Subset Sum and related problems Jesper Nederlof Eindhoven
Subset Sum (and related problems) Jesper Nederlof (Eindhoven University) Disclaimer: No turtles or hares were hurt in this research
• 0100111100100 00010100001101010010 Subset Sum Algo i’th bit? 0/1
Floyd: Cycle Finding BCM (FOCS 13): Shuffle function HS (JACM 72): Mit. M BCM: Element Distinctness Crypto: List merging List Disjointness (with small freqs 2) AKKN (STACS 16): Subset Sum distribution is smooth LN(STOC 10): Save Hash mod p space with DFT Subset Sum (many distinct sums) Subset Sum (few distinct sums) Subset Sum
Floyd: Cycle Finding BCM: Element Distinctness BCM (FOCS 13): Shuffle function
Element Distinctness (ED): BCM(FOCS 13) •
• s Floyd’s Cycle Finding
• Floyd’s Cycle Finding i s j • T = #steps turtle in first round (6 in ex) • p= stem-length (3 in ex), q=cycle length (6 in ex) • 2 T=T+xq -> T=xq -> T+p=xq+p
• Floyd’s Cycle Finding i s j
`Shuffling’ f
Floyd: Cycle Finding BCM: Element Distinctness BCM (FOCS 13): Shuffle function Crypto: List merging List Disjointness (with small freqs 2)
List Disjointness •
• List Disjointness
Floyd: Cycle Finding BCM (FOCS 13): Shuffle function HS (JACM 72): Mit. M BCM: Element Distinctness Crypto: List merging List Disjointness (with small freqs 2) Subset Sum (many distinct sums)
Meet in the Middle •
Meet in the Middle • new
Floyd: Cycle Finding BCM (FOCS 13): Shuffle function HS (JACM 72): Mit. M BCM: Element Distinctness Crypto: List merging AKKN (STACS 16): Subset Sum distribution is smooth List Disjointness (with small freqs 2) Subset Sum (many distinct sums)
Subset Sum Distribution is smooth (AKKN) Histogram 0 0 0 1 1 2 4 8 16 32 1 2 3 4 5 16
Subset Sum Distribution is smooth (AKKN) • Proof of Lemma uses a connection to UDCP’s
Uniquely Decodable Code Pairs (UDCP) • 1011100 1101101 0000000 1010011 0101010 1111101 1100111 1 0 0 1 1 1 0 ? ? 2 1 0011001 1010101 00110110 0 1 1 0 1 0 0
Uniquely Decodable Code Pairs (UDCP) •
Subset Sum Distribution is smooth (AKKN)
Floyd: Cycle Finding BCM (FOCS 13): Shuffle function HS (JACM 72): Mit. M BCM: Element Distinctness Crypto: List merging List Disjointness (with small freqs 2) AKKN (STACS 16): Subset Sum distribution is smooth LN(STOC 10): Save Hash mod p space with DFT Subset Sum (many distinct sums) Subset Sum (few distinct sums) Subset Sum
Subset Sum with few Distinct Sums
Floyd: Cycle Finding BCM (FOCS 13): Shuffle function HS (JACM 72): Mit. M BCM: Element Distinctness Crypto: List merging List Disjointness (with small freqs 2) AKKN (STACS 16): Subset Sum distribution is smooth LN(STOC 10): Save Hash mod p space with DFT Random Subset Sum k-Sum (many distinct sums) Subset Sum (few distinct sums) Subset Sum Knapsack & Binary Linear Programming Nvd. Zv. L (MFCS 12): Reduce without adding variables
Further Results •
Reducing a Target Interval to a Few Exact Queries (MFCS’ 12 Nv. Lvd. Z) – There is an algorithm that given • weights , and • non-negative integers with , – returns a set such that • for , and is an integer, • is at most , and • for every set it holds that if and only if there exist an index such that .
Representation Method [HJ’ 10] needle Only consider
Representation Method [HJ’ 10]
Concluding Remarks •
- Slides: 32