Let P=(p1,p2,…,pn) be a polygonal chain. The stretch factor of
P is the ratio between the total length of P and the distance of its
endpoints, ∑i=1n−1∣pipi+1∣/∣p1pn∣. For a parameter c≥1, we call P a c-chain if ∣pipj∣+∣pjpk∣≤c∣pipk∣, for
every triple (i,j,k), 1≤i<j<k≤n. The stretch factor is a global
property: it measures how close P is to a straight line, and it involves all
the vertices of P; being a c-chain, on the other hand, is a
fingerprint-property: it only depends on subsets of O(1) vertices of the
chain.
We investigate how the c-chain property influences the stretch factor in
the plane: (i) we show that for every ε>0, there is a noncrossing
c-chain that has stretch factor Ω(n1/2−ε), for
sufficiently large constant c=c(ε); (ii) on the other hand, the
stretch factor of a c-chain P is O(n1/2), for every
constant c≥1, regardless of whether P is crossing or noncrossing; and
(iii) we give a randomized algorithm that can determine, for a polygonal chain
P in R2 with n vertices, the minimum c≥1 for which P is
a c-chain in O(n2.5polylogn) expected time and
O(nlogn) space.Comment: 16 pages, 11 figure