Homework 7

Date: December 8, 1998

This homework is not graded. Please do this to your own satisfaction by this week. We will post the solution next week.

R-7.2. (page 294)
R-7.6.
R-7.9. (page 295)
R-7.11.
R-7.12.
C-7.2 (page 295). HINT: Assume that S and T contain distinct and disjoint sets of keys. Say S=(x₁< x₂< ... < x_n), and T=(y₁< y₂< ... < y_n). For each element x_i define its T-rank to be j if y_j < x_i < y_j+1. Write t(i) for the T-rank of x_i. Similarly, for each element y_i define its S-rank to be j if x_j < y_i < x_j+1. Write s(i) for the S-rank of y_i.
Suppose i+t(i)=k. Then the k-th smallest in S union T is x_i. Similarly, if j+s(j)=k, then the k-th smallest in S union T is y_j. Our goal is to find the smallest i such that i+t(i) ≥ k. Similarly, we can find the smallest j such that j+s(j) ≥ k.
C-7.5. (page 296)
C-7.10. HINT: suppose each node stores a new value called "size" which is equal to the number of nodes in the subtree at that node. Modify AVL inserts and deletes to maintain this information. How does this help our problem of counting number of keys in the range from k₁ from k₂

Show that the space usage of the adjacency list repreentation of graphs has complexity big-Theta of m+n.
R-9.3. (page 389)
R-9.5.
Prove by contradiction: if (u,v) is a back-edge in the DFS traversal of an undirected graph G, then v is an ancestor of u. HINT: if the "exploration" of a node is not yet complete, then it remains on path from the current node to the root of the tree. This path is precisely the sequence of nodes on the recursion stack.
Let G be an undirected graph where each node u has an even degree. Prove by induction on the number m (of edges) that G has an Euler Tour (this is a path that goes through each edge of G examply once).