Faster Pattern Matching with Mismatches in Compressed Texts Karl - - PowerPoint PPT Presentation

Few Matches or Almost Periodicity:

Faster Pattern Matching with Mismatches in Compressed Texts

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz

Max Planck Institute for Informatics, Saarland Informatics Campus (SIC), Saarbrücken, Germany

April 25, 2020

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Pattern Matching with Mismatches Pattern Matching

Given a text t and a pattern p, is p a substring of t? Finding ANPAN, k = 2 t p Finding CAKE P A N C A K E C A K E

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Pattern Matching with Mismatches

Pattern Matching with Mismatches Given a text t, a pattern p, and an integer k, does t have a length-|p| substring with Hamming-distance at most k to p? t p Finding ANPAN, k = 2 P A N C A K E A N P A N

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Pattern Matching with Mismatches

Pattern Matching with Mismatches Given a text t, a pattern p, and an integer k, does t have a length-|p| substring with Hamming-distance at most k to p?

• Thm. [Gawrychowski,Uznanski’18]

Pattern matching with k mismatches on a text of length n and a pattern of length m can be solved in time

∼

O((m + k√m) · n/m).

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Pattern Matching with Mismatches

Pattern Matching with Mismatches Given a text t, a pattern p, and an integer k, does t have a length-|p| substring with Hamming-distance at most k to p?

• Thm. [Gawrychowski,Uznanski’18]

Pattern matching with k mismatches on a text of length n and a pattern of length m can be solved in time

∼

O((m + k√m) · n/m). Matching (conditional) lower bound [GU’18]

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

What if the text is much larger than the pattern? ANPANISAJAPANESESWEETROLLMOSTCOMMONLYFILLEDWITHREDBEANPASTEANPANCANALSOBEPREPAREDWITHOTHERFILLINGSINCLUDINGWHITEBEANSGRE ANPAN

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

What if the text is much larger than the pattern?

ANPANISAJAPANESESWEETROLLMOSTCOMMONLYFILLEDWITHREDBEANPASTEANPANCANALSOBEPREPAREDWITHOTHERFILLINGSINCLUDINGWHITEBEANSGREENBEANSSESAMEANDCHESTNUT

ANPAN

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

What if the text is much larger than the pattern and given in a compressed representation?

ANPANISAJAPANESESWEETROLLMOSTCOMMONLYFILLEDWITHREDBEANPASTEANPANCANALSOBEPREPAREDWITHOTHERFILLINGSINCLUDINGWHITEBEANSGREENBEANSSESAMEANDCHESTNUT

ANPAN

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Grammar Compression Straight-Line Program (SLP)

A Straight-Line Program or SLP T is a context-free grammar that generates exactly one string eval(T ).

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Grammar Compression Straight-Line Program (SLP)

An SLP T is a set of non-terminals {T1, . . . , Tn} and productions of the form Ti → σ or Ti → TℓTr, where ℓ, r < i. We write eval(T ) = eval(Tn) for the generated string.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Grammar Compression Straight-Line Program (SLP)

An SLP T is a set of non-terminals {T1, . . . , Tn} and productions of the form Ti → σ or Ti → TℓTr, where ℓ, r < i. We write eval(T ) = eval(Tn) for the generated string. T1 → A; T2 → N; T3 → P T3 P

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Grammar Compression Straight-Line Program (SLP)

An SLP T is a set of non-terminals {T1, . . . , Tn} and productions of the form Ti → σ or Ti → TℓTr, where ℓ, r < i. We write eval(T ) = eval(Tn) for the generated string. T1 → A; T2 → N; T3 → P T4 → T1T2; T5 → T4T3 A N P T1 T2 T3 T4 T5

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Grammar Compression Straight-Line Program (SLP)

An SLP T is a set of non-terminals {T1, . . . , Tn} and productions of the form Ti → σ or Ti → TℓTr, where ℓ, r < i. We write eval(T ) = eval(Tn) for the generated string. T1 → A; T2 → N; T3 → P T4 → T1T2; T5 → T4T3 T6 → T5T4 A N P T1 A T2 N T1 T2 T3 T4 T4 T5 T6

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Grammar Compression Straight-Line Program (SLP)

An SLP T is a set of non-terminals {T1, . . . , Tn} and productions of the form Ti → σ or Ti → TℓTr, where ℓ, r < i. We write eval(T ) = eval(Tn) for the generated string. T1 → A; T2 → N; T3 → P T4 → T1T2; T5 → T4T3 T6 → T5T4; T7 → T6T4 A N P T1 A T2 N T3 P T1 A T2 N T1 T2 T3 T4 T4 T4 T5 T5 T6 T7

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Known Results

Problem uncompressed LZW/LZ78 text SLP text n = Ω( √ N) n = Ω(log N) Pattern O(N + m) O(n + m) ※

∼

O(n + m) ※ Matching [KMP’77] [G’12] [J’15] PM with k

∼

O( N

m(m + k√m))

O(n√mk2)

∼

O(nm poly(k)) Mismatches [GU’18] [GS’13] [T’14,BLRS’15] N: length of uncompressed text m: length of pattern n: length of compressed text ※: allows compressed pattern k: number of mismatches

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Known Results

Problem uncompressed LZW/LZ78 text SLP text n = Ω( √ N) n = Ω(log N) Pattern O(N + m) O(n + m) ※

∼

O(n + m) ※ Matching [KMP’77] [G’12] [J’15] PM with k

∼

O( N

m(m + k√m))

O(n√mk2)

∼

O(nm poly(k)) Mismatches [GU’18] [GS’13] [T’14,BLRS’15] N: length of uncompressed text m: length of pattern n: length of compressed text ※: allows compressed pattern k: number of mismatches

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Known Results

Problem uncompressed LZW/LZ78 text SLP text n = Ω( √ N) n = Ω(log N) Pattern O(N + m) O(n + m) ※

∼

O(n + m) ※ Matching [KMP’77] [G’12] [J’15] PM with k

∼

O( N

m(m + k√m))

O(n√mk2)

∼

O(nm poly(k)) Mismatches [GU’18]

∼

O(nk4 + mk) N: length of uncompressed text m: length of pattern n: length of compressed text ※: allows compressed pattern k: number of mismatches

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Known Results

Problem uncompressed LZW/LZ78 text SLP text n = Ω( √ N) n = Ω(log N) Pattern O(N + m) O(n + m) ※

∼

O(n + m) ※ Matching [KMP’77] [G’12] [J’15] PM with k

∼

O( N

m(m + k√m))

O(n√mk2)

∼

O(nm poly(k)) Mismatches [GU’18]

∼

O(nk4 + mk) N: length of uncompressed text m: length of pattern n: length of compressed text ※: allows compressed pattern k: number of mismatches Improvement obtained via new structural insight in solution structure

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Solution Structure of Pattern Matching Fact (Folklore)

Let text t and pattern p, |t| ≤ 3

2|p|, be given such that there are ≥ 2 matches

• f p in t that together match t completely. Then, both p and t are periodic

with some period x and every match of p in t starts at a position 1 + i · |x|.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Solution Structure of Pattern Matching Fact (Folklore)

Let text t and pattern p, |t| ≤ 3

2|p|, be given such that there are ≥ 2 matches

• f p in t that together match t completely. Then, both p and t are periodic

with some period x and every match of p in t starts at a position 1 + i · |x|. p t

p p

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Solution Structure of Pattern Matching Fact (Folklore)

Let text t and pattern p, |t| ≤ 3

2|p|, be given such that there are ≥ 2 matches

• f p in t that together match t completely. Then, both p and t are periodic

with some period x and every match of p in t starts at a position 1 + i · |x|. p t x x

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Solution Structure of Pattern Matching Fact (Folklore)

Let text t and pattern p, |t| ≤ 3

2|p|, be given such that there are ≥ 2 matches

• f p in t that together match t completely. Then, both p and t are periodic

with some period x and every match of p in t starts at a position 1 + i · |x|. p t x x x

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Solution Structure of Pattern Matching Fact (Folklore)

Let text t and pattern p, |t| ≤ 3

2|p|, be given such that there are ≥ 2 matches

• f p in t that together match t completely. Then, both p and t are periodic

with some period x and every match of p in t starts at a position 1 + i · |x|. p t x x x x x

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Solution Structure of Pattern Matching Fact (Folklore)

Let text t and pattern p, |t| ≤ 3

2|p|, be given such that there are ≥ 2 matches

• f p in t that together match t completely. Then, both p and t are periodic

with some period x and every match of p in t starts at a position 1 + i · |x|. p t x x x x x x x x x x x x x

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

What is the solution structure of Pattern Matching with Mismatches?

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Solution Structure of Pattern Matching with Mismatches

If there are at least 2 k-matches of p in t, then p and t are periodic and every k-match of p starts at a position 1 + i|x|?

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Solution Structure of Pattern Matching with Mismatches

If there are at least two k-matches of p in t, then p and t are periodic and every k-match of p starts at a position 1 + i|x|?

t p A A A A B B B B A A B B · · · · · · · · · · · · · · · · · · Am Bm Am/2 Bm/2

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Solution Structure of Pattern Matching with Mismatches

If there are at least two k-matches of p in t, then p and t are periodic and every k-match of p starts at a position 1 + i|x|?

t p A A A A B B B B A A B B · · · · · · · · · · · · · · · · · · Am Bm Am/2 Bm/2 p and t not periodic, but 2k k-matches of p in t

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Solution Structure of Pattern Matching with Mismatches

If there are at least two Ω(poly(k)) k-matches of p in t, then p and t are periodic and every k-match of p starts at a position 1 + i|x|?

t p A A A A B B B B A A B B · · · · · · · · · · · · · · · · · · Am Bm Am/2 Bm/2

Insight 1

Periodicity only if number of k-matches of p in t is Ω(poly(k))

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Solution Structure of Pattern Matching with Mismatches

If there are at least Ω(poly(k)) k-matches of p in t, then p and t are periodic and every k-match

• f p starts at a position 1 + i|x|?

t p A A A A A A A A A A A A A A A · · · · · · · · · A2m Am

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Solution Structure of Pattern Matching with Mismatches

If there are at least Ω(poly(k)) k-matches of p in t, then p and t are periodic and every k-match

• f p starts at a position 1 + i|x|?

t p A A A A A A A A A A A A A A A

B at k/2 random positions each

B B B B B · · · · · · · · · A2m Am

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Solution Structure of Pattern Matching with Mismatches

If there are at least Ω(poly(k)) k-matches of p in t, then p and t are periodic and every k-match

• f p starts at a position 1 + i|x|?

t p A A A A A A A A A A A A A A A

B at k/2 random positions each

B B B B B · · · · · · · · · A2m Am O(m) k-matches of p in t, but p and t not perfectly periodic

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Solution Structure of Pattern Matching with Mismatches

If there are at least Ω(poly(k)) k-matches of p in t, then p and t are periodic periodic up to O(k) mismatches and every k-match of p starts at a position 1 + i|x|?

t p A A A A A A A A A A A A A A A

B at k/2 random positions each

B B B B B · · · · · · · · · A2m Am

Insight 2

Periodicity only up to O(k) mismatches

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Solution Structure of Pattern Matching with Mismatches

If there are at least Ω(poly(k)) k-matches of p in t, then p and t are periodic up to O(k) mismatches and every k-match of p starts at a position 1 + i|x|?

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most O(k2), or

t′: shortest substring of t such that any k-match of p in t is also a k-match in t′

Both t′ and p have HD O(k) to the same periodic string x and all k-matches of p in t′ start at a position 1 + i · |x|.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most O(k2), or

t′: shortest substring of t such that any k-match of p in t is also a k-match in t′

Both t′ and p have HD O(k) to the same periodic string x and all k-matches of p in t′ start at a position 1 + i · |x|.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t x∗

i · · · · · · tj

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj

Consider t′: shortest substring of t that contains all k-matches

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj p1 p2 pi p16k

· · · · · ·

Split p into 16k parts pi of equal length

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi

Fix a pi

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi xi xi

Consider prefix xi of pi that is also a period of pi

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi xi xi

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Find first 3k mismatches between p and x∗

i before and after pi

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi xi xi ≤ 3k mism. ≤ 3k mism.

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Find first 3k mismatches between p and x∗

i before and after pi

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj xi xi < 6k mism.

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj xi xi < 6k mism. xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi < 2 · (6 + 1)k = 14k mism.

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi xi xi ≤ 3k mism. ≤ 3k mism.

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi xi xi = 3k mism.

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi xi xi = 3k mism.

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Insight

Any k-match of p in t′ must match at least one pi’s exactly.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi xi xi = 3k mism.

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Fix a pi

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi xi xi = 3k mism.

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Fix a pi; count k-matches where pi is matched exactly

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi xi xi = 3k mism. xi xi xi xi xi xi xi xi xi xi xi

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Consider occurrences of xi in t′

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi xi xi = 3k mism. xi xi xi xi xi xi xi xi xi xi xi

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Problem

Up to O(m) exact matches of xi in t′.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi xi xi = 3k mism. xi xi xi xi xi xi xi

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Consider power stretches of xi in t′ of length ≥ |pi|

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi xi xi = 3k mism. xi xi xi xi xi xi xi

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Consider power stretches of xi in t′ of length ≥ |pi| at most 150k different power stretches

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi xi xi = 3k mism. xi xi xi xi tj

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Fix a power stretch tj of xi in t′.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi xi xi = 3k mism. xi xi xi xi tj ≥ 2k mism.

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Fix a power stretch tj of xi in t′.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi xi xi = 3k mism. xi xi xi xi tj ≥ 2k mism.

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Insight

Must align at least one mismatch.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result, Proof Overview

Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most 1000k 2, or Both t′ and p have HD < 20k to a periodic x; all k-matches start at position 1 + i · |x|.

p1

p t′ x∗

i · · · · · · tj pi xi xi = 3k mism. xi xi xi xi tj ≥ 2k mism.

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi · · · · · ·

Insight

At most O(k 4) matches: O(k) parts in p, O(k) stretches, O(k 2) matches per combination.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Main Result Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, at least one of the following holds: The number of k-matches of p in t is at most O(k2), or

t′: shortest substring of t such that any k-match of p in t is also a k-match in t′

Both t′ and p have Hamming distance O(k) to the same periodic string x and all k-matches of p in t′ start at a position 1 + i · |x|.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Faster Algorithm Theorem (Algorithm)

Pattern matching with k mismatches on a text t given by an SLP of size n and a pattern p of length m can be solved in time O(n k3 (k log k + log m) + k m).

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Faster Algorithm Theorem (Algorithm)

Pattern matching with k mismatches on a text t given by an SLP of size n and a pattern p of length m can be solved in time O(n k3 (k log k + log m) + k m). Pattern-Compressed String [GS’13] Let p be a string of length m. We call a string f = v1 . . . vq, q

i=1 |vi| ≤ 2m a

p-pattern-compressed string (pc-string) if every vi is a substring of p. We call the vi’s factors of f.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Faster Algorithm for Pattern-Compressed Strings

Pattern-Compressed String [GS’13]

Let p be a string of length m. We call a string f = v1 . . . vq, q

i=1 |vi| ≤ 2m a p-pattern-

compressed string (pc-string) if every vi is a substring of p. We call the vi’s factors of f.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Faster Algorithm for Pattern-Compressed Strings

Pattern-Compressed String [GS’13]

Let p be a string of length m. We call a string f = v1 . . . vq, q

i=1 |vi| ≤ 2m a p-pattern-

compressed string (pc-string) if every vi is a substring of p. We call the vi’s factors of f.

PC-String, inst. J1 k, p, f1 with O(k) factors PC-String, inst. J2 k, p, f2 with O(k) factors PC-String, inst. Jn k, p, fn with O(k) factors . . . SLP Instance I SLP T = T1, . . . , Tn k, p of length m T3 T2 T1 B A

O(n k3 (k log k + log m) + k m)

O(k3(k log k + log m)) T(m, k) algorithm

O(n (log m + T(m, k)) + km)

algorithm

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Faster Algorithm for Pattern-Compressed Strings

Pattern-Compressed String [GS’13]

Let p be a string of length m. We call a string f = v1 . . . vq, q

i=1 |vi| ≤ 2m a p-pattern-

compressed string (pc-string) if every vi is a substring of p. We call the vi’s factors of f.

PC-String, inst. J1 k, p, f1 with O(k) factors PC-String, inst. J2 k, p, f2 with O(k) factors PC-String, inst. Jn k, p, fn with O(k) factors . . . SLP Instance I SLP T = T1, . . . , Tn k, p of length m T3 T2 T1 B A O(k3(k log k + log m)) algorithm

O(n k3 (k log k + log m) + k m)

algorithm

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Faster Algorithm for Pattern-Compressed Strings

Pattern-Compressed String [GS’13]

Let p be a string of length m. We call a string f = v1 . . . vq, q

i=1 |vi| ≤ 2m a p-pattern-

compressed string (pc-string) if every vi is a substring of p. We call the vi’s factors of f.

PC-String, inst. J1 k, p, f1 with O(k) factors PC-String, inst. J2 k, p, f2 with O(k) factors PC-String, inst. Jn k, p, fn with O(k) factors . . . SLP Instance I SLP T = T1, . . . , Tn k, p of length m T3 T2 T1 B A O(k3(k log k + log m)) algorithm

O(n k3 (k log k + log m) + k m)

algorithm

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Faster Algorithm for Pattern-Compressed Strings

Theorem (Algorithm for pc-strings) Pattern matching with k mismatches on a pattern p of length m and a p-pc-string f of size O(k) representing at most 2m characters, can be solved in time O(k3(k log k + log m)).

(With O(km) preprocessing on p.)

Implementation of structural insight Need e.g. tools for finding first O(k) mismatches to a periodic string

• r finding all power stretches of a given string in a pc-string

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Faster Algorithm for Pattern-Compressed Strings

Theorem (Algorithm for pc-strings) Pattern matching with k mismatches on a pattern p of length m and a p-pc-string f of size O(k) representing at most 2m characters, can be solved in time O(k3(k log k + log m)).

(With O(km) preprocessing on p.)

Implementation of structural insight Need e.g. tools for finding first O(k) mismatches to a periodic string

• r finding all power stretches of a given string in a pc-string

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Faster Algorithm for Pattern-Compressed Strings

Theorem (Algorithm for pc-strings) Pattern matching with k mismatches on a pattern p of length m and a p-pc-string f of size O(k) representing at most 2m characters, can be solved in time O(k3(k log k + log m)).

(With O(km) preprocessing on p.)

Implementation of structural insight Need e.g. tools for finding first O(k) mismatches to a periodic string

• r finding all power stretches of a given string in a pc-string

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Basic Definitions and General Overview New Structural Insights Faster Algorithm

Faster Algorithm Theorem (Algorithm)

Pattern matching with k mismatches on a text t given by an SLP of size n and a pattern p of length m can be solved in time O(n k3 (k log k + log m) + k m).

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Open Problems

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Open Problems

Improve insight to O(k) mismatches in the aperiodic case

Theorem (Structural Insight′) [KW’19+]

For pattern p and text t, |t| ≤ 2|p|, it holds at least one of: The number of k-matches of p in t is at most O(k), or

t′: shortest substring of t such that any k-match of p in t is also a k-match in t′

Both t′ and p have Hamming distance O(k) to the same periodic string x and all k-matches of p in t′ start at a position 1 + i · |x|.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Open Problems

Improve insight to O(k) mismatches in the aperiodic case Improve dependence on k in the algorithm

Theorem (Algorithm)

Pattern matching with k mismatches on a text t given by an SLP of size n and a pattern p of length m can be solved in time O(n k3 (k log k + log m) + k m).

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Open Problems

Improve insight to O(k) mismatches in the aperiodic case Improve dependence on k in the algorithm Fully-compressed setting (p also given as an SLP) Pattern Matching with Errors (Edit distance instead of Hamming distance)

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Solution Structure of Pattern Matching with Mismatches

t p A2m/3−1 A A A A A A A A A A A A A A A A2m Am · · · · · · · · · · · · · · · · · ·

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Solution Structure of Pattern Matching with Mismatches

t p A2m/3−1 A A A A A A A A A A A A A A A

B at 2m/3, 4m/3 in t and the middle k + 1 positions in p

B B B B A2m/3−1 A2m/3−1 A2m/3 Bk+1 A(m−k−1)/2 A(m−k−1)/2 · · · · · · · · · · · · · · · · · ·

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Solution Structure of Pattern Matching with Mismatches

t p A2m/3−1 A A A A A A A A A A A A A A A

B at 2m/3, 4m/3 in t and the middle k + 1 positions in p

B B B B A2m/3−1 A2m/3−1 A2m/3 Bk+1 A(m−k−1)/2 A(m−k−1)/2 · · · · · · · · · · · · · · · · · · All matches start at the union of two intervals.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Solution Structure of Pattern Matching with Mismatches

t p A2m/3−1 A A A A A A A A A A A A A A A

B at 2m/3, 4m/3 in t and the middle k + 1 positions in p

B B B B A2m/3−1 A2m/3−1 A2m/3 Bk+1 A(m−k−1)/2 A(m−k−1)/2 · · · · · · · · · · · · · · · · · ·

Insight 3

Arithmetic progression only approximates all matches

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result Theorem (Structural Insight)

Given strings p of length m and t of length at most 2m, at least one of the following holds: The number of k-matches of p in t is at most O(k2). t′: shortest substring of t such that any k-match of p in t is also a k-match in t′ There is a substring x of p, with |x| = O(m/k), such that δH(p, x∗[1, m]) ≤ O(k) and δH(t′, x∗[1, |t′|]) ≤ O(k). Moreover, any k-match of p in t′ starts at a position of the form 1 + i · |x| with 0 ≤ i ≤ (|t′| − |p|)/|x| (but not every starting position 1 + i · |x| necessarily yields a k-match).

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result Theorem (Structural Insight)

Given strings p of length m and t of length at most 2m, at least one of the following holds: The number of k-matches of p in t is at most O(k2). t′: shortest substring of t such that any k-match of p in t is also a k-match in t′ There is a substring x of p, with |x| = O(m/k), such that δH(p, x∗[1, m]) ≤ O(k) and δH(t′, x∗[1, |t′|]) ≤ O(k). Moreover, any k-match of p in t′ starts at a position of the form 1 + i · |x| with 0 ≤ i ≤ (|t′| − |p|)/|x| (but not every starting position 1 + i · |x| necessarily yields a k-match).

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result Theorem (Structural Insight)

Given strings p of length m and t of length at most 2m, at least one of the following holds: The number of k-matches of p in t is at most O(k2). t′: shortest substring of t such that any k-match of p in t is also a k-match in t′ There is a substring x of p, with |x| = O(m/k), such that δH(p, x∗[1, m]) ≤ O(k) and δH(t′, x∗[1, |t′|]) ≤ O(k). Moreover, any k-match of p in t′ starts at a position of the form 1 + i · |x| with 0 ≤ i ≤ (|t′| − |p|)/|x| (but not every starting position 1 + i · |x| necessarily yields a k-match).

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result Theorem (Structural Insight)

Given strings p of length m and t of length at most 2m, at least one of the following holds: The number of k-matches of p in t is at most O(k2). t′: shortest substring of t such that any k-match of p in t is also a k-match in t′ There is a substring x of p, with |x| = O(m/k), such that δH(p, x∗[1, m]) ≤ O(k) and δH(t′, x∗[1, |t′|]) ≤ O(k). Moreover, any k-match of p in t′ starts at a position of the form 1 + i · |x| with 0 ≤ i ≤ (|t′| − |p|)/|x| (but not every starting position 1 + i · |x| necessarily yields a k-match).

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, it holds at least one of: The number of k-matches of p in t is at most O(k2), and Both t and p have HD O(k) to the same periodic string.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, it holds at least one of: The number of k-matches of p in t is at most O(k2), and Both t and p have HD O(k) to the same periodic string.

t P A N C A K E P A N Finding ANPAN, k = 2 non-periodic case t P U N R A N P A M P A N Finding PANPAN, k = 2 periodic case

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, it holds at least one of: The number of k-matches of p in t is at most O(k2), and Both t and p have HD O(k) to the same periodic string.

t P A N C A K E P A N p A N P A N A N P A N Finding ANPAN, k = 2 non-periodic case t P U N R A N P A M P A N Finding PANPAN, k = 2 periodic case

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, it holds at least one of: The number of k-matches of p in t is at most O(k2), and Both t and p have HD O(k) to the same periodic string.

t P A N C A K E P A N p A N P A N A N P A N Finding ANPAN, k = 2 non-periodic case t P U N R A N P A M P A N Finding PANPAN, k = 2 periodic case

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, it holds at least one of: The number of k-matches of p in t is at most O(k2), and Both t and p have HD O(k) to the same periodic string.

t P A N C A K E P A N p A N P A N A N P A N Finding ANPAN, k = 2 non-periodic case t P U N R A N P A M P A N p P A N P A N P A N P A N P A N P A N Finding PANPAN, k = 2 periodic case

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Theorem (Structural Insight)

For pattern p and text t, |t| ≤ 2|p|, it holds at least one of: The number of k-matches of p in t is at most O(k2), and Both t and p have HD O(k) to the same periodic string.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Theorem (Structural Insight)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, then both t and p have a HD < 20k to the same periodic string.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Theorem (Structural Insight)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, then both t and p have a HD < 20k to the same periodic string.

Main Steps: At least 1000k2 k-matches of p in t and p has a HD < 6k to a specific periodic string x ∈ x(p) = ⇒ t has a Hamming Distance < 20k to x p has HD ≥ 6k to any specific periodic string x ∈ x(p) = ⇒ Less than 1000k2 k-matches of p in t

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Theorem (Structural Insight)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, then both t and p have a HD < 20k to the same periodic string.

Main Steps: At least 1000k2 k-matches of p in t and p has a HD < 6k to a specific periodic string x ∈ x(p) = ⇒ t has a Hamming Distance < 20k to x p has HD ≥ 6k to any specific periodic string x ∈ x(p) = ⇒ Less than 1000k2 k-matches of p in t

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to a periodic string x ∈ x(p), then t has HD < 20k to x.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to a periodic string x ∈ x(p), then t has HD < 20k to x.

t p

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to a periodic string x ∈ x(p), then t has HD < 20k to x.

p1

p Split p into 16k parts pi of equal length

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to a periodic string x ∈ x(p), then t has HD < 20k to x.

p1

p

p1 p2 pi p16k

· · · · · · Split p into 16k parts pi of equal length

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to a periodic string x ∈ x(p), then t has HD < 20k to x.

p1

p

pi

Fix a pi

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to a periodic string x ∈ x(p), then t has HD < 20k to x.

p1

p

pi xi xi

Consider prefix xi of pi that is also a period of pi

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to a periodic string x ∈ x(p), then t has HD < 20k to x.

p1

p

pi xi xi

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi

Find first 3k mismatches between p and x∗

i before and after pi

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to a periodic string x ∈ x(p), then t has HD < 20k to x.

p1

p

pi xi xi ≤ 3k mism. ≤ 3k mism.

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi

Find first 3k mismatches between p and x∗

i before and after pi

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to some x∗

i , 1 ≤ i ≤ 16k, then t has HD < 20k to x∗ i .

p1

p

pi xi xi < 6k mism.

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to some x∗

i , 1 ≤ i ≤ 16k, then t has HD < 20k to x∗ i .

Claim (Proof omitted)

If there are at least 2 + 16k k-matches of p in t, all starting positions of k-matches differ by (integer) multiples of |xi|.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to some x∗

i , 1 ≤ i ≤ 16k, then all starting positions of k-matches differ by multiples of |xi|

and t has HD < 20k to x∗

i .

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to some x∗

i , 1 ≤ i ≤ 16k, then all starting positions of k-matches differ by multiples of |xi|

and t has HD < 20k to x∗

i .

x∗

i

t p

pi pi

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to some x∗

i , 1 ≤ i ≤ 16k, then all starting positions of k-matches differ by multiples of |xi|

and t has HD < 20k to x∗

i .

x∗

i

t x∗

i

p

pi pi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to some x∗

i , 1 ≤ i ≤ 16k, then all starting positions of k-matches differ by multiples of |xi|

and t has HD < 20k to x∗

i .

x∗

i

t x∗

i

p

pi pi < 6k mism. to x∗

i

< 6k mism. to x∗

i

xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to some x∗

i , 1 ≤ i ≤ 16k, then all starting positions of k-matches differ by multiples of |xi|

and t has HD < 20k to x∗

i .

x∗

i

t x∗

i

p

pi pi < 6k mism. to x∗

i

< 6k mism. to x∗

i

< 7k mism. to x∗

i

< 7k mism. to x∗

i

xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to some x∗

i , 1 ≤ i ≤ 16k, then all starting positions of k-matches differ by multiples of |xi|

and t has HD < 20k to x∗

i .

Lemma (Step 2)

Fix a pattern p of length m and a text t of length at most 2m. If the pattern p has a HD ≥ 6k to all strings x∗

i , 1 ≤ i ≤ 16k, then there are less

than 1000k2 k-matches of p in t.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 2)

Fix a pattern p of length m and a text t of length at most 2m. If the pattern p has a HD ≥ 6k to all strings x∗

i , 1 ≤ i ≤ 16k, then there are less

than 1000k2 k-matches of p in t.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 2)

Fix a pattern p of length m and a text t of length at most 2m. If the pattern p has a HD ≥ 6k to all strings x∗

i , 1 ≤ i ≤ 16k, then there are less

than 1000k2 k-matches of p in t.

p1

p

p1 p2 pi p16k

· · · · · · Recall: Split p into 16k parts pi of equal length

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 2)

Fix a pattern p of length m and a text t of length at most 2m. If the pattern p has a HD ≥ 6k to all strings x∗

i , 1 ≤ i ≤ 16k, then there are less

than 1000k2 k-matches of p in t.

p1

p

p1 p2 pi p16k

· · · · · ·

Insight

Any k-match of p in t must match at least 15k pi’s exactly.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 2)

Fix a pattern p of length m and a text t of length at most 2m. If the pattern p has a HD ≥ 6k to all strings x∗

i , 1 ≤ i ≤ 16k, then there are less

than 1000k2 k-matches of p in t.

p1

p

pi

Fix a pi

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 2)

Fix a pattern p of length m and a text t of length at most 2m. If the pattern p has a HD ≥ 6k to all strings x∗

i , 1 ≤ i ≤ 16k, then there are less

than 1000k2 k-matches of p in t.

p1

p

pi

Fix a pi; count k-matches where pi is matched exactly

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 2)

Fix a pattern p of length m and a text t of length at most 2m. If the pattern p has a HD ≥ 6k to all strings x∗

i , 1 ≤ i ≤ 16k, then there are less

than 1000k2 k-matches of p in t.

p1

p

pi xi xi

t

xi xi xi xi xi xi xi xi xi xi xi

Search for xi in t

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 2)

Fix a pattern p of length m and a text t of length at most 2m. If the pattern p has a HD ≥ 6k to all strings x∗

i , 1 ≤ i ≤ 16k, then there are less

than 1000k2 k-matches of p in t.

p1

p

pi xi xi

t

xi xi xi xi xi xi xi xi xi xi xi

Problem

Up to O(m) exact matches of xi in t.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 2)

Fix a pattern p of length m and a text t of length at most 2m. If the pattern p has a HD ≥ 6k to all strings x∗

i , 1 ≤ i ≤ 16k, then there are less

than 1000k2 k-matches of p in t.

p1

p

pi xi xi

t

xi xi xi xi xi xi xi

Search for power stretches of xi in t of length ≥ |pi|

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 2)

Fix a pattern p of length m and a text t of length at most 2m. If the pattern p has a HD ≥ 6k to all strings x∗

i , 1 ≤ i ≤ 16k, then there are less

than 1000k2 k-matches of p in t.

p1

p

pi xi xi

t

xi xi xi xi xi xi xi

Insight

Only ≤ 150k different power stretches of xi in t.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 2)

Fix a pattern p of length m and a text t of length at most 2m. If the pattern p has a HD ≥ 6k to all strings x∗

i , 1 ≤ i ≤ 16k, then there are less

than 1000k2 k-matches of p in t.

p1

p

pi xi xi

t

xi xi xi xi xi xi xi

Fix a power stretch tj of xi in t.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 2)

Fix a pattern p of length m and a text t of length at most 2m. If the pattern p has a HD ≥ 6k to all strings x∗

i , 1 ≤ i ≤ 16k, then there are less

than 1000k2 k-matches of p in t.

p1

p

pi xi xi

t

xi xi xi xi tj

Fix a power stretch tj of xi in t.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 2)

Fix a pattern p of length m and a text t of length at most 2m. If the pattern p has a HD ≥ 6k to all strings x∗

i , 1 ≤ i ≤ 16k, then there are less

than 1000k2 k-matches of p in t.

p1

p

pi xi xi

t

xi xi xi xi tj

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 2)

Fix a pattern p of length m and a text t of length at most 2m. If the pattern p has a HD ≥ 6k to all strings x∗

i , 1 ≤ i ≤ 16k, then there are less

than 1000k2 k-matches of p in t.

p1

p

pi xi xi

t

xi xi xi xi tj

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi ≥ 3k mism.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview

p1

p

pi xi xi

t

xi xi xi xi tj

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi ≥ 3k mism. ≥ 2k mism.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview

p1

p

pi xi xi

t

xi xi xi xi tj

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi ≥ 3k mism. ≥ 2k mism.

Insight

Must align at least k mismatches.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview

p1

p

pi xi xi

t

xi xi xi xi tj

x∗

i xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi xi ≥ 3k mism. ≥ 2k mism.

Insight

At most O(k4) matches: O(k) parts in p, O(k) streches, O(k2) matches per combination.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Lemma (Step 1)

Fix a pattern p of length m and a text t of length at most 2m. If the number of k-matches of p in t is at least 1000k2, and p has HD < 6k to some x∗

i , 1 ≤ i ≤ 16k, then all starting positions of k-matches differ by multiples of |xi|

and t has HD < 20k to x∗

i .

Lemma (Step 2)

Fix a pattern p of length m and a text t of length at most 2m. If the pattern p has a HD ≥ 6k to all strings x∗

i , 1 ≤ i ≤ 16k, then there are less

than 1000k2 k-matches of p in t.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Main Result, Proof Overview Theorem (Structural Insight)

Given strings p of length m and t of length at most 2m, at least one of the following holds: The number of k-matches of p in t is at most O(k2). t′: shortest substring of t such that any k-match of p in t is also a k-match in t′ There is a substring x of p, with |x| = O(m/k), such that δH(p, x∗[1, m]) ≤ O(k) and δH(t′, x∗[1, |t′|]) ≤ O(k). Moreover, any k-match of p in t′ starts at a position of the form 1 + i · |x| with 0 ≤ i ≤ (|t′| − |p|)/|x| (but not every starting position 1 + i · |x| necessarily yields a k-match).

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Faster Algorithm for Pattern-Compressed Strings

Pattern-Compressed String [GS’13]

Let p be a string of length m. We call a string f = v1 . . . vq, q

i=1 |vi| ≤ 2m a p-pattern-

compressed string (pc-string) if every vi is a substring of p. We call the vi’s factors of f.

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Faster Algorithm for Pattern-Compressed Strings

Pattern-Compressed String [GS’13]

Let p be a string of length m. We call a string f = v1 . . . vq, q

i=1 |vi| ≤ 2m a p-pattern-

compressed string (pc-string) if every vi is a substring of p. We call the vi’s factors of f.

PC-String, inst. J1 k, p, f1 with O(k) factors PC-String, inst. J2 k, p, f2 with O(k) factors PC-String, inst. Jn k, p, fn with O(k) factors . . . SLP Instance I SLP T = T1, . . . , Tn k, p of length m T3 T2 T1 B A

O(n k3 (k log k + log m) + k m)

O(k3(k log k + log m)) T(m, k) algorithm

O(n (log m + T(m, k)) + km)

algorithm

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Faster Algorithm for Pattern-Compressed Strings

Pattern-Compressed String [GS’13]

Let p be a string of length m. We call a string f = v1 . . . vq, q

i=1 |vi| ≤ 2m a p-pattern-

compressed string (pc-string) if every vi is a substring of p. We call the vi’s factors of f.

PC-String, inst. J1 k, p, f1 with O(k) factors PC-String, inst. J2 k, p, f2 with O(k) factors PC-String, inst. Jn k, p, fn with O(k) factors . . . SLP Instance I SLP T = T1, . . . , Tn k, p of length m T3 T2 T1 B A O(k3(k log k + log m)) algorithm

O(n k3 (k log k + log m) + k m)

algorithm

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Faster Algorithm for Pattern-Compressed Strings

Pattern-Compressed String [GS’13]

Let p be a string of length m. We call a string f = v1 . . . vq, q

i=1 |vi| ≤ 2m a p-pattern-

compressed string (pc-string) if every vi is a substring of p. We call the vi’s factors of f.

PC-String, inst. J1 k, p, f1 with O(k) factors PC-String, inst. J2 k, p, f2 with O(k) factors PC-String, inst. Jn k, p, fn with O(k) factors . . . SLP Instance I SLP T = T1, . . . , Tn k, p of length m T3 T2 T1 B A O(k3(k log k + log m)) algorithm

O(n k3 (k log k + log m) + k m)

algorithm

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Faster Algorithm for Pattern-Compressed Strings

Theorem (Algorithm for pc-strings) Pattern matching with k mismatches on a pattern p of length m and a p-pc-string f of size O(k) representing at most 2m characters, can be solved in time O(k3(k log k + log m)).

(With O(km) preprocessing on p.)

Implementation of structural insight Need e.g. tools for finding first O(k) mismatches to a periodic string

• r finding all power stretches of a given string in a pc-string

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Faster Algorithm for Pattern-Compressed Strings

Theorem (Algorithm for pc-strings) Pattern matching with k mismatches on a pattern p of length m and a p-pc-string f of size O(k) representing at most 2m characters, can be solved in time O(k3(k log k + log m)).

(With O(km) preprocessing on p.)

Implementation of structural insight Need e.g. tools for finding first O(k) mismatches to a periodic string

• r finding all power stretches of a given string in a pc-string

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Faster Algorithm for Pattern-Compressed Strings

Theorem (Algorithm for pc-strings) Pattern matching with k mismatches on a pattern p of length m and a p-pc-string f of size O(k) representing at most 2m characters, can be solved in time O(k3(k log k + log m)).

(With O(km) preprocessing on p.)

Implementation of structural insight Need e.g. tools for finding first O(k) mismatches to a periodic string

• r finding all power stretches of a given string in a pc-string

Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts

Navigation

Start Definition SLP Known Results Insight Ex. 1 Insight Ex. 2 Insight Ex. 3 Insight Theorem Insight Step 1 Overview Insight Step 1 Details Insight Step 2 Overview Algorithm General Idea Algorithm Overview Open Problems End Karl Bringmann, Marvin Künnemann, and Philip Wellnitz Faster Pattern Matching with Mismatches in Compressed Texts