Truly Subcubic Algorithms for Language Edit Distance and RNA Folding - - PowerPoint PPT Presentation

Truly Subcubic Algorithms for Language Edit Distance and RNA Folding via Fast Bounded-Difference Min-Plus Product

Karl Bringmann, Fabrizio Grandoni, Barna Saha, Virginia Vassilevska Williams June 11, 2017

Bounded Differences (BD) Matrices

Integer matrix 𝑁 has BD if for all 𝑗, 𝑘:

𝑁 𝑗, 𝑘 − 𝑁[𝑗, 𝑘 + 1] ≤ 1 𝑁 𝑗, 𝑘 − 𝑁[𝑗 + 1, 𝑘] ≤ 1

and

More generally: 𝑿-BD when differences are at most 𝑋

2 2 3 2 1 1 2 3 2 1 2 3 1 1 2

(min,+) Product

For 𝑜×𝑜-matrices 𝐵, 𝐶, their (min,+) product 𝐷 = 𝐵 ∗ 𝐶 is defined by

𝐷 𝑗, 𝑘 = min

7 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘]

(min,+) product is equivalent to All Pairs Shortest Paths

[Fischer,Meyer’71]

trivial algorithm: 𝑃(𝑜;) best known algorithm: 𝑜;/2?( @AB C

• )

[Williams’14]

Standard matrix multiplication:

𝐷 𝑗, 𝑘 = E 𝐵 𝑗, 𝑙 ⋅ 𝐶[𝑙, 𝑘]

• 7

time 𝑃(𝑜G) where 𝜕 ≤ 2.373

(min,+) Product

For 𝑜×𝑜-matrices 𝐵, 𝐶, their (min,+) product 𝐷 = 𝐵 ∗ 𝐶 is defined by

𝐷 𝑗, 𝑘 = min

7 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘]

(min,+) product is equivalent to All Pairs Shortest Paths

[Fischer,Meyer’71]

trivial algorithm: 𝑃(𝑜;) best known algorithm: 𝑜;/2?( @AB C

• )

[Williams’14]

Big Open Problem: Is (min,+) product in time 𝑷(𝒐𝟒O𝜻) for some 𝜻 > 𝟏? Study special cases!

(min,+) Product for Structured Matrices

𝐵S 𝑗, 𝑘 = 𝑦U[V,W] 𝐷 𝑗, 𝑘 = min

7 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘]

𝐷′ 𝑗, 𝑘 = E 𝐵S 𝑗, 𝑙 ⋅ 𝐶S[𝑙, 𝑘]

• 7

𝐷 𝑗, 𝑘 = degree of highest monomial in 𝐷S[𝑗, 𝑘] Sketch: Matrices with small entries: If 𝐵, 𝐶 have entries in −𝑈, … , 𝑈 ∪ ∞ then 𝐵 ∗ 𝐶 can be computed in time 𝑃 i(𝑈𝑜G)

[Alon,Galil,Margalit’97]

(min,+) Product for Structured Matrices

Matrices with small entries: If 𝐵, 𝐶 have entries in −𝑈, … , 𝑈 ∪ ∞ then 𝐵 ∗ 𝐶 can be computed in time 𝑃 i(𝑈𝑜G)

[Alon,Galil,Margalit’97]

Matrices with few distinct entries: If each row of 𝐵 has a small number of distinct entries, then for arbitrary 𝐶 we can compute 𝐵 ∗ 𝐶 in truly subcubic time

[Yuster’09]

Question: Is (min,+) product in time 𝑷(𝒐𝟒O𝜻) for BD matrices?

Why care about BD matrices?

1st Application: Language Edit Distance (LED)

CFG Parsing: Given a context-free grammar 𝐻 and a string 𝑡 of length 𝑜, is 𝑡 in 𝑀(𝐻)?

[L. Valiant’75]

Language Edit Distance: „error-correcting CFG parsing“ Given a CFG 𝐻 and a string 𝑡, compute minimum edit distance of 𝑡 to any string in 𝑀(𝐻) ... is in time 𝑃 i(𝑜G) insertions, deletions, substitutions for simplicity: |𝐻| = 𝑃(1)

[Aho,Peterson’72]

... is in time 𝑃(𝑜;) We show using Valiant’s approach: If (min,+) product on BD matrices is in time 𝑃(𝑜n), then LED is in time 𝑃 i(𝑜n) ~8 page proof intuitive reason for BD: LED(𝑡) and LED(𝑡𝑑) differ by ≤ 1 for any symbol 𝑑

2nd Application: RNA Folding

RNA can be seen as a sequence of symbols from {A,C,G,U} Biologists want to predict the secondary structure of RNA: A can pair with U, and C can pair with G Given an RNA sequence, find the largest set of matching pairs, such that no two pairs intersect AUUGCAG not allowed but AUUGCAG is okay

Disclaimer: No author of this paper is a biologist.

[Nussinov,Jacobson’80]

... is in time 𝑃(𝑜;) ... can be cast as a LED problem (without substitutions) If (min,+) product on BD matrices is in time 𝑃(𝑜n), then RNA Folding is in time 𝑃 i(𝑜n)

3rd Application: Optimal Stack Generation

Optimal Stack Generation: Given a string 𝑡 over alphabet Σ, determine the shortest sequence of stack

• perations push(.), emit, pop s.t. performing these operations starting from

an empty stack will emit 𝑡 and end with an empty stack for simplicity: |Σ| = 𝑃(1) ... is in time 𝑃(𝑜;) (dynamic programming)

[Tarjan’05]

We show: If (min,+) product on O(1)-BD matrices is in time 𝑃(𝑜n), then Optimal Stack Generation is in time 𝑃 i(𝑜n) 𝑡 = bab push(b) emit push(a) emit pop emit pop b b b b b b a a b a b intuitive reason for BD: OSG(𝑡) and OSG(𝑡𝑑) differ by ≤ 3 for any 𝑑 ∈ Σ

Main Result

... so we have seen that (min,+) product of BD matrices is well motivated Main Result: We can compute the (min,+) product of BD matrices Generalization: For 𝑿-BD matrix 𝐵 with 𝑋 ≪ 𝑜;OG ≈ 𝑜t.uvu and arbitrary 𝐶 we can compute their (min,+) product in randomized truly subcubic time here: 𝑷(𝒐𝟑.𝟘) in randomized time 𝑃(𝑜v.y;) and deterministic time 𝑃(𝑜v.yz)

Algorithm Sketch

Input: BD matrices 𝐵, 𝐶. Want: 𝐷 𝑗, 𝑘 = min

7 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘]

1) Compute approximation 𝐸 𝑗, 𝑘 = 𝐷 𝑗, 𝑘 ± 𝑃 𝑜t.v

compute 𝐷 𝑗, 𝑘 exactly for all 𝑗, 𝑘 that are multiples of 𝑜t.v

(𝑗, 𝑘)

set 𝐸 𝑗, 𝑘 to some 𝐷[𝑗’, 𝑘’] by rounding 𝑗, 𝑘 If 𝐵, 𝐶 are BD, then their (min,+) product is also BD

(𝑗S, 𝑘S) 𝑜t.v time 𝑃(𝑜v.u)

Algorithm Sketch

Input: BD matrices 𝐵, 𝐶. Want: 𝐷 𝑗, 𝑘 = min

7 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘]

1) Compute approximation 𝐸 𝑗, 𝑘 = 𝐷 𝑗, 𝑘 ± 𝑃 𝑜t.v 𝐵 𝑗, 𝑙 + 𝐶 𝑙, 𝑘 = 𝐷 𝑗, 𝑘 implies 𝐵 𝑗, 𝑙 + 𝐶 𝑙, 𝑘 − 𝐸 𝑗, 𝑘 ≤ 𝑃 𝑜t.v call these triples (𝑗, 𝑙, 𝑘) relevant then 𝐷 𝑗, 𝑘 = min

7:(V,7,W) •€@€•‚ƒ„ 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘]

Algorithm Sketch

Input: BD matrices 𝐵, 𝐶. Want: 𝐷 𝑗, 𝑘 = min

7 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘]

2) Cover most relevant triples: fix 𝑗∗, 𝑘∗, and define matrices 𝐵∗, 𝐶∗ 𝐵∗ 𝑗, 𝑙 ≔ 𝐵 𝑗, 𝑙 + 𝐶 𝑙, 𝑘∗ − 𝐸 𝑗, 𝑘∗ − 𝐵 𝑗∗, 𝑙 + 𝐶 𝑙, 𝑘∗ − 𝐸 𝑗∗, 𝑘∗ 𝐶∗ 𝑙, 𝑘 ≔ 𝐵 𝑗∗, 𝑙 + 𝐶 𝑙, 𝑘 − 𝐸 𝑗∗, 𝑘 (min,+) product 𝐷∗ of 𝐵∗, 𝐶∗: 𝐷∗ 𝑗, 𝑘 = min

7 𝐵∗ 𝑗, 𝑙 + 𝐶∗ 𝑙, 𝑘

can be cancelled afterwards = 𝐷 𝑗, 𝑘 − 𝐸 𝑗, 𝑘∗ + 𝐸 𝑗∗, 𝑘∗ − 𝐸 𝑗∗, 𝑘

|𝐵 𝑗, 𝑙 + 𝐶 𝑙, 𝑘 − 𝐸 𝑗, 𝑘 | ≤ 𝑃 𝑜t.v (𝑗, 𝑙, 𝑘) relevant:

1) Compute approximation 𝐸 𝑗, 𝑘 = 𝐷 𝑗, 𝑘 ± 𝑃 𝑜t.v

Algorithm Sketch

Input: BD matrices 𝐵, 𝐶. Want: 𝐷 𝑗, 𝑘 = min

7 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘]

2) Cover most relevant triples: fix 𝑗∗, 𝑘∗, and define matrices 𝐵∗, 𝐶∗ 𝐵∗ 𝑗, 𝑙 ≔ 𝐵 𝑗, 𝑙 + 𝐶 𝑙, 𝑘∗ − 𝐸 𝑗, 𝑘∗ − 𝐵 𝑗∗, 𝑙 + 𝐶 𝑙, 𝑘∗ − 𝐸 𝑗∗, 𝑘∗ 𝐶∗ 𝑙, 𝑘 ≔ 𝐵 𝑗∗, 𝑙 + 𝐶 𝑙, 𝑘 − 𝐸 𝑗∗, 𝑘 1) Compute approximation 𝐸 𝑗, 𝑘 = 𝐷 𝑗, 𝑘 ± 𝑃 𝑜t.v if 𝑗, 𝑙, 𝑘∗ , 𝑗∗, 𝑙, 𝑘∗ , 𝑗∗, 𝑙, 𝑘 are all relevant, then 𝐵∗ 𝑗, 𝑙 , 𝐶∗ 𝑙, 𝑘 = 𝑃 𝑜t.v set all 𝛻(𝑜t.v)-entries of 𝐵∗, 𝐶∗ to ∞ then (min,+) product of 𝐵∗ and 𝐶∗ can be computed in time 𝑃 i(𝑜G‡t.v) (𝑗, 𝑙, 𝑘) is „covered“ if 𝐵∗ 𝑗, 𝑙 and 𝐶∗ 𝑙, 𝑘 are 𝑃(𝑜t.v), i.e., not set to ∞

|𝐵 𝑗, 𝑙 + 𝐶 𝑙, 𝑘 − 𝐸 𝑗, 𝑘 | ≤ 𝑃 𝑜t.v (𝑗, 𝑙, 𝑘) relevant:

Algorithm Sketch

Input: BD matrices 𝐵, 𝐶. Want: 𝐷 𝑗, 𝑘 = min

7 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘]

2) Cover most relevant triples: repeat for 𝑃(𝑜t.; log 𝑜) rounds: 𝐵∗ 𝑗, 𝑙 ≔ 𝐵 𝑗, 𝑙 + 𝐶 𝑙, 𝑘∗ − 𝐸 𝑗, 𝑘∗ − 𝐵 𝑗∗, 𝑙 + 𝐶 𝑙, 𝑘∗ − 𝐸 𝑗∗, 𝑘∗ 𝐶∗ 𝑙, 𝑘 ≔ 𝐵 𝑗∗, 𝑙 + 𝐶 𝑙, 𝑘 − 𝐸 𝑗∗, 𝑘 1) Compute approximation 𝐸 𝑗, 𝑘 = 𝐷 𝑗, 𝑘 ± 𝑃 𝑜t.v set all 𝛻(𝑜t.v)-entries of 𝐵∗, 𝐶∗ to ∞

|𝐵 𝑗, 𝑙 + 𝐶 𝑙, 𝑘 − 𝐸 𝑗, 𝑘 | ≤ 𝑃 𝑜t.v (𝑗, 𝑙, 𝑘) relevant:

initialize 𝐷 ˆ 𝑗, 𝑘 ≔ ∞ pick 𝑗∗, 𝑘∗ randomly compute (min,+) product 𝐷∗ = 𝐵∗ ∗ 𝐶∗ 𝐷 ˆ 𝑗, 𝑘 ≔ min 𝐷 ˆ 𝑗, 𝑘 , 𝐷∗ 𝑗, 𝑘 + 𝐸 𝑗, 𝑘∗ − 𝐸 𝑗∗, 𝑘∗ + 𝐸 𝑗∗, 𝑘

(𝑗, 𝑙, 𝑘) is „covered“ if 𝐵∗ 𝑗, 𝑙 and 𝐶∗ 𝑙, 𝑘 are 𝑃(𝑜t.v) in some round

Lem: After 𝑃(𝑜‰ log 𝑜) rounds there are 𝑃(𝑜;O‰/; + 𝑜v.Š) uncovered relevant triples w.h.p. = 𝑃 𝑜v.‹ time 𝑃 𝑜G‡t.v = 𝑃(𝑜v.u) 𝑃 i(𝑜t.;) iterations total time 𝑃 i 𝑜v.‹

Algorithm Sketch

Input: BD matrices 𝐵, 𝐶. Want: 𝐷 𝑗, 𝑘 = min

7 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘]

2) Cover most relevant triples 1) Compute approximation 𝐸 𝑗, 𝑘 = 𝐷 𝑗, 𝑘 ± 𝑃 𝑜t.v

|𝐵 𝑗, 𝑙 + 𝐶 𝑙, 𝑘 − 𝐸 𝑗, 𝑘 | ≤ 𝑃 𝑜t.v (𝑗, 𝑙, 𝑘) relevant:

3) Enumerate uncovered relevant triples: ”for each uncovered relevant (𝑗, 𝑙, 𝑘):“ 𝐷 ˆ 𝑗, 𝑘 ≔ min 𝐷 ˆ 𝑗, 𝑘 , 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘]

(𝑗, 𝑙, 𝑘) is „covered“ if 𝐵∗ 𝑗, 𝑙 and 𝐶∗ 𝑙, 𝑘 are 𝑃(𝑜t.v) in some round

now 𝐷 ˆ is correct output

Algorithm Sketch

Input: BD matrices 𝐵, 𝐶. Want: 𝐷 𝑗, 𝑘 = min

7 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘]

2) Cover most relevant triples 1) Compute approximation 𝐸 𝑗, 𝑘 = 𝐷 𝑗, 𝑘 ± 𝑃 𝑜t.v

|𝐵 𝑗, 𝑙 + 𝐶 𝑙, 𝑘 − 𝐸 𝑗, 𝑘 | ≤ 𝑃 𝑜t.v (𝑗, 𝑙, 𝑘) relevant:

3) Enumerate uncovered relevant triples: (𝑗, 𝑘) (𝑗S, 𝑘S) for all 𝑗S, 𝑙S, 𝑘S divisible by 𝑜t.v: if 𝑗S, 𝑙S, 𝑘S is relevant and uncovered: for all 𝑗S − 𝑜t.v < 𝑗 ≤ 𝑗S, 𝑙S − 𝑜t.v < 𝑙 ≤ 𝑙S, 𝑘S − 𝑜t.v < 𝑘 ≤ 𝑘S: 𝐷 ˆ 𝑗, 𝑘 ≔ min 𝐷 ˆ 𝑗, 𝑘 , 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘] now 𝐷 ˆ is correct output

(𝑗, 𝑙, 𝑘) is „covered“ if 𝐵∗ 𝑗, 𝑙 and 𝐶∗ 𝑙, 𝑘 are 𝑃(𝑜t.v) in some round

Algorithm Sketch

Input: BD matrices 𝐵, 𝐶. Want: 𝐷 𝑗, 𝑘 = min

7 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘]

2) Cover most relevant triples 1) Compute approximation 𝐸 𝑗, 𝑘 = 𝐷 𝑗, 𝑘 ± 𝑃 𝑜t.v

|𝐵 𝑗, 𝑙 + 𝐶 𝑙, 𝑘 − 𝐸 𝑗, 𝑘 | ≤ 𝑃 𝑜t.v (𝑗, 𝑙, 𝑘) relevant:

3) Enumerate uncovered relevant triples: for all 𝑗S, 𝑙S, 𝑘S divisible by 𝑜t.v: if 𝑗S, 𝑙S, 𝑘S is relevant and uncovered: for all 𝑗S − 𝑜t.v < 𝑗 ≤ 𝑗S, 𝑙S − 𝑜t.v < 𝑙 ≤ 𝑙S, 𝑘S − 𝑜t.v < 𝑘 ≤ 𝑘S: 𝐷 ˆ 𝑗, 𝑘 ≔ min 𝐷 ˆ 𝑗, 𝑘 , 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘]

(𝑗, 𝑙, 𝑘) is „covered“ if 𝐵∗ 𝑗, 𝑙 and 𝐶∗ 𝑙, 𝑘 are 𝑃(𝑜t.v) in some round

either all or none are relevant and uncovered 𝑃(𝑜v.•) iterations time 𝑃(𝑜t.; log 𝑜) total time 𝑃(𝑜v.‹)

= number of relevant uncovered triples

total time of algorithm 𝑃 i 𝑜v.‹ now 𝐷 ˆ is correct output

Correctness

𝐵∗ 𝑗, 𝑙 ≔ 𝐵 𝑗, 𝑙 + 𝐶 𝑙, 𝑘∗ − 𝐸 𝑗, 𝑘∗ − 𝐵 𝑗∗, 𝑙 + 𝐶 𝑙, 𝑘∗ − 𝐸 𝑗∗, 𝑘∗ 𝐶∗ 𝑙, 𝑘 ≔ 𝐵 𝑗∗, 𝑙 + 𝐶 𝑙, 𝑘 − 𝐸 𝑗∗, 𝑘 set all 𝛻(𝑜t.v)-entries of 𝐵∗, 𝐶∗ to ∞

|𝐵 𝑗, 𝑙 + 𝐶 𝑙, 𝑘 − 𝐸 𝑗, 𝑘 | ≤ 𝑃 𝑜t.v (𝑗, 𝑙, 𝑘) relevant:

pick 𝑗∗, 𝑘∗ randomly

(𝑗, 𝑙, 𝑘) is „covered“ if 𝐵∗ 𝑗, 𝑙 and 𝐶∗ 𝑙, 𝑘 are 𝑃(𝑜t.v) in some round

... ... Lem: After 𝑃(𝑜‰ log 𝑜) rounds there are 𝑃(𝑜;O‰/; + 𝑜v.Š) uncovered relevant triples w.h.p. If 𝑗, 𝑙, 𝑘∗ , 𝑗∗, 𝑙, 𝑘∗ , 𝑗∗, 𝑙, 𝑘 are all relevant, then 𝑗, 𝑙, 𝑘 is covered

Correctness

|𝐵 𝑗, 𝑙 + 𝐶 𝑙, 𝑘 − 𝐸 𝑗, 𝑘 | ≤ 𝑃 𝑜t.v (𝑗, 𝑙, 𝑘) relevant: (𝑗, 𝑙, 𝑘) is „covered“ if 𝐵∗ 𝑗, 𝑙 and 𝐶∗ 𝑙, 𝑘 are 𝑃(𝑜t.v) in some round

Lem: After 𝑃(𝑜‰ log 𝑜) rounds there are 𝑃(𝑜;O‰/; + 𝑜v.Š) uncovered relevant triples w.h.p. If 𝑗, 𝑙, 𝑘∗ , 𝑗∗, 𝑙, 𝑘∗ , 𝑗∗, 𝑙, 𝑘 are all relevant, then 𝑗, 𝑙, 𝑘 is covered When are 𝑗, 𝑙, 𝑘∗ , 𝑗∗, 𝑙, 𝑘∗ , 𝑗∗, 𝑙, 𝑘 all relevant? bipartite graph 𝐻7: 𝑗 𝑘 if (𝑗, 𝑙, 𝑘) is relevant we „cover“ a relevant triple (𝑗, 𝑙, 𝑘) if 𝑗, 𝑘, 𝑗∗, 𝑘∗ form a 4-cycle in 𝐻7 need: 𝐻7 contains many 4-cycles Any bipartite graph with 𝑛 ≥ 4𝑜‘.Š edges contains Ω 𝑛•/𝑜• 4-cyles. Lem: have: many relevant triples 𝑘∗ 𝑗∗

Algorithm Recap

Input: BD matrices 𝐵, 𝐶. Want: 𝐷 𝑗, 𝑘 = min

7 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘]

1) Compute approximation 𝐸 𝑗, 𝑘 = 𝐷 𝑗, 𝑘 ± 𝑃 𝑜t.v

compute 𝐷 𝑗, 𝑘 exactly for all 𝑗, 𝑘 that are multiples of 𝑜t.v set 𝐸 𝑗, 𝑘 to some 𝐷[𝑗’, 𝑘’] by rounding 𝑗, 𝑘

2) Cover most relevant triples:

repeat for 𝑃(𝑜t.; log 𝑜) rounds: 𝐵∗ 𝑗, 𝑙 ≔ 𝐵 𝑗, 𝑙 + 𝐶 𝑙, 𝑘∗ − 𝐸 𝑗, 𝑘∗ − 𝐵 𝑗∗, 𝑙 + 𝐶 𝑙, 𝑘∗ − 𝐸 𝑗∗, 𝑘∗ 𝐶∗ 𝑙, 𝑘 ≔ 𝐵 𝑗∗, 𝑙 + 𝐶 𝑙, 𝑘 − 𝐸 𝑗∗, 𝑘 set all 𝛻(𝑜t.v)-entries of 𝐵∗, 𝐶∗ to ∞ initialize 𝐷 ˆ 𝑗, 𝑘 ≔ ∞ pick 𝑗∗, 𝑘∗ randomly compute (min,+) product 𝐷∗ = 𝐵∗ ∗ 𝐶∗ 𝐷 ˆ 𝑗, 𝑘 ≔ min 𝐷 ˆ 𝑗, 𝑘 , 𝐷∗ 𝑗, 𝑘 + 𝐸 𝑗, 𝑘∗ − 𝐸 𝑗∗, 𝑘∗ + 𝐸 𝑗∗, 𝑘

3) Enumerate uncovered relevant triples:

for all 𝑗S, 𝑙S, 𝑘S divisible by 𝑜t.v: if 𝑗S, 𝑙S, 𝑘S is relevant and uncovered: for all 𝑗S − 𝑜t.v < 𝑗 ≤ 𝑗S, 𝑙S−𝑜t.v < 𝑙 ≤ 𝑙S, 𝑘S−𝑜t.v < 𝑘 ≤ 𝑘S: 𝐷 ˆ 𝑗, 𝑘 ≔ min 𝐷 ˆ 𝑗, 𝑘 , 𝐵 𝑗, 𝑙 + 𝐶[𝑙, 𝑘]

|𝐵 𝑗, 𝑙 + 𝐶 𝑙, 𝑘 − 𝐸 𝑗, 𝑘 | ≤ 𝑃 𝑜t.v (𝑗, 𝑙, 𝑘) relevant: (𝑗, 𝑙, 𝑘) is „covered“ if 𝐵∗ 𝑗, 𝑙 and 𝐶∗ 𝑙, 𝑘 are 𝑃(𝑜t.v) in some round

Conclusion

(min,+) product of BD matrices can be solved in rand. time 𝑃(𝑜v.y;) we generalize the subcubic special cases of (min,+) matrix multiplication: this yields subcubic 𝑃(𝑜v.y;) algorithms for:

• Language Edit Distance, a classic parsing problem from ‘72
• RNA Folding, a classic bioinformatics problem from ‘80
• Optimal Stack Generation, an open problem by Tarjan

Open Problems: Conditional lower bounds imply that LED and RNA Folding are in Ω “ 𝑜G , 1) What is the right exponent? 2) Find more applications of BD (min,+) product

[Abboud,Backurs,V-Williams15]