Sequence Comparison.pdf

94 6 Anatomy of Spaced Seeds
RP(π)={i 1 = 0,i 2 ,···,i wπ = |π|−1}. (6.1)
The seed π is said to hit R at position j if and only if R[ j −|π| + i k + 1] =1for
all 1 ≤ k ≤ w π . Here, we use the ending position as the hit position following the
convention in the renewal theory.
Let A i denote the event that π hits R at position i and Ā i the complement of A i .
We use π i to denote the probability that π first hits R at position i − 1, that is,
π i = Pr[Ā 0 Ā 1 ···Ā i−2 A i−1 ].
We call π i the first hit probability. Let Π n := Π n (p) denote the probability that π
hits R[0,n − 1] and ¯Π n := 1 − Π n . We call Π n the hit probability and ¯Π n the non-hit
probability of π.
For each 0 ≤ n < |π|−1, trivially, A n = /0 and Π n = 0. Because the event
∪ 0≤i≤n−1 A i is the disjoint union of the events
∪ 0≤i≤n−2 A i
and
∪ 0≤i≤n−2 A i A n−1 = Ā 0 Ā 1 ···Ā n−2 A n−1 ,
Π n = π 1 + π 2 + ···+ π n ,
or equivalently,
¯Π n = π n+1 + π n+2 + ··· (6.2)
for n ≥|π|.
Example 6.1. Let θ be the consecutive seed of weight w. Ifθ hits random sequence
R at position n − 1, but not at position n − 2, then R[n − w,n − 1] =11···1 and
R[n − w − 1]=0. As a result, θ cannot hit R at positions n − 3,n − 4,...,n − w − 1.
This implies
θ n = Pr[Ā 0 Ā 1 ···Ā n−2 A n−1 ]=p w (1 − p) ¯Θ n−w−1 , n ≥ w + 1.
By formula (6.2), its non-hit probability satisfies the following recurrence relation
¯Θ n = ¯Θ n−1 − p w (1 − p) ¯Θ n−w−1 . (6.3)
Example 6.2. Let π be a spaced seed and k > 1. By inserting (k − 1) ∗’s between
every two consecutive positions in π, we obtain a spaced seed π ′ of the same weight
and length |π ′ | = k|π|−k + 1. It is not hard to see that π ′ hits the random sequence
R[0,n − 1]=s[0]s[1]···s[n − 1] if and only if π hits one of the following k random
sequences:
s[i]s[k + i]s[2k + i]···s[lk+ i],
i = 0,1,...,r,