1. Faster Space-Efficient Algorithms for Subset Sum
Jesper Nederlof (Eindhoven University, Netherlands)
Joint work with
Nikhil Bansal, Shashwat Garg, and Nikhil Vyas
of 11

2. Subset Sum Given ints 𝑤 1 ,…, 𝑤 𝑛 ,𝑡,
is there 𝑋⊆ 𝑛 with 𝑤 𝑋 := 𝑖∈𝑋 𝑤 𝑖 =𝑡?
Today: how efficiently can we solve this in the wost case?
NP-complete -> exponential time
Basic DP [Bellman(50’s)] runs in 𝑂 ∗ (𝑡) time and space
What if 𝑡 is huge (i.e 2 𝑛 )?
of 11

3. Meet in the Middle [HS(JACM’74)]
Let L=( 𝑤 1 ,…, 𝑤 𝑛/2 ) R=( 𝑤 𝑛/2+1 ,…, 𝑤 𝑛 )
Compute all possible 2 𝑛/2 sums
For each v∈𝑥, check v∈𝑦
Sort 𝑦 + binary search in 3: 𝑂 ∗ (2 𝑛/2 ) time and space
[SS(SICOMP’81)] improved to 𝑂 ∗ (2 𝑛/2 ) time, 𝑂 ∗ ( 2 𝑛/4 ) space
Natural question: can we beat O ∗ ( 2 𝑛 ) using polynomial space?
𝑤 1 ,…, 𝑤 𝑛/2 𝑤 𝑛/2+1 ,…, 𝑤 𝑛 𝑅 𝐿
x = 𝑤 𝑋 𝑋⊆𝐿
y = 𝑡−𝑤 𝑋 𝑋⊆𝑅
of 11

4. Our contribution
Main thm: SSS in 𝑂 ∗ ( 𝑛 ) time and poly space, if given a random oracle is given.
The oracle stores the random bits for us!
Weaker assumption than sufficiently strong PRG’s
Algo
i’th bit?
𝑝𝑜𝑙𝑦(𝑛) time
0/1
of 11

5. Our contribution
Main thm: SSS in 𝑂 ∗ ( 𝑛 ) time and poly space, if given a random oracle is given.
Generalization to Knapsack
Further generalization to Binary IP with few constraints
Random* 3SUM in 𝑂 ( 𝑛 2.5 ) time and 𝑂( log 𝑛 ) space without oracle
of 11

6. Union bound over d events
Main Idea
Consider w( 2 [𝑛] ) = {𝑤⋅𝑥 : x ∈ 0,1 𝑛 } i.e. all possible 2 𝑛 sums generated by 𝑤= 𝑤 1 ,…, 𝑤 𝑛 Let 𝑑=|𝑤( 2 [𝑛] )| i.e. # distinct possible sums Case 1: If d< 𝑛 (few distinct sums) a) Hash mod 𝑂(𝑑), which makes 𝑡= 𝑂 ∗ (𝑑) b) 𝑂 ∗ (𝑡) time DP, but in poly space [LN(STOC’10)] Interpolates 𝑝 𝑥 = 𝑖=1 𝑛 (1+ 𝑥 𝑤 𝑖 ) to find coeff. of 𝑥 𝑡
Consider w( 2 [𝑛] ) = {𝑤⋅𝑥 : x ∈ 0,1 𝑛 }
i.e. all possible 2 𝑛 sums generated by 𝑤= 𝑤 1 ,…, 𝑤 𝑛
Let 𝑑=|𝑤( 2 [𝑛] )| i.e. # distinct possible sums
Case 1: If d< 𝑛 (few distinct sums)
a) Hash mod 𝑂(𝑑), which makes t= 𝑂 ∗ (𝑑)
b) 𝑂 ∗ (𝑡) time DP, but in poly space [LN(STOC’10)]
Case 2: If d > 𝑛 (many distinct sums)
a) Upper bound max bin size via combinatorics of subset sums
b) Use Floyd’s cycle finding (and the oracle )
Union bound over d events
of 11

7. Main Idea
Consider w( 2 [𝑛] ) = {𝑤⋅𝑥 : x ∈ 0,1 𝑛 } i.e. all possible 2 𝑛 sums generated by 𝑤= 𝑤 1 ,…, 𝑤 𝑛 Let 𝑑=|𝑤( 2 [𝑛] )| i.e. # distinct possible sums Case 1: If d< 𝑛 (few distinct sums) a) Hash mod 𝑂(𝑑), which makes t= 𝑂 ∗ (𝑑) b) 𝑂 ∗ (𝑡) time DP, but in poly space [LN(STOC’10)] Case 2: If d > 𝑛 (many distinct sums) a) Upper bound max bin size via combinatorics of subset sums b) Use Floyd’s cycle finding (and the oracle )
of 11

8.  Main Idea Lemma [AKKN(STACS’16)]: 𝑑⋅ 𝑏 𝑚𝑎𝑥 ≤ 2 1.5𝑛
Case 2: If d > 𝑛 (many distinct sums)
a) Upper bound max bin size via combinatorics of subset sums 
Consider w( 2 [𝑛] ) = {𝑤⋅𝑥 : x ∈ 0,1 𝑛 } i.e. all possible 2 𝑛 sums generated by 𝑤= 𝑤 1 ,…, 𝑤 𝑛 Let 𝑑=|𝑤( 2 [𝑛] )| i.e. # distinct possible sums Case 1: If d< 𝑛 (few distinct sums) a) Hash mod 𝑂(𝑑), which makes t= 𝑂 ∗ (𝑑) b) 𝑂 ∗ (𝑡) time DP, but in poly space [LN(STOC’10)] Case 2: If d > 𝑛 (many distinct sums) a) Proof max bucket size low via combinatorics of subset sums b) Use Floyd’s cycle finding (and the oracle )
max bin size 𝑏 𝑚𝑎𝑥 = 𝑚𝑎𝑥 𝑣 |{𝑥∈{0,1 } 𝑙 :𝑤⋅𝑥=𝑣}|
Lemma [AKKN(STACS’16)]: 𝑑⋅ 𝑏 𝑚𝑎𝑥 ≤ 2 1.5𝑛 𝒘 𝒅
𝒃 𝒎𝒂𝒙
Histogram 1 32
Cool add. comb. result
Smoothness Subset Sum distrib.
Proved via simple connection with `Uniquely Decodable Code Pairs’
As d > 𝑛 , 𝑏 𝑚𝑎𝑥 ≤ 𝑛
of 11

9. Meet in the Middle [HS(JACM’74)]
Let L=( 𝑤 1 ,…, 𝑤 𝑛/2 ) R=( 𝑤 𝑛/2+1 ,…, 𝑤 𝑛 )
Compute all possible 2 𝑛/2 sums
For each v∈𝑥, check v∈𝑦
Sort 𝑦 + binary search in 3: 𝑂 ∗ (2 𝑛/2 ) time and space
Looks like the Element Distinctness problem:
Given list 𝑧 of 𝑚 ints find two positions with equal ints, if exist
Repeat
1. Define z as the concatenation of x and y
2. (Almost) uniformly sample a solution of ED instance 𝑧
3. Check if `fake’ solution or a real SSS solution
U =
𝑤 1 ,…, 𝑤 𝑛/2 𝑤 𝑛/2+1 ,…, 𝑤 𝑛 𝑅 𝐿
x = 𝑤 𝑋 𝑋⊆𝐿
y = 𝑡−𝑤 𝑋 𝑋⊆𝑅
none
How many times? 𝑂 #𝑓𝑎𝑘𝑒 𝑠𝑜𝑙𝑠 ≤𝑂( 𝑛 )
Using max bin bound!!
of 11

10. Sample solution of ED instance
A priori for even finding a solution, 𝑂( 𝑚 2 ) time seems best with polylog 𝑚 space
Crucially relies on Floyd’s rho algorithm
Thm [BCM(FOCS’13)]:
ED in 𝑂 ( 𝑚 1.5 ) time, 𝑂( log 𝑚 ) space, if given random oracle
LD: Given two lists 𝑥,𝑦∈ 𝑚 𝑁 , is there a common value?
Define 𝑓 𝑥,𝑦 = 𝑣=1 𝑚 |𝑥 −1 𝑣 ​ 2 + |𝑦 −1 𝑣 ​ 2
Refined analysis of [BCM(FOCS’13)]. Sample time 𝑂(𝑚/ 𝑓 )
Thm: LD In 𝑂 𝑚 𝑓 time and 𝑂( log 𝑚 ) space, if given f≥𝑓(𝑥,𝑦) and random oracle
of 11

11. Take-home Messages Cycle finding is a great tool low space algo’s
Win/win approach for many/few distinct sums
Similar idea useful for other problems?
To improve [HS(JACM’74)] it remains to `win’ in the case of few sums
Thanks for listening, hope to see you at our poster!
of 11