A recent data visualization literacy study shows that most people cannot read networks that use hierarchical cluster representations such as "super-noding" and "edge bundling." Other studies that compare standard node-link representations with map-like visualizations show that map-like visualizations are superior in terms of task performance, memorization and engagement. With this in mind, we propose the Zoomable Multilevel Tree (ZMLT) algorithm for map-like visualization of large graphs that is representative, real, persistent, overlap-free labeled, planar, and compact. These six desirable properties are formalized with the following guarantees: (1) The abstract and embedded trees represent the underlying graph appropriately at different level of details (in terms of the structure of the graph as well as the embedding thereof); (2) At every level of detail we show real vertices and real paths from the underlying graph; (3) If any node or edge appears in a given level, then they also appear in all deeper levels; (4) All nodes at the current level and higher levels are labeled and there are no label overlaps; (5) There are no crossings on any level; (6) The drawing area is proportional to the total area of the labels. This algorithm is implemented and we have a functional prototype for the interactive interface in a web browser.
Steiner Trees: The Steiner tree problem is one of Karp's original 21 NP-Complete problems @cite_40 . A polynomial time approximation scheme exists if the underlying graph is planar @cite_49 , but not for general graphs @cite_24 . More importantly, the Steiner tree can be efficiently approximated within a factor of @math @cite_4 . In the vertex-weighted Steiner tree problem, the cost of the tree is determined by the weights of the vertices included in the solution (rather than the weights of the edges). This version of problem is also NP-hard @cite_37 but can be approximated within a logarithmic factor @cite_41 @cite_15 . The classic Steiner tree problem has also generalized to a multi-level setting @cite_32 @cite_10 @cite_6 , where the terminals appear on different levels and must be connected by edges of appropriate levels, as described in the introduction. In particular, Ahmed . @cite_31 showed that the standard (edge-weighted) Steiner tree problem can be efficiently approximated to within a constant factor. Similarly, Darabi @cite_12 described a multi-level version of the vertex-weighted Steiner tree, and showed that it can be approximated as well as the single-level version.
@cite_37 @cite_31 @cite_4 @cite_41 @cite_32 @cite_6 @cite_24 @cite_40 @cite_49 @cite_15 @cite_10 @cite_12
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The Steiner tree problem on weighted graphs seeks a minimum weight subtree containing a given subset of the vertices (terminals). We show that it is NP-hard to approximate the Steiner tree problem within a factor 96 95. Our inapproximability results are stated in a parametric way, and explicit hardness factors would be improved automatically by providing gadgets and or expanders with better parameters. The Steiner tree problem is one of the most fundamental NP-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to the current best 1.55 [Robins,Zelikovsky-SIDMA'05]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP-relaxation for Steiner tree with integrality gap smaller than 2 [Vazirani,Rajagopalan-SODA'99]. In this paper we improve the approximation factor for Steiner tree, developing an LP-based approximation algorithm. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider a directed-component cut relaxation for the k-restricted Steiner tree problem. We sample one of these components with probability proportional to the value of the associated variable in the optimal fractional solution and contract it. We iterate this process for a proper number of times and finally output the sampled components together with a minimum-cost terminal spanning tree in the remaining graph. Our algorithm delivers a solution of cost at most ln(4) times the cost of an optimal k-restricted Steiner tree. This directly implies a ln(4)+e The Steiner tree problem is one of the most fundamental NP-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to 1.55 [Robins and Zelikovsky 2005]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP relaxation of Steiner tree with integrality gap smaller than 2 [Rajagopalan and Vazirani 1999]. In this article we present an LP-based approximation algorithm for Steiner tree with an improved approximation factor. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider an LP relaxation of the problem, which is based on the notion of directed components. We sample one component with probability proportional to the value of the associated variable in a fractional solution: the sampled component is contracted and the LP is updated consequently. We iterate this process until all terminals are connected. Our algorithm delivers a solution of cost at most ln(4) + e As a by-product of our analysis, we show that the integrality gap of our LP is at most 1.55, hence answering the mentioned open question. In this paper we study the Steiner tree problem over a dynamic set of terminals. We consider the model where we are given an @math -vertex graph @math with positive real edge weights, and our goal is to maintain a tree which is a good approximation of the minimum Steiner tree spanning a terminal set @math , which changes over time. The changes applied to the terminal set are either terminal additions (incremental scenario), terminal removals (decremental scenario), or both (fully dynamic scenario). Our task here is twofold. We want to support updates in sublinear @math time, and keep the approximation factor of the algorithm as small as possible. We show that we can maintain a @math -approximate Steiner tree of a general graph in @math time per terminal addition or removal. Here, @math denotes the stretch of the metric induced by @math . For planar graphs we achieve the same running time and the approximation ratio of @math . Moreover, we show faster algorithms for incremental and decremental scenarios. Finally, we show that if we allow higher approximation ratio, even more efficient algorithms are possible. In particular we show a polylogarithmic time @math -approximate algorithm for planar graphs. One of the main building blocks of our algorithms are dynamic distance oracles for vertex-labeled graphs, which are of independent interest. We also improve and use the online algorithms for the Steiner tree problem.