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A Simple Near-Linear Pseudopolynomial Time Randomized Algorithm for Subset Sum Ce Jin , Hongxun Wu Tsinghua University SOSA 2019 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  1. A Simple Near-Linear Pseudopolynomial Time Randomized Algorithm for Subset Sum Ce Jin , Hongxun Wu Tsinghua University SOSA 2019 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 1 / 15

  2. Subset Sum Subset Sum Given a (multi)set S of n positive integers and a target integer t , does there exist a subset R ⊆ S that sums to exactly t ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 2 / 15

  3. Subset Sum Subset Sum Given a (multi)set S of n positive integers and a target integer t , does there exist a subset R ⊆ S that sums to exactly t ? Example: S = { 1 , 1 , 3 , 10 } , t = 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 2 / 15

  4. Subset Sum Subset Sum Given a (multi)set S of n positive integers and a target integer t , does there exist a subset R ⊆ S that sums to exactly t ? Example: S = { 1 , 1 , 3 , 10 } , t = 12 YES! R = { 1 , 1 , 10 } , sum( R ) = 1 + 1 + 10 = 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 2 / 15

  5. Subset Sum Subset Sum Given a (multi)set S of n positive integers and a target integer t , does there exist a subset R ⊆ S that sums to exactly t ? Example: S = { 1 , 1 , 3 , 10 } , t = 12 YES! R = { 1 , 1 , 10 } , sum( R ) = 1 + 1 + 10 = 12 A classic NP-hard problem. O (2 n / 2 ) algorithm [Horowitz and Sahni, JACM 1974] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 2 / 15

  6. Pseudopolynomial time algorithm Pseudopolynomial time algorithm: in poly( n , t ) time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 3 / 15

  7. Pseudopolynomial time algorithm Pseudopolynomial time algorithm: in poly( n , t ) time. Dynamic programming algorithm [Bellman, 1957]: ( S = { s 1 , . . . , s n } ) f [ i , x ] := f [ i − 1 , x ] OR f [ i − 1 , x − s i ] f [0 , x ] := [ x == 0] Output f [ n , t ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 3 / 15

  8. Pseudopolynomial time algorithm Pseudopolynomial time algorithm: in poly( n , t ) time. Dynamic programming algorithm [Bellman, 1957]: ( S = { s 1 , . . . , s n } ) f [ i , x ] := f [ i − 1 , x ] OR f [ i − 1 , x − s i ] f [0 , x ] := [ x == 0] Output f [ n , t ] O ( nt ) time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 3 / 15

  9. Pseudopolynomial time algorithm O ( nt ) [Bellman, 1957] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 4 / 15

  10. Pseudopolynomial time algorithm O ( nt ) [Bellman, 1957] O ( n + √ nt ) deterministic algorithm. [Koiliaris and Xu, SODA 2017] ˜ ( ˜ O ( T ) stands for O ( T poly log T )) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 4 / 15

  11. Pseudopolynomial time algorithm O ( nt ) [Bellman, 1957] O ( n + √ nt ) deterministic algorithm. [Koiliaris and Xu, SODA 2017] ˜ ( ˜ O ( T ) stands for O ( T poly log T )) ˜ O ( n + t ) randomized algorithm. [Bringmann, SODA 2017] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 4 / 15

  12. Pseudopolynomial time algorithm O ( nt ) [Bellman, 1957] O ( n + √ nt ) deterministic algorithm. [Koiliaris and Xu, SODA 2017] ˜ ( ˜ O ( T ) stands for O ( T poly log T )) ˜ O ( n + t ) randomized algorithm. [Bringmann, SODA 2017] ▶ more precisely, O ( n + t log t log 4 n ) time (failure probability 1 n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 4 / 15

  13. Pseudopolynomial time algorithm O ( nt ) [Bellman, 1957] O ( n + √ nt ) deterministic algorithm. [Koiliaris and Xu, SODA 2017] ˜ ( ˜ O ( T ) stands for O ( T poly log T )) ˜ O ( n + t ) randomized algorithm. [Bringmann, SODA 2017] ▶ more precisely, O ( n + t log t log 4 n ) time (failure probability 1 n ) ▶ uses color coding, layer splitting, FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 4 / 15

  14. Pseudopolynomial time algorithm O ( nt ) [Bellman, 1957] O ( n + √ nt ) deterministic algorithm. [Koiliaris and Xu, SODA 2017] ˜ ( ˜ O ( T ) stands for O ( T poly log T )) ˜ O ( n + t ) randomized algorithm. [Bringmann, SODA 2017] ▶ more precisely, O ( n + t log t log 4 n ) time (failure probability 1 n ) ▶ uses color coding, layer splitting, FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 4 / 15

  15. Pseudopolynomial time algorithm O ( nt ) [Bellman, 1957] O ( n + √ nt ) deterministic algorithm. [Koiliaris and Xu, SODA 2017] ˜ ( ˜ O ( T ) stands for O ( T poly log T )) ˜ O ( n + t ) randomized algorithm. [Bringmann, SODA 2017] ▶ more precisely, O ( n + t log t log 4 n ) time (failure probability 1 n ) ▶ uses color coding, layer splitting, FFT Near-optimal! (No t 1 − ε 2 o ( n ) time algorithm for any ε > 0, unless SETH fails) [Abboud, Bringmann, Hermelin, Shabtay, SODA 2019] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 4 / 15

  16. Our result We present a randomized algorithm for Subset Sum in ˜ O ( n + t ) time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 5 / 15

  17. Our result We present a randomized algorithm for Subset Sum in ˜ O ( n + t ) time. Our techniques are simple and quite different from Bringmann’s algorithm. generating functions FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 5 / 15

  18. Our result We present a randomized algorithm for Subset Sum in ˜ O ( n + t ) time. Our techniques are simple and quite different from Bringmann’s algorithm. generating functions FFT n + t ) is O ( n + t log 2 t ). 1 Precise running time (with failure probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 5 / 15

  19. Our result We present a randomized algorithm for Subset Sum in ˜ O ( n + t ) time. Our techniques are simple and quite different from Bringmann’s algorithm. generating functions FFT n + t ) is O ( n + t log 2 t ). 1 Precise running time (with failure probability Can be improved to O ( n + t log t ) time (slightly less simple). faster than Bringmann’s algorithm by a log 4 n factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 5 / 15

  20. Our result We present a randomized algorithm for Subset Sum in ˜ O ( n + t ) time. Our techniques are simple and quite different from Bringmann’s algorithm. generating functions FFT n + t ) is O ( n + t log 2 t ). 1 Precise running time (with failure probability Can be improved to O ( n + t log t ) time (slightly less simple). faster than Bringmann’s algorithm by a log 4 n factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 5 / 15

  21. Our approach: generating functions ( S = { s 1 , s 2 , . . . , s n } , target integer = t ) Consider the generating function of this instance ∏ (1 + x s i ) A ( x ) = 1 ≤ i ≤ n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 6 / 15

  22. Our approach: generating functions ( S = { s 1 , s 2 , . . . , s n } , target integer = t ) Consider the generating function of this instance ∏ (1 + x s i ) A ( x ) = 1 ≤ i ≤ n Expand it ∑ a i x i A ( x ) = 1 + i ≥ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ce Jin , Hongxun Wu (Tsinghua University) SOSA 2019 6 / 15

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