≈ 0 : 000; + Pr[class (1 ; 0)] ≈ 0 : 000; − Pr[class (1 ; 1)] ≈ 0 : 057; + Pr[class (1 ; 2)] ≈ 0 : 000; + Pr[class (2 ; 1)] ≈ 0 : 000; − Pr[class (2 ; 2)] ≈ 0 : 008; + Right column is sign of a S;T .

  18. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 2 negations and 92 diffusions: Pr[class (0 ; 0)] ≈ 0 : 918; + Pr[class (0 ; 1)] ≈ 0 : 001; + Pr[class (1 ; 0)] ≈ 0 : 000; − Pr[class (1 ; 1)] ≈ 0 : 059; + Pr[class (1 ; 2)] ≈ 0 : 001; + Pr[class (2 ; 1)] ≈ 0 : 000; − Pr[class (2 ; 2)] ≈ 0 : 022; + Right column is sign of a S;T .

  19. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 3 negations and 138 diffusions: Pr[class (0 ; 0)] ≈ 0 : 897; + Pr[class (0 ; 1)] ≈ 0 : 001; + Pr[class (1 ; 0)] ≈ 0 : 000; − Pr[class (1 ; 1)] ≈ 0 : 058; + Pr[class (1 ; 2)] ≈ 0 : 002; + Pr[class (2 ; 1)] ≈ 0 : 000; + Pr[class (2 ; 2)] ≈ 0 : 042; + Right column is sign of a S;T .

  20. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 4 negations and 184 diffusions: Pr[class (0 ; 0)] ≈ 0 : 873; + Pr[class (0 ; 1)] ≈ 0 : 001; + Pr[class (1 ; 0)] ≈ 0 : 000; − Pr[class (1 ; 1)] ≈ 0 : 054; + Pr[class (1 ; 2)] ≈ 0 : 002; + Pr[class (2 ; 1)] ≈ 0 : 000; + Pr[class (2 ; 2)] ≈ 0 : 070; + Right column is sign of a S;T .

  21. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 5 negations and 230 diffusions: Pr[class (0 ; 0)] ≈ 0 : 838; + Pr[class (0 ; 1)] ≈ 0 : 001; + Pr[class (1 ; 0)] ≈ 0 : 001; − Pr[class (1 ; 1)] ≈ 0 : 054; + Pr[class (1 ; 2)] ≈ 0 : 003; + Pr[class (2 ; 1)] ≈ 0 : 000; + Pr[class (2 ; 2)] ≈ 0 : 104; + Right column is sign of a S;T .

  22. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 6 negations and 276 diffusions: Pr[class (0 ; 0)] ≈ 0 : 800; + Pr[class (0 ; 1)] ≈ 0 : 001; + Pr[class (1 ; 0)] ≈ 0 : 001; − Pr[class (1 ; 1)] ≈ 0 : 051; + Pr[class (1 ; 2)] ≈ 0 : 006; + Pr[class (2 ; 1)] ≈ 0 : 000; + Pr[class (2 ; 2)] ≈ 0 : 141; + Right column is sign of a S;T .

  23. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 7 negations and 322 diffusions: Pr[class (0 ; 0)] ≈ 0 : 758; + Pr[class (0 ; 1)] ≈ 0 : 002; + Pr[class (1 ; 0)] ≈ 0 : 001; − Pr[class (1 ; 1)] ≈ 0 : 047; + Pr[class (1 ; 2)] ≈ 0 : 007; + Pr[class (2 ; 1)] ≈ 0 : 000; + Pr[class (2 ; 2)] ≈ 0 : 184; + Right column is sign of a S;T .

  24. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 8 negations and 368 diffusions: Pr[class (0 ; 0)] ≈ 0 : 708; + Pr[class (0 ; 1)] ≈ 0 : 003; + Pr[class (1 ; 0)] ≈ 0 : 001; − Pr[class (1 ; 1)] ≈ 0 : 046; + Pr[class (1 ; 2)] ≈ 0 : 007; + Pr[class (2 ; 1)] ≈ 0 : 000; + Pr[class (2 ; 2)] ≈ 0 : 234; + Right column is sign of a S;T .

  25. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 9 negations and 414 diffusions: Pr[class (0 ; 0)] ≈ 0 : 658; + Pr[class (0 ; 1)] ≈ 0 : 003; + Pr[class (1 ; 0)] ≈ 0 : 001; − Pr[class (1 ; 1)] ≈ 0 : 042; + Pr[class (1 ; 2)] ≈ 0 : 009; + Pr[class (2 ; 1)] ≈ 0 : 000; + Pr[class (2 ; 2)] ≈ 0 : 287; + Right column is sign of a S;T .

  26. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 10 negations and 460 diffusions: Pr[class (0 ; 0)] ≈ 0 : 606; + Pr[class (0 ; 1)] ≈ 0 : 003; + Pr[class (1 ; 0)] ≈ 0 : 002; − Pr[class (1 ; 1)] ≈ 0 : 037; + Pr[class (1 ; 2)] ≈ 0 : 013; + Pr[class (2 ; 1)] ≈ 0 : 000; + Pr[class (2 ; 2)] ≈ 0 : 338; + Right column is sign of a S;T .

  27. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 11 negations and 506 diffusions: Pr[class (0 ; 0)] ≈ 0 : 547; + Pr[class (0 ; 1)] ≈ 0 : 004; + Pr[class (1 ; 0)] ≈ 0 : 003; − Pr[class (1 ; 1)] ≈ 0 : 036; + Pr[class (1 ; 2)] ≈ 0 : 015; + Pr[class (2 ; 1)] ≈ 0 : 001; + Pr[class (2 ; 2)] ≈ 0 : 394; + Right column is sign of a S;T .

  28. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 12 negations and 552 diffusions: Pr[class (0 ; 0)] ≈ 0 : 491; + Pr[class (0 ; 1)] ≈ 0 : 004; + Pr[class (1 ; 0)] ≈ 0 : 003; − Pr[class (1 ; 1)] ≈ 0 : 032; + Pr[class (1 ; 2)] ≈ 0 : 014; + Pr[class (2 ; 1)] ≈ 0 : 001; + Pr[class (2 ; 2)] ≈ 0 : 455; + Right column is sign of a S;T .

  29. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 13 negations and 598 diffusions: Pr[class (0 ; 0)] ≈ 0 : 436; + Pr[class (0 ; 1)] ≈ 0 : 005; + Pr[class (1 ; 0)] ≈ 0 : 003; − Pr[class (1 ; 1)] ≈ 0 : 026; + Pr[class (1 ; 2)] ≈ 0 : 017; + Pr[class (2 ; 1)] ≈ 0 : 000; + Pr[class (2 ; 2)] ≈ 0 : 513; + Right column is sign of a S;T .

  30. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 14 negations and 644 diffusions: Pr[class (0 ; 0)] ≈ 0 : 377; + Pr[class (0 ; 1)] ≈ 0 : 006; + Pr[class (1 ; 0)] ≈ 0 : 004; − Pr[class (1 ; 1)] ≈ 0 : 025; + Pr[class (1 ; 2)] ≈ 0 : 022; + Pr[class (2 ; 1)] ≈ 0 : 001; + Pr[class (2 ; 2)] ≈ 0 : 566; + Right column is sign of a S;T .

  31. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 15 negations and 690 diffusions: Pr[class (0 ; 0)] ≈ 0 : 322; + Pr[class (0 ; 1)] ≈ 0 : 005; + Pr[class (1 ; 0)] ≈ 0 : 004; − Pr[class (1 ; 1)] ≈ 0 : 021; + Pr[class (1 ; 2)] ≈ 0 : 023; + Pr[class (2 ; 1)] ≈ 0 : 001; + Pr[class (2 ; 2)] ≈ 0 : 623; + Right column is sign of a S;T .

  32. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 16 negations and 736 diffusions: Pr[class (0 ; 0)] ≈ 0 : 270; + Pr[class (0 ; 1)] ≈ 0 : 006; + Pr[class (1 ; 0)] ≈ 0 : 005; − Pr[class (1 ; 1)] ≈ 0 : 017; + Pr[class (1 ; 2)] ≈ 0 : 022; + Pr[class (2 ; 1)] ≈ 0 : 001; + Pr[class (2 ; 2)] ≈ 0 : 680; + Right column is sign of a S;T .

  33. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 17 negations and 782 diffusions: Pr[class (0 ; 0)] ≈ 0 : 218; + Pr[class (0 ; 1)] ≈ 0 : 007; + Pr[class (1 ; 0)] ≈ 0 : 005; − Pr[class (1 ; 1)] ≈ 0 : 015; + Pr[class (1 ; 2)] ≈ 0 : 024; + Pr[class (2 ; 1)] ≈ 0 : 001; + Pr[class (2 ; 2)] ≈ 0 : 730; + Right column is sign of a S;T .

  34. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 18 negations and 828 diffusions: Pr[class (0 ; 0)] ≈ 0 : 172; + Pr[class (0 ; 1)] ≈ 0 : 006; + Pr[class (1 ; 0)] ≈ 0 : 005; − Pr[class (1 ; 1)] ≈ 0 : 011; + Pr[class (1 ; 2)] ≈ 0 : 029; + Pr[class (2 ; 1)] ≈ 0 : 001; + Pr[class (2 ; 2)] ≈ 0 : 775; + Right column is sign of a S;T .

  35. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 19 negations and 874 diffusions: Pr[class (0 ; 0)] ≈ 0 : 131; + Pr[class (0 ; 1)] ≈ 0 : 007; + Pr[class (1 ; 0)] ≈ 0 : 006; − Pr[class (1 ; 1)] ≈ 0 : 008; + Pr[class (1 ; 2)] ≈ 0 : 030; + Pr[class (2 ; 1)] ≈ 0 : 002; + Pr[class (2 ; 2)] ≈ 0 : 816; + Right column is sign of a S;T .

  36. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 20 negations and 920 diffusions: Pr[class (0 ; 0)] ≈ 0 : 093; + Pr[class (0 ; 1)] ≈ 0 : 007; + Pr[class (1 ; 0)] ≈ 0 : 007; − Pr[class (1 ; 1)] ≈ 0 : 007; + Pr[class (1 ; 2)] ≈ 0 : 027; + Pr[class (2 ; 1)] ≈ 0 : 002; + Pr[class (2 ; 2)] ≈ 0 : 857; + Right column is sign of a S;T .

  37. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 21 negations and 966 diffusions: Pr[class (0 ; 0)] ≈ 0 : 062; + Pr[class (0 ; 1)] ≈ 0 : 007; + Pr[class (1 ; 0)] ≈ 0 : 006; − Pr[class (1 ; 1)] ≈ 0 : 004; + Pr[class (1 ; 2)] ≈ 0 : 030; + Pr[class (2 ; 1)] ≈ 0 : 001; + Pr[class (2 ; 2)] ≈ 0 : 890; + Right column is sign of a S;T .

  38. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 22 negations and 1012 diffusions: Pr[class (0 ; 0)] ≈ 0 : 037; + Pr[class (0 ; 1)] ≈ 0 : 008; + Pr[class (1 ; 0)] ≈ 0 : 007; − Pr[class (1 ; 1)] ≈ 0 : 002; + Pr[class (1 ; 2)] ≈ 0 : 034; + Pr[class (2 ; 1)] ≈ 0 : 001; + Pr[class (2 ; 2)] ≈ 0 : 910; + Right column is sign of a S;T .

  39. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 23 negations and 1058 diffusions: Pr[class (0 ; 0)] ≈ 0 : 017; + Pr[class (0 ; 1)] ≈ 0 : 008; + Pr[class (1 ; 0)] ≈ 0 : 007; − Pr[class (1 ; 1)] ≈ 0 : 002; + Pr[class (1 ; 2)] ≈ 0 : 034; + Pr[class (2 ; 1)] ≈ 0 : 002; + Pr[class (2 ; 2)] ≈ 0 : 930; + Right column is sign of a S;T .

  40. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 24 negations and 1104 diffusions: Pr[class (0 ; 0)] ≈ 0 : 005; + Pr[class (0 ; 1)] ≈ 0 : 007; + Pr[class (1 ; 0)] ≈ 0 : 007; − Pr[class (1 ; 1)] ≈ 0 : 000; + Pr[class (1 ; 2)] ≈ 0 : 030; + Pr[class (2 ; 1)] ≈ 0 : 002; + Pr[class (2 ; 2)] ≈ 0 : 948; + Right column is sign of a S;T .

  41. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 25 negations and 1150 diffusions: Pr[class (0 ; 0)] ≈ 0 : 000; + Pr[class (0 ; 1)] ≈ 0 : 008; + Pr[class (1 ; 0)] ≈ 0 : 008; − Pr[class (1 ; 1)] ≈ 0 : 000; + Pr[class (1 ; 2)] ≈ 0 : 031; + Pr[class (2 ; 1)] ≈ 0 : 001; + Pr[class (2 ; 2)] ≈ 0 : 952; + Right column is sign of a S;T .

  42. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 26 negations and 1196 diffusions: Pr[class (0 ; 0)] ≈ 0 : 002; − Pr[class (0 ; 1)] ≈ 0 : 008; + Pr[class (1 ; 0)] ≈ 0 : 008; − Pr[class (1 ; 1)] ≈ 0 : 000; − Pr[class (1 ; 2)] ≈ 0 : 035; + Pr[class (2 ; 1)] ≈ 0 : 002; + Pr[class (2 ; 2)] ≈ 0 : 945; + Right column is sign of a S;T .

  43. 12 Classify ( S; T ) according to (#( S ∩ { p; q } ) ; #( T ∩ { p; q } )); reduce a to low-dim vector. Analyze evolution of this vector. e.g. n = 15, r = 1024, after 27 negations and 1242 diffusions: Pr[class (0 ; 0)] ≈ 0 : 011; − Pr[class (0 ; 1)] ≈ 0 : 007; + Pr[class (1 ; 0)] ≈ 0 : 007; − Pr[class (1 ; 1)] ≈ 0 : 001; − Pr[class (1 ; 2)] ≈ 0 : 034; + Pr[class (2 ; 1)] ≈ 0 : 003; + Pr[class (2 ; 2)] ≈ 0 : 938; + Right column is sign of a S;T .

  44. 13 Data structures Moving from D ( S ) to D ( T ): dominated by O (1) evaluations of f if f is extremely slow. But usually f is not so slow.

  45. 13 Data structures Moving from D ( S ) to D ( T ): dominated by O (1) evaluations of f if f is extremely slow. But usually f is not so slow. Store set S and multiset f ( S ) in, e.g., hash tables?

  46. 13 Data structures Moving from D ( S ) to D ( T ): dominated by O (1) evaluations of f if f is extremely slow. But usually f is not so slow. Store set S and multiset f ( S ) in, e.g., hash tables? Minor problem: time to hash S is huge for some sets S .

  47. 13 Data structures Moving from D ( S ) to D ( T ): dominated by O (1) evaluations of f if f is extremely slow. But usually f is not so slow. Store set S and multiset f ( S ) in, e.g., hash tables? Minor problem: time to hash S is huge for some sets S . Fix: randomize hash function (1979 Carter–Wegman), and specify big enough time for whole algorithm to be reliable.

  48. 14 Major problem: hash table depends on history, not just on S . Algorithm fails horribly. Need history-independent D ( S ).

  49. 14 Major problem: hash table depends on history, not just on S . Algorithm fails horribly. Need history-independent D ( S ). 2003 Ambainis: “combination of a hash table and a skip list”. Several pages of analysis.

  50. 14 Major problem: hash table depends on history, not just on S . Algorithm fails horribly. Need history-independent D ( S ). 2003 Ambainis: “combination of a hash table and a skip list”. Several pages of analysis. 2013 Bernstein–Jeffery–Lange– Meurer: radix tree. Simplest radix tree: Left subtree stores { x : (0 ; x ) ∈ S } if nonempty. Right subtree stores { x : (1 ; x ) ∈ S } if nonempty.

  51. 15 Caveats The 2 2 n= 3 analysis assumes cheap random access to memory. Justified by simplicity, not realism.

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