The size of the constructed DAA's state space is (m + 1)|Σ|^m and grows exponentially with the pattern length, making the application for long patterns infeasible in practice. However, depending on the particular circumstances (i.e., algorithm and pattern analyzed), the constructed DAA can often be substantially reduced by state space minimization [21]. For example, for B(N)DM, both cost and shift of an examined window depend only on the longest factor of p that is a suffix of the window. Since there are only O(m²) different factors, it is reasonable that |Σ|^m can be replaced by O(m²), for a total state space of size O(m³). Therefore, for each algorithm, a specialized construction may exist that directly constructs the minimal state space whose size may only grow polynomially with m. For the Horspool algorithm, it is known that the state space has a size of only O(m²), as the construction of Tsai [13] can be adapted to construct a DAA according to our definition. However, we have been unable to provide a direct construction of the minimal DAA applicable to all window-based pattern matching algorithms.

Hopcroft's algorithm [21] minimizes a DFA in O(| | log | |) time by iteratively refining a partition of the state set. In the beginning, all states are partitioned into two distinct sets: one containing the accepting states, and the other containing the non-accepting states. This partition is iteratively refined whenever a reason for non-equivalence of two states in the same set is found. Upon termination, the states are partitioned into sets of equivalent states. Refer to [22] for an in-depth explanation of Hopcroft's algorithm.

The algorithm can straightforwardly be adapted to minimize DAAs by choosing the initial state set partition appropriately. In our case, each DAA state is associated with the same operation. The only differences in state's behavior thus stem from different emissions. Therefore, Hopcroft's algorithm can be initialized by the partition induced by the emissions and then continued as usual.

As we exemplify in Section 7, this leads to a considerable reduction of the number of states.

A finite-memory text model is a tuple ( , c₀, Σ, φ), where is a finite state space (called context space), c₀ ∈ a start context, Σ an alphabet, and φ : × Σ × → [0, 1] a transition function with Σ_{σ∈Σ,c′∈} φ(c, σ, c′) = 1 for all c ∈ . The random variable giving the text model state after t steps is denoted C_t with C₀ :≡ c₀. A probability measure is now induced by stipulating ℙ ( S 0 … S n - 1 = s , C 1 = c 1 , … , C n = c n ) : = ∏ i = 0 n - 1 φ ( c i , s [ i ] , c i + 1 )for all n ∈ ℕ₀, s ∈ Σⁿ, and (c₁, …, c_n) ∈ ⁿ.

The idea is that the model given by ( , c₀, Σ, φ) generates a random text by moving from context to context and emitting a character at each transition, where φ(c, σ, c′) is the probability of moving from context c to context c′ and thereby generating the letter σ.

Note that the probability ℙ(S₀ … S_|s|−1 = s) is obtained by marginalization over all context sequences that generate s. This can be efficiently done, using the decomposition of the following lemma.

Let ( , c₀, Σ, φ) be a finite-memory text model. Then, ℙ ( S 0 … S n = s σ , C n + 1 = c ) = ∑ c ′ ∈ C ℙ ( S 0 … S n - 1 = s , C n = c ′ ) ⋅ φ ( c ′ , σ , c )for all n ∈ ℕ₀, s ∈ Σⁿ, σ ∈ Σ and c ∈ .

We have ℙ ( S 0 … S n = s σ , C n + 1 = c ) = ∑ c 1 , … , c n ℙ ( S 0 … S n = s σ , C 1 = c 1 , … , C n = c n , C n + 1 = c ) = ∑ c 1 , … , c n ∏ i = 0 n - 1 φ ( c i , s [ i ] , c i + 1 ) ⋅ ( c n , σ , c ) = ∑ c n ∈ C ( ∑ c 1 , … , c n - 1 ∏ i = 0 n - 1 φ ( c i , s [ i ] , c i + 1 ) ) ⋅ φ ( c n , σ , c ) = ∑ c n ∈ C ℙ ( S 0 … S n - 1 = s , C n = c n ) ⋅ φ ( c n , σ , c )

Renaming c_n to c′ yields the claimed result.

Similar text models are used in [23], where they are called probability transducers. In the following, we refer to a finite-memory text model ( , c₀, Σ, φ) simply as text model, as all text models considered in this article are special cases of Definition 3.

For an i.i.d. model, we set = {ε} and φ(ε, σ, ε) = p_σ for each σ ∈ Σ, where p_σ is the occurrence probability of letter σ (and ε may be interpreted as an empty context). For a Markovian text model of order r, the distribution of the next character depends on the r preceding characters (fewer at the beginning); thus C : = ∪ i = 0 r ∑ i. This notion of text models also covers variable order Markov chains as introduced in [24], which can be converted into equivalent models of fixed order. Text models as defined above have the same expressive power as character-emitting HMMs, that means, they allow to construct the same probability distributions.

A probabilistic arithmetic automaton is a tuple = ( , q₀, T, , v₀, ℰ, μ = (μ_q)_q∈ , θ = (θ_q)_q∈ ), where , q₀, , v₀, ℰ and θ have the same meaning as for a DAA, each μ_q is a state-specific probability distribution on the emissions ℰ, and T : × → [0, 1] is a transition function, such that T(q, q′) gives the probability of a transition from state q to state q′, i.e., (T(q, q′))_{q, q′∈} is a stochastic matrix.

A PAA induces three stochastic processes: (1) the state process (Q_t)_t∈ℕ with values in , (2) the emission process (E_t)_t∈ℕ with values in ℰ, and (3) the value process (V_t)_t∈ℕ with values in such that ₀ :≡ v₀ and _t ≔ θ_Qt (V_t−1, E_t).

We now restate the PAA recurrences from [1] to compute the state-value distribution after t steps. For the sake of a shorter notation, we define f_t(q, v) ≔ ℙ(Q_t = q, V_t = v). Since we are generally only interested in the value distribution, note that it can be obtained by marginalization: ℙ(V_t = v) = Σ_q∈ f_t(q, v).

The state-value distribution is given by f₀(q, v) = 1 if q = q₀ and v = v₀, and f₀(q, v) = 0 otherwise. For t ≥ 0, (1) f t + 1 ( q , v ) = ∑ q ′ ∈ Q ∑ ( v ′ , e ) ∈ θ q - 1 ( v ) f t ( q ′ , v ′ ) ⋅ T ( q ′ , q ) ⋅ μ q ( e ) ,where θ q - 1 ( v ) denotes the inverse image set of v under θ_q.

The recurrence in Lemma 2 resembles the Forward recurrences known from HMMs.

Note that the range of V_t is finite for each t, even when is infinite, as V_t is a function of the states and emissions up to time t, and state set and emission set ℰ are finite. We define _t ≔ range V_t and ϑ_n ≔ max_0≤t≤n | _t|. Clearly ϑ_n ≤ (| | · |ℰ|)ⁿ. Therefore all actual computations are on finite sets. When analyzing the number of character accesses of a pattern matching algorithm, we have _t ⊂ {0, …, m(n − m + 1)}, as at most (n − m + 1) search windows are processed, each causing at most m character accesses. Thus, ϑ_n ∈ O(n · m).

Let a text model M = ( , c₀, Σ, φ) and a DAA D = ( Q D , q 0 D , ∑ , δ , V , v 0 , ℰ , ( η q ) q ∈ Q D , ( θ q D ) q ∈ Q D ) over the same alphabet Σ be given. Then, we define the PAA induced by and M by giving

a state space ≔ × ,

Note that states having zero probability of being reached from q₀ may be omitted from and T without changing the PAA's state, emission or value process. The next lemma states that the PAA given by Definition 3 indeed reflects the probabilistic behavior of the input DAA acting on a random text generated by the text model. Furthermore, it gives the runtime required to compute the distribution of DAA values via dynamic programming.

Let a text model M = ( , c₀, Σ, φ) and a DAA D = ( Q D , q 0 D , ∑ , δ , V , v 0 , ℰ , ( η q ) q ∈ Q D , ( θ q D ) q ∈ Q D )be given and let = ( , q₀, T, , v₀, ℰ, μ = (μ_q)_q∈ , θ = (θ_q)_q∈ ) be the PAA given by Definition 5. Then,

ℒ(V_t) = ℒ(value (S₀ … S_t−1)) for all t ∈ ℕ₀, where S is a random text according to the text model M,

As in Section 5.2, we define f_t(q, v) ≔ ℙ(Q_t = q, V_t = v). To prove 1, we show that (3) f t ( ( q D , c ) , v ) = ∑ s ∈ ∑ t ⟦ δ ^ ( ( q 0 D , v 0 ) , s ) = ( q D , v ) ⟧ ⋅ ℙ ( S 0 … S t - 1 = s , C t = c )for all q ∈ , c ∈ , v ∈ , and t ∈ ℕ₀. For t = 0, Equation (3) is satisfied by definitions of PAAs, DAAs and text models. For t > 0 we prove it inductively. Assume Equation (3) to be correct for all t′ with 0 ≤ t′ < t. Then (4) f t ( ( q D , c ) ︸ = : q , v ) (5) = ∑ q ′ ∈ Q ∑ ( v ′ , e ) ∈ θ q - 1 ( v ) f t - 1 ( q ′ , v ′ ) ⋅ T ( q ′ , q ) ⋅ μ q ( e ) (6) = ∑ q ′ ∈ Q ∑ ( v ′ , e ) ∈ V × ℰ ⟦ θ q D D ( v ′ , e ) = v ⟧ ⋅ f t - 1 ( q ′ , v ′ ) ⋅ T ( q ′ , q ) ⋅ ⟦ η q D = e ⟧ (7) = ∑ q ′ D ∈ Q D ∑ c ′ ∈ C ∑ ( v ′ , e ) ∈ V × ℰ ⟦ θ q D D ( v ′ , e ) = v ⟧ ⋅ ⟦ η q D = e ⟧ ⋅ f t - 1 ( q ′ , v ′ ) ⋅ ∑ σ ∈ ∑ ⟦ δ ( q ′ D , σ ) = q D ⟧ ⋅ φ ( c ′ , σ , c ) (8) = ∑ s ∈ ∑ t - 1 ∑ σ ∈ ∑ ∑ q ′ D ∈ Q D ∑ c ′ ∈ C ∑ ( v ′ , e ) ∈ V × ℰ ⟦ θ q D D ( v ′ , e ) = v ⟧ ⋅ ⟦ η q D = e ⟧ ⋅ ⟦ δ ( q ′ D , σ ) = σ D ⟧ ⋅ ⟦ δ ^ ( ( q 0 D , v 0 ) , s ) = ( q ′ D , v ′ ) ⟧ ⋅ ℙ ( S 0 … S t - 2 = s , C t - 1 = c ′ ) ⋅ φ ( c ′ , σ , c ) (9) = ∑ s σ ∈ ∑ t ∑ q ′ D ∈ Q D ∑ ( v ′ , e ) ∈ V × ℰ ⟦ θ q D D ( v ′ , e ) = v ⟧ ⋅ ⟦ η q D = e ⟧ ⋅ ⟦ δ ^ ( ( q 0 D , v 0 ) , s ) = ( q ′ D , v ′ ) ⟧ ⋅ ⟦ δ ( q ′ D , σ ) = σ D ⟧ ⋅ ℙ ( S 0 … S t - 2 = s σ , C t = c ) (10) = ∑ s σ ∈ ∑ t ⟦ δ ^ ( ( q 0 D , v 0 ) , s σ ) = ( q D , v ) ⟧ ⋅ ℙ ( S 0 … S t - 1 = s σ , C t = c ) .

In the above derivation, step (4)→(5) follows from (1). Step (5)→(6) follows from the definitions of θ_q and μ_q. Step (6)→(7) uses the definitions of T and in Lemma 3. Step (7)→(8) uses the induction assumption. Step (8)→(9) uses Lemma 1. The final step (9)→(10) follows by combining the four Iverson brackets summed over q′ and (v′, e) into a single Iverson bracket.

To compute the table f_n containing f_n(q, v) for all q ∈ and v ∈ , we start with f₀ and perform n update steps. The runtime bounds given in 2. and 3. can be verified by considering a “push” algorithm: When computing f_t+1, we initialize the table with zeros and iterate over all q ∈ , v ∈ and q′ ∈ {q″ ∈ : T(q, q″) > 0}; for each combination of q, v, and q′ with T(q, q′) > 0, we add f_t(q, v) · T(q, q′) to f_t+1(q′, θ_q′(v, η_q′)).

As a direct consequence of the above lemma and of the DAA construction from Section 4, we arrive at our main theorem.

Let a finite-memory text model ( , c₀, Σ, φ), a window-based pattern matching algorithm A, a pattern p with |p| = m, and the functions shift^A,p and ξ^A,p be given. Then, the cost distribution ℒ ( X n A , p ) can be computed using O(n² · m · | | · | |² · |Σ|) time and O(| | · | | · n · m) space. If for all c ∈ and σ ∈ Σ, there exists at most one c′ ∈ such that φ(c, σ, c′) > 0, a factor of | | can be dropped from the runtime bounds.

Using optimal algorithm-dependent DAA constructions schemes (e.g., the O(m²) construction for the Horspool algorithm by Tsai [13]) allows to replace | | by a polynomial in m, instead of O(m|Σ|^m).

Let w_i, …, w_k be the sequence of windows examined by algorithm A when processing the text s ∈ Σⁿ. In the beginning, the rightmost position of the current window is at position m − 1 and is moved by shift^A,p(w_i) after processing window w_i. After processing all windows it is beyond the end of the text. Formally, n ≤ m - 1 + ∑ i = 1 k shift A , p ( w i ) < n + m

By using the assumption that shift^A,p(w) = m − ξ^A,p(w) + 1, we obtain n ≤ m - 1 + ∑ i = 1 k ( m - ξ A , p ( w i ) + 1 ) < n + m ⇔ n ≤ m - 1 + k ( m + 1 ) - ξ A , p ( s ) < n + m ⇔ k ( m + 1 ) < ξ A , p ( s ) + n + 1 ≤ k ( m + 1 ) + mwhich means that ξ^A,p(s) + n + 1 lies strictly between k-times and (k + 1)-times m + 1 and thus cannot be a multiple of m + 1.

The probability that one pattern matching algorithm is faster than another depends on the pattern. Using the technique introduced in Section 6, we can quantify the strength of this effect. Figure 4 shows distributions of cost differences for different patterns and algorithms. That means, the probability that the first algorithm is faster is represented by the area under the curve left of zero. For the pattern ACCCCC and a random text of length 100, for example, there is a 53.9% probability that Horspool's algorithm needs fewer character accesses than B(N)DM (for the same second order Markovian model as used before), while for ACGTAC, the probability is only 0.0016%.

Worth noting and perhaps surprising is the fact that there is a non-zero probability of BOM being faster than B(N)DM although, shift^B(N)DM,p(w) ≥ shift^BOM,p(w) for all window contents w. The explanation, of course, is that a shorter (say, first) shift for BOM leads to a different window content than for B(N)DM for the second window, which may have a larger shift value. This effect depends on the pattern: For the pattern ACCCCC, there is a 52.4% probability that BOM is at least as fast as B(N)DM (in terms of character accesses), while it is 4.9% for ACGTAC, again on texts of length 100. An example text where BOM is faster than B(N)DM while searching for the pattern ACCCCC is shown in Figure 5. Both algorithms read two characters of the first window but prescribe different shifts. The first window ends on TC. BOM recognizes that TC is not a substring of the pattern an shifts the window by five positions, just far enough to exclude this substring. In contrast, B(N)DM determines that neither C nor TC are prefixes of the pattern and shifts the window by six positions. It turns out, however, that a shorter shift of five positions was beneficial in this case as BOM can process the second window with one character access, while B(N)DM uses two character accesses.

To assess the effect of DAA minimization before constructing PAAs, we constructed minimized DAAs for all 21840 patterns of lengths 2 to 7 over Σ = { A, C, G, T}. The minimum, average, and maximum state counts are shown in Table 1. For length 6, Figure 6 contains a detailed histogram. These statistics show that construction and minimization as given in this article lead to smaller automata (and thus better runtimes) than the constructions given in the conference version of this article [18]. It may be conjectured that the worst-case size of the minimal state space grows only polynomially with m for all of these algorithms, as has been previously proven for the Horspool algorithm [13].

The algorithms were implemented in JAVA and are available as part of the MoSDi software package available at http://mosdi.googlecode.com. They were run on an Intel Core 2 Dual CPU at 2.1 GHz. Computing the distributions shown in Figure 3 took 0.5 to 1.3 seconds for each distribution. Distributions of differences as in Figure 4 were computed in 56 to 97 seconds.