MOTIVATION: Tumor trees, which depict the evolutionary process of cancer, provide a backbone for discovering recurring evolutionary processes in cancer. While they are not the primary information extracted from genomic data, they are valuable for this purpose. One such extraction method involves summarizing multiple trees into a single representative tree, such as consensus trees or supertrees. RESULTS: We define the "weighted centroid tree problem" to find the centroid tree of a set of single-labeled rooted trees through the following steps: (i) mapping the given trees into the Euclidean space, (ii) computing the weighted centroid matrix of the mapped trees, and (iii) finding the nearest mapped tree (NMTP) to the centroid matrix. We show that this setup encompasses previously studied parent-child and ancestor-descendent metrics as well as the GraPhyC and TuELiP consensus tree algorithms. Moreover, we show that, while the NMTP problem is polynomial-time solvable for the adjacency embedding, it is NP-hard for ancestry and distance mappings. We introduce integer linear programs for NMTP in different setups where we also provide a new algorithm for the case of ancestry embedding called 2-AncL2, that uses a novel weighting scheme for ancestry signals. Our experimental results show that 2-AncL2 has a superior performance compared to available consensus tree algorithms. We also illustrate our setup's application on providing representative trees for a large real breast cancer dataset, deducing that the cluster centroid trees summarize reliable evolutionary information about the original dataset. AVAILABILITY AND IMPLEMENTATION: https://github.com/vasei/WAncILP.

We follow two main objectives in this article. On the one hand, we introduce a security model called LORBACPA+ for self-synchronized stream ciphers which is stronger than the blockwise LOR-IND-CPA, where we show that standard constructions as delayed CBC or similar existing self-synchronized modes of operation are not secure in this stronger model. Then, on the other hand, following contributions of Millérioux et al., we introduce a new self-synchronized stream cipher and prove its security in LORBACPA+ model.

Let G be a simple undirected uniquely vertex k-colorable graph, or a k-UCG for short. M. Truszczyński [Some results on uniquely colorable graphs, in Finite and Infinite Sets, North-Holland, Amsterdam, 1984, pp. 733--748] introduced $e^{^{*}}(G)=|V(G)|(k-1)-{k \choose 2}$ as the minimum number of edges for a k-UCG and S. J. Xu [J. Combin. Theory Ser. B, 50 (1990), pp. 319--320] conjectured that any minimal k-UCG contains a K k as a subgraph. In this paper, first we introduce a technique called forcing. Then by applying this technique in conjunction with a feedback structure we construct a k-UCG with clique number k-t, for each $t \geq 1$ and each k, when k is large enough. This also improves some known results for the case t=1. Second, we analyze the parameter $\Lambda(G)=|E(G)|-e^{^{*}}(G)$ for our constructions, and we obtain some bounds for the functions \lambda_{_{t}}(k)= \min \{\Lambda(G) \ : \ G \ {\rm is \ a} \ k\mbox{\rm -UCG and} \ cl(G)=k-t \}, $$ $$ \nu_{_{t}}(k)= \min \{|V(G)| \ : \ G \ {\rm is \ a} \ k\mbox{\rm -UCG and} \ cl(G)=k-t \}. Also, we introduce some possible applications of the technique in cryptography and data compression.