Complexity Theory of Polynomial-Time Problems Lecture 6: 3SUM Part - PowerPoint PPT Presentation

Complexity Theory of Polynomial-Time Problems Lecture 6: 3SUM Part I Karl Bringmann

3SUM given sets 𝐵,𝐶,𝐷 of 𝑜 integers are there 𝑏 ∈ 𝐵, 𝑐 ∈ 𝐶,𝑑 ∈ 𝐷 such that 𝑏 + 𝑐 + 𝑑 = 0 ? (we assume that we can add/subtract/compare input integers in constant time) trivial algorithm: 𝑃(𝑜 / ) well-known: 𝑃(𝑜 1 ) Conjecture: no 𝑃 𝑜 123 algorithm → 3SUM-Hardness [Gajentaan,Overmars’95]

More Known Algorithms trivial: 𝑃(𝑜 / ) well-known: 𝑃(𝑜 1 ) using FFT: 𝑃(𝑜 + 𝑉 polylog𝑉) for numbers in {−𝑉, …, 𝑉} 789 : ; (789 789 <) : using Word RAM bit-tricks: 𝑃(𝑜 1 ⋅ ; ) , 𝑃(𝑜 1 ⋅ ) 789 : < (cell size 𝑥 = Ω(log𝑜) , [Baran,Demaine,Patrascu’05] each number fits in a cell) (789 789 <) : no bit-tricks: 𝑃(𝑜 1 ⋅ ) [Gronlund,Pettie’14] 789 < we prove a simplified version: Without bit-tricks, 3SUM is in time 𝑃(𝑜 1 ⋅ =87> 789 789 < Thm: ) 789 <

Equivalent Variants given sets 𝐵,𝐶,𝐷 of 𝑜 integers 1) are there 𝑏 ∈ 𝐵, 𝑐 ∈ 𝐶,𝑑 ∈ 𝐷 such that 𝑏 + 𝑐 + 𝑑 = 0 ? replace 𝐷 by −𝑑 𝑑 ∈ 𝐷} given sets 𝐵,𝐶,𝐷 of 𝑜 integers 2) are there 𝑏 ∈ 𝐵, 𝑐 ∈ 𝐶,𝑑 ∈ 𝐷 such that 𝑏 + 𝑐 = 𝑑 ? ⇔ 𝑏 + 𝑐 − 𝑑 = 0 given sets 𝐵,𝐶,𝐷 of 𝑜 integers and target 𝑢 replace 𝐷 by 𝑑 − 𝑢 𝑑 ∈ 𝐷} 3) are there 𝑏 ∈ 𝐵, 𝑐 ∈ 𝐶,𝑑 ∈ 𝐷 such that 𝑏 + 𝑐 + 𝑑 = 𝑢 ? ⇔ 𝑏 + 𝑐 + (𝑑 − 𝑢) = 0 4) given a set 𝑌 of 𝑜 integers are there 𝑦, 𝑧, 𝑨 ∈ 𝑌 such that 𝑦 + 𝑧 + 𝑨 = 0 ? ↑ : set 𝐵, 𝐶, 𝐷 ≔ 𝑌 ↓ : set 𝑌 ≔ 𝑏 + 4𝑉 | 𝑏 ∈ 𝐵 ∪ 𝐶 ∪ {𝑑 − 4𝑉 | 𝑑 ∈ 𝐷} where 𝐵,𝐶,𝐷 ⊆ {−𝑉,. . , 𝑉}

Outline 1) algorithm for small universe 2) quadratic algorithm 3) small decision tree 4) logfactor improvement 5) some 3SUM-hardness results

Algorithm for Small Numbers 𝑃(𝑜 + 𝑉 polylog𝑉) for numbers in {−𝑉, …, 𝑉} add 𝑉 to each number, then numbers are in {0,. . , 2𝑉} and we want 𝑏 ∈ 𝐵, 𝑐 ∈ 𝐶,𝑑 ∈ 𝐷 such that 𝑏 + 𝑐 + 𝑑 = 3𝑉 𝑦 ^ define polynomials 𝑞 \ 𝑦 ≔ ∑ and similarly 𝑞 _ 𝑦 , 𝑞 ` 𝑦 ^∈\ have degree at most 2𝑉 𝑦 ^ 𝑦 b 𝑦 c compute 𝑟 𝑦 ≔ 𝑞 \ 𝑦 ⋅ 𝑞 _ 𝑦 ⋅ 𝑞 ` 𝑦 = (∑ )(∑ )(∑ ) ^∈\ c∈` b∈_ ( 𝑦 ^ ⋅ 𝑦 b ⋅ 𝑦 c = 𝑦 ^fbfc ) what is the coefficient of 𝑦 /d in 𝑟(𝑦) ? it is the number of (𝑏, 𝑐, 𝑑) summing to 3𝑉 use efficient polynomial multiplication (via Fast Fourier Transform): polynomials of degree 𝑒 can be multiplied in time 𝑃(𝑒 polylog𝑒)

Outline 1) algorithm for small universe 2) quadratic algorithm 3) small decision tree 4) logfactor improvement 5) some 3SUM-hardness results

Quadratic Algorithm given a set 𝐵 of 𝑜 integers are there 𝑏, 𝑐, 𝑑 ∈ 𝐵 such that 𝑏 + 𝑐 + 𝑑 = 0 ? sort 𝐵 in increasing order: 𝐵 = {𝑏 g ,… , 𝑏 < } for each 𝑑 ∈ 𝐵 : check whether there are 𝑏, 𝑐 ∈ 𝐵 s.t. 𝑏 + 𝑐 + 𝑑 = 0 initialize 𝑗 = 𝑜,𝑘 = 1 𝑏 g 𝑏 1 𝑏 / … 𝑏 < 𝑏 g while 𝑗 > 0 and 𝑘 ≤ 𝑜 : 𝑏 1 if 𝑏 m + 𝑏 n = −𝑑 : return (𝑏 m , 𝑏 n ,𝑑) 𝑏 / if 𝑏 m + 𝑏 n > −𝑑 : 𝑗 ≔ 𝑗 − 1 if 𝑏 m + 𝑏 n < −𝑑 : 𝑘 ≔ 𝑘 + 1 … return “no solution” 𝑏 <

Quadratic Algorithm given a set 𝐵 of 𝑜 integers are there 𝑏, 𝑐, 𝑑 ∈ 𝐵 such that 𝑏 + 𝑐 + 𝑑 = 0 ? sort 𝐵 in increasing order: 𝐵 = {𝑏 g ,… , 𝑏 < } for each 𝑑 ∈ 𝐵 : check whether there are 𝑏, 𝑐 ∈ 𝐵 s.t. 𝑏 + 𝑐 + 𝑑 = 0 initialize 𝑗 = 𝑜,𝑘 = 1 𝑏 g 𝑏 1 𝑏 / … 𝑏 < 𝑏 g while 𝑗 > 0 and 𝑘 ≤ 𝑜 : 𝑏 1 if 𝑏 m + 𝑏 n = −𝑑 : return (𝑏 m , 𝑏 n ,𝑑) 𝑏 / if 𝑏 m + 𝑏 n > −𝑑 : 𝑗 ≔ 𝑗 − 1 if 𝑏 m + 𝑏 n < −𝑑 : 𝑘 ≔ 𝑘 + 1 … return “no solution” 𝑏 <

Quadratic Algorithm given a set 𝐵 of 𝑜 integers are there 𝑏, 𝑐, 𝑑 ∈ 𝐵 such that 𝑏 + 𝑐 + 𝑑 = 0 ? sort 𝐵 in increasing order: 𝐵 = {𝑏 g ,… , 𝑏 < } for each 𝑑 ∈ 𝐵 : check whether there are 𝑏, 𝑐 ∈ 𝐵 s.t. 𝑏 + 𝑐 + 𝑑 = 0 initialize 𝑗 = 𝑜,𝑘 = 1 𝑏 g 𝑏 1 𝑏 / … 𝑏 < 𝑏 g while 𝑗 > 0 and 𝑘 ≤ 𝑜 : 𝑏 1 if 𝑏 m + 𝑏 n = −𝑑 : return (𝑏 m , 𝑏 n ,𝑑) 𝑏 / if 𝑏 m + 𝑏 n > −𝑑 : 𝑗 ≔ 𝑗 − 1 if 𝑏 m + 𝑏 n < −𝑑 : 𝑘 ≔ 𝑘 + 1 … return “no solution” 𝑏 <

Quadratic Algorithm given a set 𝐵 of 𝑜 integers are there 𝑏, 𝑐, 𝑑 ∈ 𝐵 such that 𝑏 + 𝑐 + 𝑑 = 0 ? sort 𝐵 in increasing order: 𝐵 = {𝑏 g ,… , 𝑏 < } for each 𝑑 ∈ 𝐵 : check whether there are 𝑏, 𝑐 ∈ 𝐵 s.t. 𝑏 + 𝑐 + 𝑑 = 0 initialize 𝑗 = 𝑜,𝑘 = 1 𝑏 g 𝑏 1 𝑏 / … 𝑏 < 𝑏 g while 𝑗 > 0 and 𝑘 ≤ 𝑜 : 𝑏 1 if 𝑏 m + 𝑏 n = −𝑑 : return (𝑏 m , 𝑏 n ,𝑑) 𝑏 / if 𝑏 m + 𝑏 n > −𝑑 : 𝑗 ≔ 𝑗 − 1 if 𝑏 m + 𝑏 n < −𝑑 : 𝑘 ≔ 𝑘 + 1 … return “no solution” 𝑏 <

Quadratic Algorithm given a set 𝐵 of 𝑜 integers are there 𝑏, 𝑐, 𝑑 ∈ 𝐵 such that 𝑏 + 𝑐 + 𝑑 = 0 ? sort 𝐵 in increasing order: 𝐵 = {𝑏 g ,… , 𝑏 < } for each 𝑑 ∈ 𝐵 : check whether there are 𝑏, 𝑐 ∈ 𝐵 s.t. 𝑏 + 𝑐 + 𝑑 = 0 initialize 𝑗 = 𝑜,𝑘 = 1 𝑏 g 𝑏 1 𝑏 / … 𝑏 < 𝑏 g while 𝑗 > 0 and 𝑘 ≤ 𝑜 : 𝑏 1 if 𝑏 m + 𝑏 n = −𝑑 : return (𝑏 m , 𝑏 n ,𝑑) 𝑏 / if 𝑏 m + 𝑏 n > −𝑑 : 𝑗 ≔ 𝑗 − 1 if 𝑏 m + 𝑏 n < −𝑑 : 𝑘 ≔ 𝑘 + 1 … return “no solution” 𝑏 <

Quadratic Algorithm given a set 𝐵 of 𝑜 integers are there 𝑏, 𝑐, 𝑑 ∈ 𝐵 such that 𝑏 + 𝑐 + 𝑑 = 0 ? sort 𝐵 in increasing order: 𝐵 = {𝑏 g ,… , 𝑏 < } for each 𝑑 ∈ 𝐵 : check whether there are 𝑏, 𝑐 ∈ 𝐵 s.t. 𝑏 + 𝑐 + 𝑑 = 0 initialize 𝑗 = 𝑜,𝑘 = 1 𝑏 g 𝑏 1 𝑏 / … 𝑏 < 𝑏 g while 𝑗 > 0 and 𝑘 ≤ 𝑜 : 𝑏 1 if 𝑏 m + 𝑏 n = −𝑑 : return (𝑏 m , 𝑏 n ,𝑑) 𝑏 / if 𝑏 m + 𝑏 n > −𝑑 : 𝑗 ≔ 𝑗 − 1 if 𝑏 m + 𝑏 n < −𝑑 : 𝑘 ≔ 𝑘 + 1 … return “no solution” 𝑏 <

Quadratic Algorithm given a set 𝐵 of 𝑜 integers are there 𝑏, 𝑐, 𝑑 ∈ 𝐵 such that 𝑏 + 𝑐 + 𝑑 = 0 ? sort 𝐵 in increasing order: 𝐵 = {𝑏 g ,… , 𝑏 < } for each 𝑑 ∈ 𝐵 : check whether there are 𝑏, 𝑐 ∈ 𝐵 s.t. 𝑏 + 𝑐 + 𝑑 = 0 initialize 𝑗 = 𝑜,𝑘 = 1 𝑏 g 𝑏 1 𝑏 / … 𝑏 < 𝑏 g while 𝑗 > 0 and 𝑘 ≤ 𝑜 : 𝑏 1 if 𝑏 m + 𝑏 n = −𝑑 : return (𝑏 m , 𝑏 n ,𝑑) 𝑏 / if 𝑏 m + 𝑏 n > −𝑑 : 𝑗 ≔ 𝑗 − 1 if 𝑏 m + 𝑏 n < −𝑑 : 𝑘 ≔ 𝑘 + 1 … return “no solution” 𝑏 <

Quadratic Algorithm given a set 𝐵 of 𝑜 integers are there 𝑏, 𝑐, 𝑑 ∈ 𝐵 such that 𝑏 + 𝑐 + 𝑑 = 0 ? sort 𝐵 in increasing order: 𝐵 = {𝑏 g ,… , 𝑏 < } for each 𝑑 ∈ 𝐵 : check whether there are 𝑏, 𝑐 ∈ 𝐵 s.t. 𝑏 + 𝑐 + 𝑑 = 0 initialize 𝑗 = 𝑜,𝑘 = 1 𝑏 g 𝑏 1 𝑏 / … 𝑏 < 𝑏 g while 𝑗 > 0 and 𝑘 ≤ 𝑜 : 𝑏 1 if 𝑏 m + 𝑏 n = −𝑑 : return (𝑏 m , 𝑏 n ,𝑑) 𝑏 / if 𝑏 m + 𝑏 n > −𝑑 : 𝑗 ≔ 𝑗 − 1 if 𝑏 m + 𝑏 n < −𝑑 : 𝑘 ≔ 𝑘 + 1 … return “no solution” 𝑏 <

Quadratic Algorithm given a set 𝐵 of 𝑜 integers are there 𝑏, 𝑐, 𝑑 ∈ 𝐵 such that 𝑏 + 𝑐 + 𝑑 = 0 ? sort 𝐵 in increasing order: 𝐵 = {𝑏 g ,… , 𝑏 < } for each 𝑑 ∈ 𝐵 : check whether there are 𝑏, 𝑐 ∈ 𝐵 s.t. 𝑏 + 𝑐 + 𝑑 = 0 initialize 𝑗 = 𝑜,𝑘 = 1 𝑏 g 𝑏 1 𝑏 / … 𝑏 < 𝑏 g while 𝑗 > 0 and 𝑘 ≤ 𝑜 : 𝑏 1 if 𝑏 m + 𝑏 n = −𝑑 : return (𝑏 m , 𝑏 n ,𝑑) 𝑏 / if 𝑏 m + 𝑏 n > −𝑑 : 𝑗 ≔ 𝑗 − 1 if 𝑏 m + 𝑏 n < −𝑑 : 𝑘 ≔ 𝑘 + 1 … return “no solution” 𝑏 <

Quadratic Algorithm given a set 𝐵 of 𝑜 integers are there 𝑏, 𝑐, 𝑑 ∈ 𝐵 such that 𝑏 + 𝑐 + 𝑑 = 0 ? sort 𝐵 in increasing order: 𝐵 = {𝑏 g ,… , 𝑏 < } for each 𝑑 ∈ 𝐵 : check whether there are 𝑏, 𝑐 ∈ 𝐵 s.t. 𝑏 + 𝑐 + 𝑑 = 0 initialize 𝑗 = 𝑜,𝑘 = 1 𝑏 g 𝑏 1 𝑏 / … 𝑏 < 𝑏 g while 𝑗 > 0 and 𝑘 ≤ 𝑜 : 𝑏 1 if 𝑏 m + 𝑏 n = −𝑑 : return (𝑏 m , 𝑏 n ,𝑑) 𝑏 / if 𝑏 m + 𝑏 n > −𝑑 : 𝑗 ≔ 𝑗 − 1 otherwise: 𝑘 ≔ 𝑘 + 1 … return “no solution” time 𝑃(𝑜) per 𝑑 ∈ 𝐵 time 𝑃(𝑜 1 ) overall 𝑏 <

Outline 1) algorithm for small universe 2) quadratic algorithm 3) small decision tree 4) logfactor improvement 5) some 3SUM-hardness results

Decision Tree Complexity 3SUM has a decision tree of depth 𝑃(𝑜 //1 log𝑜) Thm: