Description
Problem 1 Let G = (V, E) be a simple undirected graph with weights w : E → ZZ+. The
inductivity of a vertex ordering (permutation Π of V )
D
vΠ(1), vΠ(2), . . . , vΠ(n)
E
is defined by
max
2≤j≤n
X
1≤Π(i)<Π(j)
w(vΠ(i)vΠ(j)
). (1)
Use (even if not completely covered yet) Fibonacci heaps to obtain a O(|E| + |V | log |V |)-time
algorithm (present pseudocode) to produce a least-inductivity vertex ordering of G, together with
the proof of correctness. That is, find the permutation Π of V that minimizes Formula (1)
Hint: Use a greedy strategy paying attention to nodes with smallest weighted degree in G.
Give a O(E| + |V |)-time algorithm for the unweighted case (all the weights are 1).
Problem 2 Describe a binary search tree on n nodes such that the average depth of a node in
the tree is Θ(lg n) but the height of the tree is not O(lg n). How large can the height of an n-node
binary search tree be if the average depth of a node is Θ(lg n)?
Problem 3 Assume every node in a binary search tree has a pointer to its parent, in addition to
pointers to the left and right child. Design an algorithm (write pseudocode), which, given a node
v, finds w, the node-successor of v in inorder (the element of w is also the successor of the element
of v in the sorted order of elements).
Analyze the running time of s consecutive calls to successor (that is, w is given as the argument
to the next call, and so on) in terms of s and h, the height of the tree. A tight (within a constant)
analysis is worth one third of the points.
Problem 4 Suppose we wish not only to increment a binary number, but also to reset it to zero
(i.e., make all bits in it 0). Counting the cost to examine or modify a bit as 1, show how to
implement a binary number as an array of bits so that any sequence of n INCREMENT and
RESET operations costs O(n) on an initially zero number.
Hint: Keep a pointer to the high-order 1.
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