Finding a minimum weight K-link path in graphs with Monge property and applications

Abstract

Let G be a weighted, complete, directed acyclic graph (DAG), whose edge weights obey the Monge condition. We give an efficient algorithm for finding the minimum weight K-link path between a given pair of vertices for any given K. The time complexity of our algorithm is O(n~=) for the concave case and O (ncr (n) log3 n) for the convex case. Our algorithm uses some properties of DAGs with Monge property together with a refined parametric search technique. We apply our algorithm (for the concave case) to get efficient solutions for the following problems, improving on previous results:(1) Finding the largest K-gon contained in a given polygon.(2) Finding the smallest K-gon that is the intersection of K halfplanes out of of given set of halfplanes defining an n-gon. (3) Computing maximum K-cliques of an interval graph. (4) Computing length limited Huffman codes. (5) Computing optimal discrete quantization.

The directional p-median problem: Definition, complexity, and algorithms

“…, n} and arc weights w(i, j) for arc (i, j) from vertex i to j, 0 6 i < j. Solving DPM1 is equivalent to finding a minimum weight p-link path from vertex 0 to n in G. Further, we show that the arc weights in the DPM1 graph representation obey the concave Monge property, allowing a solution in time Oðn ffiffiffiffiffiffiffiffiffiffiffiffiffi p log n p Þ [1]. These results are stated in the following two lemmas.…”

Polyline Fitting of Planar Points Under Min-Sum Criteria

“…In the restricted case where the vertex set or the set of lines supporting edges of the output polyline is required to be a subset of that of the input polyline, the problem is reduced to the k-link shortest path problem in a graph. In particular, if the input polyline is convex, this problem is related to matrix searching (see [2]). However, for the general case the authors are not aware of an efficient algorithm, and it is an interesting research problem.…”

“…The preprocessing time is O(n log n). Now, our algorithm for finding y(t) is as follows: Given t, we first compute all the intersections between X = t and the lines of Ψ k , sort them to have a list y (1) , y (2) , . .…”

A Generic Top-Down Dynamic-Programming Approach to Prefix-Free Coding

“…Some examples of such prefix-free coding problems are Length-limited coding e.g, [16,17,3,18], Unequalcost coding e.g., [6,13,7,12] Mixed-radix coding [10], Reserved-length coding [4], and One-ended coding [5,8,9], The major observation is that all of the best algorithms known for these problems use some form of dynamic programming (DP) to build an optimal (mincost) coding tree that corresponds to an optimal code. These DPs primarily differ in whether they build the tree from the bottom-up or the top-down.…”