Download Midterm 2 Exam Study Guide

Midterm 2 Exam Study Guide
The Midterm is on Thursday, April 4, 2002 at the regular lecture time and place.
1. Analysis of recursion
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Recurrence relations
 Finding a closed form by repeated substitution
 Proving the closed form by induction
2. Linear data structures. Arrays, lists, stacks, queues. Array and linked structure
implementations of lists, stacks, queues. Array of nodes and dynamic pointer
implementations of linked structures.
3. Trees. General and binary trees. Representations and traversals. General trees as
binary trees. Binary search trees. Applications.
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Trees
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Definition and terminology
 e.g. degree of a node, leaf, child, parent, sibling, path, length of a
path, level or depth of a node, height of a node, height of a tree,
ancestor vs. proper ancestor, descendent vs. proper descendent
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Tree traversals
 Breadth-first traversal
 Depth-first traversal
 Preorder, postorder, inorder
Expression trees
4. Algorithm design techniques. Greedy methods, priority queue search, exhaustive
search, divide and conquer, dynamic programming. Recursion. The influence of data
structure on algorithm performance.
5. Graphs and digraphs. Representions. Breadth and depth first searches. Connectivity
algorithms. Shortest path. Minimal spanning tree. The union find problem. Hamiltonian
path and travelling salesperson problems..
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Directed acyclic graphs (DAG's)
Undirected graphs
Terminology differences between undirected and directed graphs
 Edges both emanate from and are incident on vertices
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Cycle Detection using Depth-first Search
Spanning trees, and fact that DFS and BFS can be used to construct spanning
trees
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Degree = in-degree = out-degree
Vertex- and edge-weighted graphs
Connectedness and how to test for it
 Connectedness of an undirected graph
 Strong connectedness of a directed graph
 Weak connectedness of a directed graph
 Underlying undirected graph of a directed graph
Graph implementation
 Adjacency matrix
 Adjacency lists
 When matrix is better than list, when list is better than matrix
Graph traversal
 Depth-first traversal
 How algorithm works
 Running time for adjacency lists, matrices
 Breadth-first traversal
 How algorithm works
 Running time for adjacency lists, matrices
Topological order
 How algorithm works
 Running time for adjacency lists, matrices
Other applications of depth-first traversal
 Testing for cycles in a directed graph
 Testing for connectedness of an undirected graph
 Testing for strong connectedness of a directed graph
 Testing for weak connectedness of a directed graph
Given a problem, design an algorithm for solving it. For example,
o Given a graph Gand a vertex ss find all vertices at distance k from s.
o Given a graph G and two vertices s and t, find a path of length at most k
between s and t.
o Given an undirected graph G, decide whether G has a cycle.
o Given a graph G, a set of vertices S, and a vertex t
(t not in S), find the vertex in S that is closest to t.
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Given a graph G and a vertex s, find the vertex that is farthest from s.
Given two binary trees, decide whether they are the same.
Compute the time complexity of an algorithm. For example, compute the time
complexity of the above algorithms.
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Dijkstra's algorithm builds shortest paths from a given vertex s to the other
vertices in the graph. These paths are computed in increasing order of length.
Why is it more difficult to build the paths in decreasing order of length?
Given a graph show how Dijkstra's algorithm finds shortest paths.
Binary Tree ADT
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Recursive Definition
Concept that this is a position-based ADT
Compare merits and simulate operations on both array-based and
reference-based representations.
How to save and restore a binary tree to its original shape
Binary Seach Tree ADT
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Recursive Definition
Concept that this is a value-based ADT
What are the basic operations allowed (and which operations of the
binary tree are NOT allowed)?
Why did we need KeyedItem? (What possible violation of the ADT
does this solve.)
The BST Search Algorithm: be able to simulate it.
Insertion and how this uses the search algorithm.
Deletion: how it uses the search algorithm; handling the three cases.
Quantitative facts about Binary Seach Trees (height as a function of
number of values in the tree). Memorize if you must, but you are safer
being able to re-derive the formulas from diagrams (as in Figure 1030).
Balancing, and problems that arise when data is inserted in nonrandom
order
How to save and restore a binary search tree such that it becomes
balanced
Traversals
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Pre-, In-, and Post-order traversals of binary trees, including
expression trees and binary search trees, and what these produce.
Implementation of traversals using an iterator (just the basic
concept)
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Nonrecursive traversals will not be tested (though it's worth reading
to see the relationship to stacks).
Representation of General Trees as Binary Trees.
Eulerian and Hamiltonian Graphs