Let T = (V,E) be a tree, and letr, r′ ∈ V be any two nodes....

Let T = (V,E) be a tree, and letr, r′ ∈ V be any two nodes. Prove that the height of the rooted tree (T, r) is at most twice the height of the rooted tree (T, r′).

Expert Solution

Let T = (V,E) be a tree, and letr, r′ ∈ V be any two nodes. Prove that the height of the rooted tree (T, r) is at most twice the height of the rooted tree (T, r′).

Colby Messinger answered 1 year ago

ii. Let G = (V, E) be a tree. Prove G has |V | − 1...

ii. Let G = (V, E) be a tree. Prove G has |V | − 1 edges using strong induction. Hint: In the inductive step, choose an edge (u, v) and partition the set vertices into two subtrees, those that are reachable from u without traversing (u, v) and those that are reachable from v without traversing (u, v). You will have to reason why these subtrees are distinct subgraphs of G. iii. What is the total degree of a...

a.) Let T be a binary tree with n nodes. Define the lowest common ancestor (LCA) between...

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Let G = (V, E) be a directed graph, with source s ∈ V, sink t...

Let G = (V, E) be a directed graph, with source s ∈ V, sink t ∈ V, and nonnegative edge capacities {ce}. Give a polynomial-time algorithm to decide whether G has a unique minimum s-t cut (i.e., an s-t of capacity strictly less than that of all other s-t cuts).

1. Let V and W be vector spaces over R. a) Show that if T: V...

1. Let V and W be vector spaces over R. a) Show that if T: V → W and S : V → W are both linear transformations, then the map S + T : V → W given by (S + T)(v) = S(v) + T(v) is also a linear transformation. b) Show that if R: V → W is a linear transformation and λ ∈ R, then the map λR: V → W is given by (λR)(v) =...

(V) Let A ⊆ R, B ⊆ R, A 6= ∅, B 6= ∅ be two...

(V) Let A ⊆ R, B ⊆ R, A 6= ∅, B 6= ∅ be two bounded subset of R. Define a set A − B := {a − b : a ∈ A and b ∈ B}. Show that sup(A − B) = sup A − inf B and inf(A − B) = inf A − sup B

Indicate the net charge for the peptide, C-H-A-V-E-C-A-R-R-I-S-T-H-E-G-R-E-A-T-E-S-T, at the given pH values: (a) pH 1,...

Indicate the net charge for the peptide, C-H-A-V-E-C-A-R-R-I-S-T-H-E-G-R-E-A-T-E-S-T, at the given pH values: (a) pH 1, net charge: (b) pH 5, net charge: (c) pH 8, net charge: (d) pH 14, net charge:

For any n ≥ 1 let Kn,n be the complete bipartite graph (V, E) where V...

For any n ≥ 1 let Kn,n be the complete bipartite graph (V, E) where V = {xi : 1 ≤ i ≤ n} ∪ {yi : 1 ≤ i ≤ n} E = {{xi , yj} : 1 ≤ i ≤ n, 1 ≤ j ≤ n} (a) Prove that Kn,n is connected for all n ≤ 1. (b) For any n ≥ 3 find two subsets of edges E 0 ⊆ E and E 00 ⊆ E such...

Let T : V → V be a linear map. A vector v ∈ V is...

Let T : V → V be a linear map. A vector v ∈ V is called a fixed point of T if Tv = v. For example, 0 is a fixed point for every linear map T. Show that 1 is an eigenvalue of T if and only if T has nonzero fixed points, and that these nonzero fixed points are the eigenvectors of T corresponding to eigenvalue 1

Create a kD tree with: x-nodes and y-nodes -- and maintain the following two properties: The...

Create a kD tree with: x-nodes and y-nodes -- and maintain the following two properties: The children of an x-node are y-nodes The children of a y-node are x-nodes Each type of node uses a different comparison to order points. This causes different levels of the tree to compare points differently, using the following rules: For every x-node: All descendants in the left subtree have a smaller x-coordinate than the point stored at the node. Visually, all descendant points are...

If G = (V, E) is a graph and x ∈ V , let G \...

If G = (V, E) is a graph and x ∈ V , let G \ x be the graph whose vertex set is V \ {x} and whose edges are those edges of G that don’t contain x. Show that every connected finite graph G = (V, E) with at least two vertices has at least two vertices x1, x2 ∈ V such that G \ xi is connected.

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