Skip to main content
Fig. 2 | Algorithms for Molecular Biology

Fig. 2

From: A linear-time algorithm that avoids inverses and computes Jackknife (leave-one-out) products like convolutions or other operators in commutative semigroups

Fig. 2

A (rootless) segment tree. This figure illustrates the rootless segment tree constructed in the upward phase of the Jackknife Product algorithm. The commutative semigroup \(\left( {G, \circ } \right)\) illustrated is the set of nonnegative integers under addition. The bottom row of \(n^{*} = 2^{m}\) squares (\(m = 3\)) contains \(L_{0} \left[ j \right] = g_{j}\) (\(0 \le j < n^{*}\)). In the next row up, as indicated by the arrow pairs leading into each circle, the array \(L_{1}\) contains consecutive sums of consecutive disjoint pairs in \(L_{0}\), e.g., \(L_{1} \left[ 0 \right] = 13 = 5 + 8\). The rest of the segment tree is constructed recursively upward to \(L_{m - 1}\), just as \(L_{1}\) was constructed from \(L_{0}\). Here, 2 copies of the additive identity \(e = 0\) pad out \(L_{0}\) on the right. Padded on the right, the copies contribute literally nothing to the segment tree above them. Their non-contributions have dotted outlines

Back to article page