Consider the set S = {a, b, c, d}. Consider the following 4 partitions $$\,{\pi _1},\,{\pi _2},\,{\pi _3},\,{\pi _4}$$ on $$S:\,{\pi _1} = \left\{ {\overline {a\,b\,c\,d} } \right\},\,{\pi _2} = \left\{ {\overline {a\,b\,} ,\,\overline {c\,d} } \right\},\,{\pi _3} = \left\{ {\overline {a\,b\,c\,} ,\,\overline d } \right\},\,{\pi _4} = \left\{ {\overline {a\,} ,\,\overline b ,\,\overline c ,\,\overline d } \right\}.$$ Let $$ \prec $$ be the partial order on the set of partitions $$S' = \{ {\pi _1},\,{\pi _2},\,{\pi _3},\,{\pi _4}\} $$ defined as follows: $${\pi _i} \prec \,\,{\pi _j}$$ if and only if $${\pi _i} $$ refines $${\pi _j}$$. The poset diagram for $$(S',\, \prec )$$ is

If all the edge weights of an undirected graph are positive, then any subject of edges that connects all the vertices and has minimum total weight is a

Consider a weighted complete graph $$G$$ on the vertex set $$\left\{ {{v_1},\,\,\,{v_2},....,\,\,\,{v_n}} \right\}$$ such that the weight of the edge $$\left( {{v_i},\,\,\,\,{v_j}} \right)$$ is $$2\left| {i - j} \right|$$. The weight of a minimum spanning tree of $$G$$ is

Consider the polynomial $$P\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3},$$ where $${a_i} \ne 0,\forall i$$. The minimum number of multiplications needed to evaluate $$p$$ on an input $$x$$ is