Shape-Preserving Pattern Matching Algorithms
This paper discusses shape-preserving pattern matching algorithms that identify sequences shape-isomorphic to a given pattern. Two linear-time algorithms are presented, offering solutions through reduction and online techniques. Various images provide visual representations and examples of the process.
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Shape-Preserving Pattern Matching Domenico Cantone, Simone Faro and M.Oguzhan K lekci, 21st Italian Conference on Theoretical Computer Science, 14-16 September 2020, Ischia, Italy Presenter: Wen-Wen Liao Date: May. 2, 20231
Abstract Two sequences of integers x and z of the same length m 2 are shape- isomorphic if, up to a positive proportional factor, the sequences of the distances between consecutive elements of x and z are the same, i.e., if for some > 0 one has x[i + 1] x[i] = (z[i + 1] z[i]), for each 1 i m 1. In this paper we present two linear-time algorithms which, given a text y and a pattern x over an integer alphabet, finds all the factors of y that are shape-isomorphic to x. Our first solution is a two steps algorithm based on a reduction to the standard exact string matching problem, while our second solution is an online algorithm based on the well-known Knuth-Morris-Pratt string matching algorithm. We call this problem shape-preserving pattern-matching problem. 2
Shape-Preserving Pattern Matching Finer than order-isomorphism. Proportionality factors is 3 and 2, respectively. 6 6 9 4 4 6 2 2 3 3
Reducing SPPM to Exact Pattern Matching y = 14 18 15 18 12 21 15 25 26 x = 6 8 4 10 6 find k to separate ?1and ?2, ? = max( ?: ? > 0 ??? ? ? ? ? + 1 k = 4 ?1= 6 8 4 10 => prefix ?2= 10 6 => suffix 0 ) 4
Reducing SPPM to Exact Pattern Matching Delta function: for 1 ? ? 1, ? ? ? ? + 1 ? ? Last Non-Zero Function: for 1 ? ? 1,? ? ? ? ? ? ??? ? = max({1 < ? + 1:?[ ] 0} {0}) ?[?+1] ? ? [?] Ratios Function: for 1 ? ? 1, ? ? 5
Reducing SPPM to Exact Pattern Matching y = 14 18 15 18 12 21 15 25 26 x = 6 8 4 10 6 ?1= 6 8 4 10 ?2= 10 6 i 1 2 2 -2 2 -4 2 3 3 6 6 i 1 6 6 2 -4 6 (?1) [i] ( (?1) )[i] ( (?1) )[i] (?2) [i] ( (?2))[i] ( (?2) )[i] 2 3 6
Reducing SPPM to Exact Pattern Matching y = 14 18 15 18 12 21 15 25 26 x = 6 8 4 10 6 1= occurrences of (?1) in (?) 2= occurrences of ((?2)) in (?) i 1 2 3 i 1 2 (?1) [i] ( (?1) )[i] ( (?1) )[i] 2 -4 6 (?2) [i] ( (?2))[i] ( (?2) )[i] 6 -4 2 2 6 6 6 2 -2 3 3 i 1 4 4 2 -3 4 3 4 3 3 4 4 -6 4 9 4 5 9 9 6 -6 9 10 9 7 8 1 (y) [i] ( (y) )[i] ( (y) )[i] 10 10 1 10 10 3 6 6 9 4 4 7
Reducing SPPM to Exact Pattern Matching y = 14 18 15 18 12 21 15 25 26 x = 6 8 4 10 6 = {? ? + 2:? 2??? ? ? + 2 1??? ? ? 1 ? ? ? => i = 3 > 0} i 1 4 4 2 -3 4 3 4 3 3 4 4 -6 4 9 4 5 9 9 6 -6 9 10 9 7 8 1 (y) [i] ( (y) )[i] ( (y) )[i] 10 10 1 10 10 3 6 6 9 4 4 8
A KMP based Algorithm for the SPPM Problem x = 4 2 10 6 22 14 13 17 shape-border table i 1 4 0 - 2 2 1 0 3 4 6 2 2 5 6 7 8 -2 -4 x[i] [i] [i] 10 1 22 3 2 14 4 2 13 2 0.5 17 3 0.5 0 9
A KMP based Algorithm for the SPPM Problem x = 4 2 10 6 22 14 13 17 shape-border table i 1 4 0 - 2 2 1 0 3 4 6 2 2 5 6 7 8 8 16 x[i] [i] [i] 10 1 22 3 2 14 4 2 13 2 0.5 17 3 0.5 0 10
A KMP based Algorithm for the SPPM Problem y = 14 18 15 18 12 21 15 25 26 x = 6 8 4 10 6 i 1 6 0 - 2 8 1 0 3 4 1 4 5 6 3 1 x[i] [i] [i] 10 2 3 0 11
A KMP based Algorithm for the SPPM Problem i 1 6 0 - 2 8 1 0 3 4 1 4 5 6 3 1 y = 14 18 15 18 12 21 15 25 26 x = 6 8 4 10 6 x[i] [i] [i] 10 2 3 0 14 18 15 18 12 21 15 25 26 6 8 4 10 6 1 2 [2] = 1 1 2-1=1 6 8 4 10 6 0 1 1 [1] = 0 0 1-0=1 6 8 4 10 6 1 12
Conclusion Reducing SPPM to Exact Pattern Matching Time complexity: ?(? + ?) A KMP based Algorithm for the SPPM Problem Time complexity: ?(?) 13
