By Jost J., Xin Y. L.

We receive a Bernstein theorem for distinct Lagrangian graphs in for arbitrary simply assuming bounded slope yet no quantitative limit.

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**Extra info for A Bernstein theorem for special Lagrangian graphs**

**Example text**

8 Notes WEISS [122] contains an excellent treatment of the merge-find data structure and heaps. Dijkstra’s shortestpath algorithm and Floyd’s algorithm are described in most books on algorithms and data structures. page_45 Page 46 This page intentionally left blank. 1 Bipartite graphs A graph G is said to be bipartite if V(G) can be divided into two sets X and Y such that each edge has one end in X and one end in Y . 1. 1 Two bipartite graphs The maximum number of edges in a simple bipartite graph in which X and Y are the two sides of the bipartition is clearly | X |·| Y |.

If T is not a minimum tree, then we can proceed as we did in Prim’s algorithm. Let T consist of edges e 1, e 2,…, e n−1, chosen in that order. Select a minimum tree T* which contains e 1, e 2,…, ek, but not ek+1, where k is as large as possible. Consider the iteration in which ek+1= xy was selected. T* +xy contains a fundamental cycle Cxy, which must contain another edge ab incident on Tx. Since Kruskal’s algorithm chooses edges in order of their weight, WT(xy) ≤WT (ab) . Then T′ =T* +xy−ab is a spanning tree for which WT(T′) ≤WT (T*) .

In this algorithm, PQu will stand for a priority queue which can be merged. The Cheriton-Tarjan algorithm can be described as follows. It stores a list Tree of the edges of a minimum spanning tree. The components of the spanning forest are represented as Tu and the priority queue of edges incident on vertices of Tu is stored as PQu. 1 Prove that the Cheriton-Tarjan algorithm constructs a minimum spanning tree. 2 Show that a heap is best stored as an array. What goes wrong when the attempt is made to store a heap with pointers?