Finding a Length-Constrained Maximum-Density Path in a Tree Rung-Ren Lin, Wen-Hsiung Kuo, and Kun-Mao Chao

Overview  Introduction of the problem.  A brief discussion of one-dimensional sequence.  Two approaches of finding the maximum- density path in a tree.

Input & Output  Input ： A weighted tree with n edges and a lower bound L.  The lower bound is necessary.  Output ： A maximum-density path with length at least L.

The Density of a Path  The density of a path is defined as follow ：  Given a path with k edges E={e 1, e 2, …, e k }, and w(e) is an edge weight function. The density of such path is defined as

Lower Bound L = Max-Density = ( ) / 5 = 6

The Efficiency  We propose two efficient algorithms reach O(nL) time.  One of them is further modified to solve some special cases such as full m-ary trees in linear time.

One-Dimensional Sequence  The maximum-average segment problem arises naturally in several areas of sequence analysis.  For example, given a DNA sequence, which segment of the sequence of length at least L has the highest GC ratio.  The efficiency of the naïve algorithm of one- dimensional sequence is O(n 2 ).

Lower Bound L  There exists an optimal segment of length at most 2L-1.  It can be proved by a counter argument.  The naïve algorithm is O(nL) now. AB

The Relation between Two Overlapped Sequences  Let P[x] denotes the maximum-density of those segments that start from x.  For a given j<i ≦ P[i]<P[j], the density of (i,P[i]) is greater than the density of (j,P[j]) i P[i] jP[j] XYZ Y

Right-Skew Segment  A sequence S={s 1, s 2, …, s k } is right-skew if and only if the average of any prefix {s 1, s 2, …, s i ) is never larger than the average of the remaining subsequence {s i+1, …, s k ).  According to the tricks mentioned above, Goldwasser et al. proposed a linear time algorithm for one-dimensional sequence.

Right-skew Decomposition Decreasing right-skew decomposition is unique

Finding Maximum-Density Path In a Tree  The path in a tree is similar to the segment of an one-dimensional sequence. That is, there exists a maximum-density path of length at most 2L-1.  We propose two approaches reach the time complexity of O(nL).

Downward & Upward Paths  We classify the paths that start from node K into two types called downward and upward paths.  One is to stretch downward to its children only, called downward paths. And the other is to include at least its parent, called upward paths.

An upward path of node K K A downward path of node K A general tree

Notation  Let denotes the maximum-density downward path of node K of length i.  Let denotes the second-best one. If there is a tie, choose an arbitrary path.  Similarly, represent the maximum upward path of node K of length i.

A

Contributor  The node which determines its parent’s maximum-density downward path is called contributor.  Each maximum-density downward path of a given length has its own contributor.  The upward path has no contributor because each node has at most one parent.

A B C D node D node C node D node B

Downward & Upward Table  The downward table of node K is composed of and, where 1 ≦ i ≦ 2L-1.  Note that there exists an optimal path with length at most 2L-1  Similarly, the upward table of node K is composed of, where 1 ≦ i ≦ 2L-1.

Constructing Downward Table  For a given internal node K with m children {K 1, K 2, …, K m }. Let e j denotes the edge (K, K j ).  We can construct the downward table of node K by bottom-up dynamic programming.  All contributors should be recorded.

Constructing Upward Table  Suppose K’ is the parent of node K, and e’ denote the edge (K’, K).  Upward Table of node K can be constructed by top-down dynamic programming.

Time Complexity  Deciding a given length of downward and upward path of any node is O(1).  Thus, constructing downward and upward table of a node takes O(L).  There are totally n nodes in a tree, so it take O(nL) time to complete all downward and upward tables.

Approach I ： Finding the Path from its End Node  Once the downward and upward table of each node are constructed, then we can determine the maximum-density path of a given node with length from L to 2L-1 in O(L).  Therefore, it takes O(nL) time to check all nodes since there are n nodes in such tree.

Approach II ： Finding the Path from its LCA Node  LCA stands for least common ancestor.  We are now combining two downward paths together.