Then (S, T) can be described as - 2· trace(R· J), and trace(R· J) is maximized when R = VU T, where UΛ V is the singular value decomposition (SVD) of J. Thus (S, T) can be obtained from J in constant time, as J is a 3 × 3 matrix and the SVD can be computed in O d) time for a d × d matrix [ 33].
Basically, extra decoding complexity is needed since the FF decoding procedure is performed l o times for JID.
Secondly, for a set of n points in the plane, we design a query data structure for half-plane closest-pair queries that can be built in O(n2log2n) time using O(nlogn) space and answers a query in O(n1/2+ε) time, for any ε>0.
Depending on the topology of T, computing the hash tables of all nodes takes between O(n) time for a completely skewed tree and O(n) time for a balanced tree.
Whereas the original algorithm requires O(n) time (for n = | S|), the sparsified variant requires O(n L) time where L is the total size for all candidate lists of some i i.e. L := max i ∑ j ', d j ', j | L i (j ', d j '. j ) |. Obviously, L ≤ n.
As each vector has length at most n, we have that Vec≤ n, meaning a cost of O(n) time for the calls to line 4. Finally, we have that line 3 runs in O(n) time, therefore Algorithm 2 runs in O(n) time.
Since the algorithm runs in O(n) time for an n-vertex graph, we immediately obtain a randomized FPT Monte-Carlo algorithm, running in 2 O (k log k ) n time.
Our algorithm takes O(n2logn) time for an original polygonal chain with n vertices.
Our algorithms take O(mlgcnlglgn) time for some small constant c (where c≤2).
Considering the nodes in CD of each layer, it takes at most O(N) time for all layers.
