text: We give a succinct data-structure that stores a tree with color for each node, and allows finding, given a node x and a color alpha, the nearest node to x with color alpha. This results improves the O(n log n)-bits structure of [CPM 2016].
  @cite_4: 
  @cite_0: 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.
  @cite_2: We investigate the problem of approximate Nearest-Neighbor Search (NNS) in graphical metrics: The task is to preprocess an edge-weighted graph G=(V,E) on m vertices and a small "dataset" D V of size n << m, so that given a query point q V, one can quickly approximate dist(q,D) (the distance from q to its closest vertex in D) and find a vertex a D within this approximated distance. We assume the query algorithm has access to a distance oracle, that quickly evaluates the exact distance between any pair of vertices. For planar graphs G with maximum degree Delta, we show how to efficiently construct a compact data structure -- of size O(n(Delta+1 epsilon)) -- that answers (1+epsilon)-NNS queries in time O(Delta+1 epsilon). Thus, as far as NNS applications are concerned, metrics derived from bounded-degree planar graphs behave as low-dimensional metrics, even though planar metrics do not necessarily have a low doubling dimension, nor can they be embedded with low distortion into l_2. We complement our algorithmic result by lower bounds showing that the access to an exact distance oracle (rather than an approximate one) and the dependency on Delta (in query time) are both essential.
  @cite_5: Consider an undirected weighted graph G=(V,E) with |V|=n and |E|=m, where each vertex v is assigned a label from a set L of labels. We show how to construct a compact distance oracle that can answer queries of the form: "what is the distance from v to the closest lambda-labeled node" for a given node v in V and label lambda in L. This problem was introduced by Hermelin, Levy, Weimann and Yuster [ICALP 2011] where they present several results for this problem. In the first result, they show how to construct a vertex-label distance oracle of expected size O(kn^ 1+1 k ) with stretch (4k - 5) and query time O(k). In a second result, they show how to reduce the size of the data structure to O(kn ^ 1 k ) at the expense of a huge stretch, the stretch of this construction grows exponentially in k, (2^k-1). In the third result they present a dynamic vertex-label distance oracle that is capable of handling label changes in a sub-linear time. The stretch of this construction is also exponential in k, (2 3^ k-1 +1). We manage to significantly improve the stretch of their constructions, reducing the dependence on k from exponential to polynomial (4k-5), without requiring any tradeoff regarding any of the other variables. In addition, we introduce the notion of vertex-label spanners: subgraphs that preserve distances between every node v and label lambda. We present an efficient construction for vertex-label spanners with stretch-size tradeoff close to optimal.
  @cite_12: We consider distance queries in vertex-labeled planar graphs. For any fixed @math we show how to preprocess a directed planar graph with vertex labels and arc lengths into a data structure that answers queries of the following form. Given a vertex @math and a label @math return a @math -approximation of the distance from @math to its closest vertex with label @math . For a directed planar graph with @math vertices, such that the ratio of the largest to smallest arc length is bounded by @math , the preprocessing time is @math , the data structure size is @math , and the query time is @math . We also point out that a vertex label distance oracle for undirected planar graphs suggested in an earlier version of this paper is incorrect.
  @cite_11: We consider vertex-labeled graphs, where each vertex v is attached with a label from a set of labels. The vertex-to-label distance query desires the length of the shortest path from the given vertex to the set of vertices with the given label. We show how to construct an oracle for a vertex-labeled planar graph, such that (O( 1 n n) ) storing space is needed, and any vertex-to-label query can be answered in (O( 1 n ) ) time with stretch 1 + e. Here, Δ is the hop-diameter of the given graph. For the case that Δ = O(logn), we construct a distance oracle that achieves (O( 1 n) ) query time, without changing space usage.
summary: Another related problem is to find an approximate nearest node with color @math . This problem has been studied in general graphs @cite_5 @cite_11 @cite_0 and planar graphs @cite_2 @cite_4 @cite_12 .