Fast algorithms for convex cost flow problems on circles, lines, and trees

Abstract

We develop efficient algorithms to solve convex cost flow problems where the underlying graph is a circle, a line, or a tree. Each node i has an associated supply/demand b(i ). The cost of sending flow on arc (i , j ) is a piecewise linear convex function f ij defined over R. Let n be the number of nodes and m = O(n) be the total number of pieces of all the convex functions. A flow x is feasible if the imbalances on all nodes are nonnegative. Excess e i (x ) stored on node i has an associated linear cost c i × e i (x ). We show that the problem on a circle can be transformed into an equivalent problem on a line in O(n) time. Thereafter, we develop an algorithm that solves the problem on a line in O(sort(n) + nα(n)) time, where sort(n) is the time to sort n real numbers and α(n) is the inverse Ackermann function. We also prove that when the nodes lie on a tree, the problem can be solved in O(n log n) time using the dynamic tree data structure. We describe applications in areas such as distributed computing, lot-sizing, computational biology, computational music, and transportation.

A New Approach to GraphMaps, a System Browsing Large Graphs as Interactive Maps

“…This is quite fast since the maximum zoom level is a small number, i.e., at most 10, in practice. We reduce the problem to the problem of computing a minimum cost maximum flow problem, where the edge costs can be convex (Orlin, 1993;Orlin and Vaidyanathan, 2013), e.g., quadratic function of the flow passing through the edge. Figure 9(a) depicts a set of points on a line, where the associated tiles are shown using rectangular regions.…”

“…If the resulting mapping does not satisfy the rank condition, then the instance of BALANCED VISUALIZA-TION does not have any affirmative solution. The best known running time for solving a convex cost network flow problem on a network of size O(τ) is O(τ 2 log 2 τ) (Orlin, 1993;Orlin and Vaidyanathan, 2013). Besides, it is straightforward to compute the corresponding node assignments in O(n log n) time augmenting the merge sort technique with basic data structures.…”

Simple linear flow decomposition algorithms on trees, circles, and augmented trees

“…Vaidyanathan and Ahuja [2], and Orlin and Vaidyanathan [3] prove that minimum cost flow problems (when total supply is not equal to the total demand) on lines, circles, and trees can be solved in O ( n log n ) time using specialized implementations of the successive shortest path algorithm. To adapt these algorithms to solve assignment problems, many‐to‐one‐matching problems, and transportation problems on lines, circles, or trees, we need to decompose the optimal solution of the minimum cost flow problem.…”

“…This is an important step in DNA shotgun sequencing and the problem has to be repeatedly solved millions of times (Ben Dor et al [7], Collannino and Toussaint [8], Collanino et al [9]). The current fastest run‐time for (i) the transportation problem on a circle or line is O ( n log n ) (Aggarwal et al [10]); (ii) the many‐to‐one matching problem on a line is O ( n log n ) (Collanino et al [9]); (iii) the many‐to‐one matching problem on a circle is O ( n 2 ) (minimum cost flow algorithm due to [3] followed by flow decomposition); (iv) the assignment, transportation, and the many‐to‐one matching problem on a tree is O ( n 2 ) (minimum cost flow algorithm due to [3] followed by flow decomposition). The flow decomposition algorithm described in this article can be used to improve the run‐times of (iii) and (iv) to O ( n log n ) .…”

“…The O ( n 2 ) run‐time can be improved to O ( n log n ) using the dynamic tree data structure (the algorithm is similar to the algorithm for determining a blocking flow on an acyclic network described in Sleator and Tarjan [15]). This data structure stores path fragments and can perform certain operations on arcs in a path efficiently ([2,3,16,17,18]). However, an algorithm that uses complicated data structures has associated overheads which affect its practical performance.…”