### Fundamentals of Data Structures

Problem 1: [15%] ( [GTG] Exercise R-8.11, page 350 )
Draw an arithmetic expression tree that has four external nodes, storing the
numbers 1, 5, 6, and 7 (with each number stored in a distinct external node,
but not necessarily in this order), and has three internal nodes, each storing
an operator from the set { +, –, *, / }, so that the value of the root is 21.
Any of these operators may be used more than once, and they are real (not
integer) operators, i.e., they may operate on and return fractions.
Problem 2: [35%] ( [GTG] Exercise C-8.57, page 355 )
Let � be a given binary tree with � nodes. The distance between two nodes
� and � in � is the number of edges along the unique simple path between
� and �, i.e., ��+ �� – 2��, where � is the lowest common ancestor (LCA)
of � and �, and �� denotes the depth of node � in �. The diameter of � is
the maximum distance between two nodes in � (i.e., the distance between
the farthest pair of nodes in �). Give an efficient algorithm for computing
the diameter of � and analyze its running time.
Problem 3: [20%] ( [GTG] Exercise C-9.38, page 398 )
Tamarindo Airlines wants to give a first-class upgrade coupon to the top log �
of their frequent flyers, based on the number of miles accumulated, where
� is the number of the airlines’ frequent flyers. The algorithm they currently
use, which runs in �(� log �) time, sorts the flyers by the number of miles
flown and then scans the sorted list to pick the top log � flyers.
They have hired you as their chief software engineer.
Give an algorithm that identifies the top log � flyers in �(�) time.
Problem 4: [30%] Dynamic Median Finder (DMF):
We want to design a DMF ADT that maintains a collection of comparable
elements and supports the following operations on the collection:
• insert(e): inserts a given element � in �(log �) time,
• getMed(): returns the median in �(1) time,
• removeMed(): removes and returns the median in �(log �) time,
where � denotes the current number of elements in the collection.
Give an implementation of the DMF ADT using two heaps as the only
instance variables.

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