Algorithms for Problems on Maximum Density Segment

Abstract

Let A be a sequence of n ordered pairs of real numbers $(a_i, l_i)$ $(i=1, \ldots , n)$ with $l_i>0$, and L and U be two positive real numbers with $0 < L\leqslant U$. A segment, denoted by $A[i, j], 1\leqslant i\leqslant j \leqslant n$, of A is a consecutive subsequence of A between the indices i and j (i and j included). The length l[i, j], sum s[i, j] and density d[i, j] of a segment A[i, j] are $l[i, j]=\sum ^j_{t=i}{l_t}$, $s[i, j]=\sum ^j_{t=i}{a_t}$ and $d[i, j]=\frac{s[i, j]}{ l[i, j]}$ respectively. A segment A[i, j] is feasible if $L\leqslant l[i, j] \leqslant U$. The length-constrained maximum density segment problem is to find a feasible segment of maximum density. We present a simple geometric algorithm for this problem for the uniform length case ($l_i = 1$ for all i), with time and space complexities in O(n) and $O(U-L+1)$ respectively. The k length-constrained maximum density segments problem is to find the k most dense length-constrained segments. For the uniform length case, we propose an algorithm for this problem with time complexity in $O(\min \{nk, n\lg (U-L+1) + k\lg ^2(U-L+2), n(U-L+1)\})$.

A. Mukhopadhyay—Research supported by an NSERC discovery grant awarded to this author.