Faster Space-Efficient PowerPoint Presentation

Faster Space-Efficient Algorithms for Subset Sum, k-Sum and related problems

Nikhil Bansal, Shashwat Garg, Jesper Nederlof, Nikhil Vyas

(available at arXiv:1612.02788)

disclaimer: no turtles or hares were harmed during this research

Given: integers

Asked:

is there an

with

?Focus on instances with small The `classic’ results: time, space (trivial) time, space HS(JACM72)Introduces `Meet-in-the-Middle’ (MitM) approach time, space SS(SICOMP81)

Subset Sum

Main Result: There is a Monte Carlo algorithm for Subset Sum using

time and space,assuming random read-only access to random bits

Algo

010011110010000010100001100001101010010

0/1

i’th

bit?

time

Subset Sum (many distinct sums)

BCM:Element

Distinctness

Floyd:

Cycle FindingAKKN (STACS16): Subset Sum distribution is smoothHS (JACM72): MitMList Disjointness(with small freqs2)

Crypto: List merging

Subset Sum

(few distinct sums)

BCM (FOCS13):Shuffle function

Hash mod p

LN(STOC10): Save space with DFT

Subset Sum

BCM:Element

Distinctness

Floyd:

Cycle Finding

BCM (FOCS13):Shuffle function

Element Distinctness (ED) by BCM

Given list

with

Asked

if all values are distinctIn time, space, time, space Theorem(BCM): time, space, assuming random read-only access to random bits

To prove, let’s first assume

, and let

Thus we seek with  

Floyd’s Cycle Finding

Finds such using little space

Basic algo in crypto, much more obscure in TCS

View

as digraph (with arcs ) 

s

T = #steps turtle (6 in ex)

p= stem-length (3 in ex), q=cycle length (6 in ex)2T=T+xq -> T=xq ->

T+p=xq+p

Floyd’s Cycle Finding

Finds such using little spaceBasic algo in crypto, much more obscure in TCSView as digraph (with arcs ) 

i

j

s

Floyd’s Cycle Finding

Finds such using little space

Basic algo in crypto, much more obscure in TCS

View

as digraph (with arcs ) 

i

j

Only works if start outside cycle!

Works well if

is random:

Probly

reached after

steps (birthday paradox)

s

`Shuffling’ f

What if is not random? `shuffle’

Let

be a random function

Cannot remember , but use assumed oracleDefine Use Floyd to sample such that is a bad pair if but Expect at most bad pairsUsing

samples, expect to see real solution

Theorem(BCM):

time, space, assuming random read-only access to random bits

BCM:Element

Distinctness

Floyd:

Cycle Finding

BCM (FOCS13):Shuffle functionList Disjointness(with small freqs2) Crypto: List merging

Given two lists

,

Asked

do they share a common value? Very similar to ED; but want values from different listsDefine i.e p Counts number of pseudo-solutions

List

Disjointness

Theorem: There is an time, space algorithm for List Disjointness, if given

assuming

random read-only access to random bits

Define

by

merging

E.g. just concatenate

and In paper we set or with prob .5Sample such that as beforeIf and , also check or Need samplesExpect

vertices needed for a sample

List

DisjointnessTheorem: There is an time, space algorithm for List Disjointness

, if given

assuming

random read-only access to random bits

BCM:Element

Distinctness

Floyd:

Cycle Finding

Subset Sum (many distinct sums)HS (JACM72): MitMList Disjointness(with small freqs2) Crypto: List merging

BCM (FOCS13):Shuffle function

Meet in the Middle

,

LD instance solved in

time, which is

. Also uses

space

Ints

. Denote w

Reduce SSS on

integers to List

Disjointness

on lists of length

run the sorting

algo

Meet in the Middle

Ints

. Denote w

Reduce SSS on

integers to List Disjointness on lists of length run the sorting algo.  space,

time

If

,

and no improvement

How do instances with

look?

new

,

Subset Sum (many distinct sums)

BCM:Element

Distinctness

Floyd:

Cycle FindingAKKN (STACS16): Subset Sum distribution is smoothHS (JACM72): MitMList Disjointness(with small freqs2)

Crypto: List merging

BCM (FOCS13):

Shuffle function

Histogram

0 0 0 0 0

1

1 2 4

8 16321 2 3 4 516

Histogram

0 0 0 0 0

1

1 2 4 8 16321 2 3 4 516Subset Sum Distribution is smooth (AKKN)Lemma:

We use this as follows:Suppose

then

and

thus  Lemma: If , can solve SSS in

time and

space,

assuming random read-only access to random bits

 Lemma:

Subset Sum Distribution is smooth (AKKN)

Proof sketch:

There exists a

frequent sum

s.t

.

;

Let

be such that for all

implies

Then |

:

Suppose

(add in

)

Thus

Subset Sum Distribution is smooth (AKKN)

Lemma

:

Subset Sum (many distinct sums)

BCM:Element

Distinctness

Floyd:

Cycle FindingAKKN (STACS16): Subset Sum distribution is smoothHS (JACM72): MitMList Disjointness(with small freqs2)

Crypto: List merging

Subset Sum

(few distinct sums)

BCM (FOCS13):Shuffle function

Hash mod p + DFT

Subset Sum

Subset Sum with few Distinct Sums

Lemma: Can solve instance

in time

)

and space.  Done by hashing numbers mod a prime of order and run time space algorithm, that uses DFT.Combining with previous lemma we obtain 

Main Result’: There is a Monte Carlo for Subset Sum using

time and

space

,assuming random read-only access to random bits  Left out many optimization to get

Subset Sum (many distinct sums)

BCM:Element

Distinctness

Floyd:

Cycle FindingAKKN (STACS16): Subset Sum distribution is smoothHS (JACM72): MitMList Disjointness(with small freqs2)

Crypto: List merging

Subset Sum

(few distinct sums)

BCM (FOCS13):Shuffle function

Subset Sum

Random k-Sum

Knapsack & Binary Linear Programming

NvdZvL

(MFCS12):Reduce without adding variables

Hash mod p + DFT

Further Results

Using reduction to Subset SumAlso time/space tradeoffs for List Disjointness using methods of BCM

List Disjointness in

time given

space.  Theorem: Binary LP on vars and constraints in time and space where is max integer,assuming random read-only access to random bits 

Further Research

How strong is random bits assumptions exactly?Weaker than the existence of sufficiently strong PRG’sStill don’t know the exact (low-space) complexity of ED!!Can we do something problem specific?

Solve Subset Sum in time

Great open question, progress made recently (

AKKN)Study Subset Sum combinatoricsConnection by AKKN relates this to UDCP’sIf spike of size for some constant , upper bound for some depending on  

Take-home MessagesCycle finding is a great tool low space

algo’sWin/win approach for many/few distinct sums

Thanks for listening!!Slides available at

http://www.win.tue.nl/~

jnederlo/;paper available at arXiv