Fig. 2From: A linear-time algorithm that avoids inverses and computes Jackknife (leave-one-out) products like convolutions or other operators in commutative semigroupsA (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 outlinesBack to article page