1. Nikhil Bansal, Shashwat Garg, Jesper Nederlof, Nikhil Vyas
Faster Space-Efficient Algorithms for Subset Sum, k-Sum and related problems
(available at arXiv: )
Nikhil Bansal, Shashwat Garg,
Jesper Nederlof, Nikhil Vyas
disclaimer: no turtles or hares were harmed during this research

2. Subset Sum Algo i’th bit? 0/1
Subset Sum
Given integers 𝑤 1 ,…, 𝑤 𝑛 ,𝑡 find 𝑋⊆ 𝑛 with 𝑖∈𝑋 𝑤 𝑖 =𝑡
𝑂(𝑛𝑡) time by DP
𝑂 𝑛+𝑡 randomized time B(SODA’17)
𝑂 (𝑛𝑡) randomized time, poly space B(SODA’17)
𝑂 ∗ (2 𝑛 ) time, poly space ( 𝑂 ∗ omits poly factors in input)
𝑂 ∗ (2 𝑛/2 ) time, 𝑂 ∗ ( 2 𝑛/2 ) space HS(JACM72)
𝑂 ∗ (2 𝑛/2 ) time, 𝑂 ∗ ( 2 𝑛/4 ) space SS(SICOMP81)
Can solve any instance in either 𝑂 ∗ ( 𝑛 ) time or 𝑂 ∗ ( 𝑛 ) time and poly space AKKN(STACS’16)
Algo
i’th bit?
𝑝𝑜𝑙𝑦(𝑛) time
0/1
Main Result: There is a Monte Carlo algorithm for Subset Sum using 𝑂 ∗ ( 𝑛 ) time and poly space, assuming random read-only access to random bits

3. BCM:Element Distinctness Crypto: List merging
Floyd:
Cycle Finding
BCM (FOCS13):
Shuffle function
HS (JACM72): MitM
BCM:Element Distinctness
Crypto: List merging
AKKN (STACS16): Subset Sum distribution is smooth
LN(STOC10): Save space with DFT
Hash mod p
List Disjointness
(with small freqs2)
Subset Sum
(many distinct sums)
Subset Sum
(few distinct sums)
Subset Sum

4. BCM:Element Distinctness
Floyd:
Cycle Finding
BCM (FOCS13):
Shuffle function
BCM:Element Distinctness

5. Element Distinctness (ED): BCM(FOCS13)
Given z∈ 𝑚 𝑛 with 𝑚≤𝑝𝑜𝑙𝑦(𝑛), are all values distinct?
Unlimited space: 𝑂 (𝑛) time (sort)
𝑂( log 𝑛 ) space: 𝑂 ( 𝑛 2 ) time (brute-force)
Suppose 𝑧 ∈ 𝑅 𝑛 𝑛 ; find a repeated value-pair whp
Unlimited space: 𝑂 ( 𝑛 ) time ( 𝑂 ( 𝑛 ) samples)
𝑂( 𝑙𝑜𝑔 𝑛 ) space: 𝑂 (𝑛) time ( 𝑂 (1) samples and compare)
Theorem(BCM): 𝑂 (𝑛 1.5 ) time, 𝑂( log 𝑛 ) space, assuming random read-only access to random bits
We’ll first see 𝑂 ( 𝑛 ) time, 𝑂( log 𝑛 ) space if 𝑧 ∈ 𝑅 𝑛 𝑛
Approach: set 𝑓 𝑖 = 𝑧 𝑖 , and use cycle finding to find 𝑖≠𝑗 such that 𝑓 𝑖 =𝑓 𝑗

6. Floyd’s Cycle Finding s Finds such 𝑖,𝑗 using little space
Basic algo in crypto, much more obscure in TCS
View 𝑓 as digraph (with arcs (𝑖,𝑓(𝑖))) s

7. Floyd’s Cycle Finding i s j Finds such 𝑖,𝑗 using little space
Basic algo in crypto, much more obscure in TCS
View 𝑓 as digraph (with arcs (𝑖,𝑓(𝑖))) i s j
T = #steps turtle in first round (6 in ex)
p= stem-length (3 in ex), q=cycle length (6 in ex)
2T=T+xq -> T=xq -> T+p=xq+p

8. Floyd’s Cycle Finding i s j Finds such 𝑖,𝑗 using little space
Basic algo in crypto, much more obscure in TCS
View 𝑓 as digraph (with arcs (𝑖,𝑓(𝑖))) i s j
Only works if start outside cycle!
Works well assuming 𝑓 is random:
𝑝 and 𝑞 are 𝜃( 𝑛 ) whp (birthday paradox)

9. `Shuffling’ f
Theorem(BCM): 𝑂 (𝑛 1.5 ) time, 𝑂( log 𝑛 ) space, assuming random read-only access to random bits
What if 𝑓 is not random? Answer: `shuffle’ 𝑓
Let ℎ: 𝑚 →[𝑛] be a random function
Cannot remember ℎ, but use assumed oracle
Define 𝑓 𝑖 =ℎ 𝑧 𝑖
Use Floyd to sample 𝑖≠𝑗 such that 𝑓 𝑖 =𝑓(𝑗)
Call 𝑖,𝑗 a bad pair if ℎ 𝑧 𝑖 =ℎ( 𝑧 𝑗 ) but 𝑧 𝑖 ≠ 𝑧 𝑗
Expect at most 𝑛 2 /𝑛 bad pairs
Using 𝑂 𝑛 samples, expect to see real solution

10. BCM:Element Distinctness
Floyd:
Cycle Finding
BCM (FOCS13):
Shuffle function
List Disjointness
(with small freqs2)
Crypto: List merging
BCM:Element Distinctness

11. List Disjointness Given two lists 𝑥,𝑦∈ 𝑚 𝑛 , 𝑚≤𝑝𝑜𝑙𝑦(𝑛)
Asked do they share a common value?
𝑥=(1, 9, 2, 4, 9, 4, 6, 5, 2) 𝑦=(7, 8, 5, 0, 3, 7, 3, 0, 8)
Very similar to ED; but want values from different lists
Define 𝑝 𝑥 = 𝑣∈[𝑚] |𝑥 −1 𝑣 ​ 2 ; i.e p 𝑦 =1+4⋅ 𝑝 𝑥,𝑦 =𝑝 𝑥 +𝑝 𝑦 Counts number of pseudo-solutions
Theorem: There is an 𝑂 n 𝑝 time, 𝑂( log 𝑛 ) space algorithm for List Disjointness, if given 𝑝 𝑥,𝑦 ≤𝑝
assuming random read-only access to random bits

12. List Disjointness Define 𝑧∈ 𝑚 𝑛 by merging 𝑥,𝑦
Theorem: There is an 𝑂 n 𝑝 time, 𝑂( log 𝑛 ) space algorithm for List Disjointness, if given 𝑝 𝑥,𝑦 ≤𝑝
assuming random read-only access to random bits
Define 𝑧∈ 𝑚 𝑛 by merging 𝑥,𝑦
E.g. just concatenate 𝑥 and 𝑦
In paper we set 𝑧 𝑖 := 𝑥 𝑖 or 𝑧 𝑖 := 𝑦 𝑖 with prob 1/2
Sample 𝑖,𝑗 such that 𝑓 𝑖 =𝑓(𝑗) as before
If 𝑧 𝑖 =𝑧(𝑗) and 𝑖≠𝑗, also check 𝑥 𝑖 = 𝑦 𝑗 or 𝑥 𝑗 = 𝑦 𝑖
Need 𝑝(𝑥,𝑦) samples
Expect 𝑂(𝑛/ 𝑝(𝑥,𝑦) ) vertices needed for a sample

13. BCM:Element Distinctness
Floyd:
Cycle Finding
BCM (FOCS13):
Shuffle function
HS (JACM72): MitM
BCM:Element Distinctness
Crypto: List merging
List Disjointness
(with small freqs2)
Subset Sum
(many distinct sums)

14. 𝐿 𝑅 Meet in the Middle Ints 𝑤 1 ,…, 𝑤 𝑛 ,𝑡. Denote w 𝑋 ≔ 𝑖∈𝑋 𝑤 𝑖
Reduce SSS on 𝑛 integers to List Disjointness on lists of length 2 𝑛/2 ; run the sorting algo 𝐿 𝑅
𝑤 1 ,…, 𝑤 𝑛/2 , 𝑤 𝑛/2+1 ,…, 𝑤 𝑛
𝑥= 𝑤 𝑋 𝑋⊆𝐿
𝑦= 𝑡−𝑤 𝑋 𝑋⊆𝑅
LD instance solved in 𝑂( 2 𝑛/2 polylog ( 2 𝑛/2 ) ) time, which is O ∗ ( 2 𝑛/2 ). Also uses 2 𝑛/2 space

15. 𝐿 𝑅 Meet in the Middle new Ints 𝑤 1 ,…, 𝑤 𝑛 ,𝑡. Denote w 𝑋 ≔ 𝑖∈𝑋 𝑤 𝑖
Reduce SSS on 𝑛 integers to List Disjointness on lists of length 2 𝑛/2 ; run the sorting algo.
new 𝐿 𝑅
𝑤 1 ,…, 𝑤 𝑛/2 , 𝑤 𝑛/2+1 ,…, 𝑤 𝑛
𝑥= 𝑤 𝑋 𝑋⊆𝐿
𝑦= 𝑡−𝑤 𝑋 𝑋⊆𝑅
𝑂 ∗ 1 space, 𝑂 ∗ ( 2 𝑛/2 𝑝 ) time
𝑝 𝑥 = {𝑋,𝑌⊆𝐿:𝑤 𝑋 =𝑤 𝑌 } ≤ 2 𝑛
If 𝑤 𝑖 =0, 𝑝 𝑥 ,𝑝 𝑦 = 2 𝑛 -> 2 𝑛 time 
How do instances with 𝑝 𝑥 ≥ 𝑛 look?

16. BCM:Element Distinctness Crypto: List merging
Floyd:
Cycle Finding
BCM (FOCS13):
Shuffle function
HS (JACM72): MitM
BCM:Element Distinctness
Crypto: List merging
AKKN (STACS16): Subset Sum distribution is smooth
List Disjointness
(with small freqs2)
Subset Sum
(many distinct sums)

17. Subset Sum Distribution is smooth (AKKN)
Lemma: 𝑝 𝑥 ⋅|{𝑤 𝑋 :𝑋⊆𝐿}|≤ 6 𝑛/2
Note: 6 𝑛/2 < 2 1.3𝑛
𝒘 𝑳
𝒑(𝒙)
|{𝒘 𝑿 :𝑿⊆𝑳}|
Histogram
32 2 1
32⋅ 1 2 32
6⋅ ⋅ ⋅ 3 2 =76 16

18. Subset Sum Distribution is smooth (AKKN)
Lemma: 𝑝 𝑥 ⋅|{𝑤 𝑋 :𝑋⊆𝐿}|≤ 6 𝑛/2
Note: 6 𝑛/2 < 2 1.3𝑛
We use this as follows:
Suppose 𝑤 𝑋 :𝑋⊆ 𝑛 ≥ 𝑛 then
𝑤 𝑋 :𝑋⊆𝐿 ≥ 𝑛
and thus 𝑝 𝑥 ≤ 6 𝑛/2 / 𝑛 ≤ 𝑛
Cor: If 𝑤 𝑋 :𝑋⊆ 𝑛 ≥ 𝑛 , can solve SSS in 𝑂 ∗ (2 𝑛/ 𝑛 )= 𝑂 ∗ ( 𝑛 ) time and poly space,
assuming random read-only access to random bits
Proof of Lemma uses a connection to UDCP’s

19. Uniquely Decodable Code Pairs (UDCP)
Pair 𝒞 1 , 𝒞 2 ⊆{0,1 } 𝑑 s.t. 𝒞 1 + 𝒞 2 = 𝒞 1 | 𝒞 2 |.
𝒞 1 + 𝒞 2 ={x+y: x,y ∈ 𝒞 1 × 𝒞 2 },
x+y is addition over ℤ 𝑑 . 1
𝒞 1 ?? 2 1
𝒞 2 1

20. Uniquely Decodable Code Pairs (UDCP)
Pair 𝒞 1 , 𝒞 2 ⊆{0,1 } 𝑑 s.t. 𝒞 1 + 𝒞 2 = 𝒞 1 | 𝒞 2 |.
𝒞 1 ={10,01}, 𝒞 2 ={00,01,11} is UDCP: 𝒞 1 + 𝒞 2 ={10,11,21,01,02,12}
𝒞 1 + 𝒞 2 ≤ { 0,1,2} 𝑑 ≤ 3 𝑑
Side remark:
In general 𝒞 1 || 𝒞 2 ≤ 2 1.5𝑑 is best known (elegant upper bound, might tell afterwards)

21. Subset Sum Distribution is smooth (AKKN)
Lemma: 𝑝 𝑥 ⋅|{𝑤 𝑋 :𝑋⊆𝐿}|≤ 6 𝑛/2
There exists a frequent sum 𝑖 s.t.
B v ={𝑥∈ 0,1 𝑛/2 :𝑤⋅𝑥 =𝑣}; 𝑝 𝑥 ≤ 𝐵 𝑣 2 𝑛/2
Let 𝐴⊆{ 0,1} 𝑛/2 be such that for all 𝑎 1 , 𝑎 2 ∈𝐴 𝑎 1 ⋅𝑤= 𝑎 2 ⋅𝑤 implies 𝑎 1 = 𝑎 2
Then |𝐴+ 𝐵 𝑣 |= 𝐴 ⋅| 𝐵 𝑣 | (i.e. 𝐴, 𝐵 𝑣 is a UDCP):
Suppose 𝑎 1 + 𝑏 1 = 𝑎 2 + 𝑏 2 (add in ℤ 𝑛/2 )
Thus 𝐴 ⋅ 𝐵 𝑣 = 𝐴+ 𝐵 𝑣 ≤ 3 𝑛/2
lemma follows: 𝑝 𝑥 𝐴 ≤ 𝐵 𝑣 2 𝑛/2 𝐴 ≤ 6 𝑛/2
𝑤⋅(𝑎 1 + 𝑤⋅𝑏 1 )= 𝑤⋅(𝑎 2 +𝑤⋅ 𝑏 2 )
𝑤⋅ 𝑎 1 + 𝑤⋅𝑏 1 )= 𝑤⋅ 𝑎 2 +𝑤⋅ 𝑏 2 )
𝑤⋅ 𝑎 1 + 𝑤⋅𝑦 1 )= 𝑤⋅ 𝑎 2 +𝑤⋅ 𝑦 2 )
𝑤⋅(𝑎 1 + 𝑤⋅𝑏 1 )= 𝑤⋅(𝑎 2 +𝑤⋅ 𝑏 2 )
→ 𝑎 1 = 𝑎 2 → 𝑏 1 = 𝑏 2

22. BCM:Element Distinctness Crypto: List merging
Floyd:
Cycle Finding
BCM (FOCS13):
Shuffle function
HS (JACM72): MitM
BCM:Element Distinctness
Crypto: List merging
AKKN (STACS16): Subset Sum distribution is smooth
List Disjointness
(with small freqs2)
Hash mod p + DFT
Subset Sum
(many distinct sums)
Subset Sum
(few distinct sums)
Subset Sum

23. Subset Sum with few Distinct Sums
Lemma: Can solve instance 𝑤 1 ,…, 𝑤 𝑛 ,𝑡 in time O ∗ ( 𝑤 𝑋 :𝑋⊆ 𝑛 ) and poly space.
Done by hashing numbers mod a prime of order 𝑤 𝑋 :𝑋⊆ 𝑛 and run 𝑂 ∗ 𝑡 time 𝑂 ∗ (1) space algorithm, that uses DFT.
Combining with previous lemma we obtain
Main Result’: There is a Monte Carlo for Subset Sum using 𝑂 ∗ ( 𝑛 ) time and poly space, assuming random read-only access to random bits
Omitted optimization to get the 𝑂 ∗ ( 𝑛 )

24. BCM:Element Distinctness Crypto: List merging
Floyd:
Cycle Finding
BCM (FOCS13):
Shuffle function
HS (JACM72): MitM
BCM:Element Distinctness
Crypto: List merging
AKKN (STACS16): Subset Sum distribution is smooth
List Disjointness
(with small freqs2)
Hash mod p + DFT
Random k-Sum
Subset Sum
(many distinct sums)
Subset Sum
(few distinct sums)
Subset Sum
Knapsack & Binary Linear Programming
NvdZvL (MFCS12):Reduce without adding variables

25. Further Results Using reduction to Subset Sum NvLvdZ (MFCS’12)
Theorem: Binary LP on 𝑛 vars, 𝑑 constraints, max int 𝑚 in time 𝑂 ∗ 𝑛 log 𝑚𝑛 𝑛 𝑂 𝑑 and poly space,
assuming random read-only access to random bits
Using reduction to Subset Sum NvLvdZ (MFCS’12)
Established by simple recursive rounding scheme
Time/space tradeoffs for List Disjointness
By extending techniques from BCM
List Disjointness in 𝑂 (𝑛 𝑝/𝑠 ) time given 𝑠≤ 𝑛 2 /𝑝 space
Theorem: Random 3-Sum in 𝑂 ( 𝑛 2.5 ) time and log 𝑛 space if ints independently and u.a.r from [𝑚] with 𝑚≤𝑝𝑜𝑙𝑦(𝑛) being a multiple of 𝑛

26. Further Research How strong is random oracle assumption exactly?
Weaker than the existence of sufficiently strong PRG’s
Still don’t know exact (low-space) complexity of ED!!
Can we do something specific for SSS?
Solve Subset Sum in time 𝑂 ∗ −𝜖 𝑛
Can restrict to 𝑤 1 ,…, 𝑤 𝑛, ,𝑡≤ 𝑛 AKKN (STACS’16)
Show 𝒞 1 𝒞 2 ≤ 𝑑 for UDCP 𝒞 1 , 𝒞 2 ∈{0,1 } 𝑑
True if 𝒞 1 ≥ 𝑑 AKKN (ISIT’16)
Big open question in information theory
Study Subset Sum combinatorics
If spike of size 2 𝜖𝑛 for some constant 𝜖>0, bound |{𝑤 𝑋 :𝑋⊆[𝑛]}|≤ 2 1− 𝜖 ′ 𝑛 for some 𝜖 ′ ≔𝑓(𝜀)>0

27. Take-home Messages Cycle finding is a great tool low space algo’s
Several worst-case algo’s for SSS are inspired on techniques from the literature of average-case complexity of SSS:
Cycle finding in poly space setting (this work)
Howgrave-Graham approach in exp. space setting (AKKN)
Win/win approach for many/few distinct sums
In exponential space setting it remains to `win’ in the case of few sums
Thanks for listening!!