Subtree Isomorphism Revisited

Abstract

The Subtree Isomorphism problem asks whether a given tree is contained in another given tree. The problem is of fundamental importance and has been studied since the 1960s. For some variants, e.g., ordered trees , near-linear time algorithms are known, but for the general case truly subquadratic algorithms remain elusive. Our first result is a reduction from the Orthogonal Vectors problem to Subtree Isomorphism, showing that a truly subquadratic algorithm for the latter refutes the Strong Exponential Time Hypothesis (SETH). In light of this conditional lower bound, we focus on natural special cases for which no truly subquadratic algorithms are known. We classify these cases against the quadratic barrier, showing in particular that: • Even for binary, rooted trees, a truly subquadratic algorithm refutes SETH. • Even for rooted trees of depth O (log log n ), where n is the total number of vertices, a truly subquadratic algorithm refutes SETH. • For every constant d , there is a constant ε d > 0 and a randomized, truly subquadratic algorithm for degree- d rooted trees of depth at most (1+ ε d ) log d n . In particular, there is an O (min { 2.85 h , n 2 }) algorithm for binary trees of depth h . Our reductions utilize new “tree gadgets” that are likely useful for future SETH-based lower bounds for problems on trees. Our upper bounds apply a folklore result from randomized decision tree complexity.

Simulating branching programs with edit distance and friends: or: a polylog shaved is a lower bound made

“…For example, the polynomial method [41] has allowed for superpolylogarithmic shavings for APSP, and more recently to two other problems that are more closely related to ours, namely Longest Common Substring [9], and Hamming Nearest Neighbors [10]. A natural open question [41,9,10] is whether these techniques can lead to an n 2 / log ω (1) n algorithm for Edit-Distance as well. The lower bound of Backurs and Indyk is not sufficient to address this question, and only a much faster n 2 /2 ω(log n/ log log n) would have been required to improve the current CNF-SAT algorithms.…”

“…Williams showed that faster-than-trivial Circuit-SAT algorithms for many circuit classes C would imply interesting new lower bounds against that class [40,42]. Via this connection, and known reductions from certain circuit families to CNF formulas, it is possible to show that refuting SETH implies a new circuit lower bound [29]: E NP cannot be solved by linear-size series-parallel circuits 1 . However, this is a very weak lower bound consequence.…”

“…Thus, we get that if LCS can be solved in O(n 2 / log 1000 n) time, then SAT on formulas of size O(n 5 ) can be solved in O(2 n /n 15 ) time, which would imply that E NP does not have such formulas. We obtain that solving Edit-Distance or LCS in n 2 / log ω (1) n time still implies a major circuit lower bound, namely that NTIME [2 O(n) ] is not in non-uniform NC 1 . Corollary 2.…”

“…Other interesting recent results show that under SETH, the current algorithms for many central problems in diverse areas of computer science are optimal, up to n o (1) factors. These areas include pattern matching [7,2,18,1], graph algorithms [35,3,5,8,6], parameterized complexity [32,2], computational geometry [17,19], and the list is growing by the day.…”

“…Other interesting recent results show that under SETH, the current algorithms for many central problems in diverse areas of computer science are optimal, up to n o (1) factors. These areas include pattern matching [7,2,18,1], graph algorithms [35,3,5,8,6], parameterized complexity [32,2], computational geometry [17,19], and the list is growing by the day. Bringmann and Künnemann [18] generalize many of the previous SETH lower bounds [7,17,12,2] into one framework; they prove that the problem of computing any similarity measure δ over two sequences (of bits, symbols, points, etc) will require quadratic time, as long as the similarity measure has a certain property (namely, if δ admits alignment gadgets).…”