Thematic maps visually communicate statistical information about spatial units such as countries or states. They must balance the individual readability of those map elements that carry the statistical information and the overall cartographic context. Nowadays, most maps are not static images, but must flexibly respond to a range of device types and display sizes. Current approaches to responsive thematic mapping are limited: they are labor-intensive for practitioners and often rely on combining disjointed visual encodings to cover different device types. In this paper we introduce the first algorithmic framework to efficiently compute responsive thematic maps that smoothly adapt to different display sizes. A key component of our framework is the layout guide: a combinatorial structure which encodes the two essential aspects of a thematic map. The first aspect are the visual requirements of each statistical map element (at least their desired width and height), the second aspect is the cartographic context in the form of relative positions of map elements. Our main algorithmic contribution is the map arranger which takes a visual container as input and returns a suitable layout guide. The map arranger does so in a stable and consistent manner: if the container changes only a little, then so does the layout guide, and the same input container always results in the same layout guide. To use our framework, one needs three ingredients: $(1)$ a reference layout, which corresponds to the ``ideal'' thematic map, $(2)$ a total vertical and horizontal order for all map elements (the desired layouts for containers with extreme aspect ratios), and $(3)$ a thematic mapping algorithm that can construct a thematic map from a layout guide. We demonstrate our framework on two types of thematic maps, namely rectangular and Demers cartograms.

For a positive integer $k$ and an ordered set of $n$ points in the plane, define its \textit{k-sector ordered Yao graphs} as follows. Divide the plane around each point into $k$ equal sectors and draw an edge from each point to its closest predecessor in each of the $k$ sectors. We analyze several natural parameters of these graphs. Our main results are as follows: I Let $d_k(n)$ be the maximum integer so that for every $n$-element point set in the plane, there exists an order such that the corresponding $k$-sector ordered Yao graph has maximum degree at least $d_k(n)$. We show that $d_k(n)=n-1$ if $k=4$ or $k \ge 6$, and provide some estimates for the remaining values of $k$. Namely, we show that $d_1(n) = \Theta( \log {n} )$; $\frac{1}{2}(n-1) \le d_3(n) \le 5\left\lceil\frac{n}{6}\right\rceil-1$; $\frac{2}{3}(n-1) \le d_5(n) \le n-1$; II Let $e_k(n)$ be the minimum integer so that for every $n$-element point set in the plane, there exists an order such that the corresponding $k$-sector ordered Yao graph has at most $e_k(n)$ edges. Then $e_k(n)=\left\lceil\frac{k}{2}\right\rceil\cdot n-o(n)$. III Let $w_k$ be the minimum integer so that for every point set in the plane, there exists an order such that the corresponding $k$-sector ordered Yao graph has clique number at most $w_k$. Then $\lceil\frac{k}{2}\rceil \le w_k\le \lceil\frac{k}{2}\rceil+1$. All the orders mentioned above can be constructed effectively.