lec5-1.au
Find the missing integer from 0 to n using O(n)
``is bit[j] in A[i]'' queries.
Note - there are a total of 
bits, so we are not allowed to
read the entire input!
Also note, the problem is asking us to minimize the number of bits we read. We can spend as much time as we want doing other things provided we don't look at extra bits.
How can we find the last bit of the missing integer?
Ask all the n integers what their last bit is and see whether 0 or 1 is the bit which occurs less often than it is supposed to. That is the last bit of the missing integer!
How can we determine the second-to-last bit?
Ask the 
numbers which ended with the correct last bit!
By analyzing the bit patterns of the numbers from 0 to n
which end with this bit.
By recurring on the remaining candidate numbers, we get the answer in T(n) = T(n/2) + n =O(n), by the Master Theorem.
lec15-1.au Graphs
A graph G consists of a set of vertices V together with a set E of vertex pairs or edges.
Graphs are important because any binary relation is a graph, so graphs can be used to represent essentially any relationship.
Example: A network of roads, with cities as vertices and roads between cities as edges.
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Example: An electronic circuit, with junctions as vertices as components as edges.
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To understand many problems, we must think of them in terms of graphs! lec15-2.au The Friendship Graph
Consider a graph where the vertices are people, and there is an edge between two people if and only if they are friends.
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This graph is well-defined on any set of people: SUNY SB, New York, or the world.
What questions might we ask about the friendship graph?
A graph is undirected if (x,y) implies (y,x). Otherwise the graph is directed. The ``heard-of'' graph is directed since countless famous people have never heard of me! The ``had-sex-with'' graph is presumably undirected, since it requires a partner.
An edge of the form (x,x) is said to be a loop. If x is y's friend several times over, that could be modeled using multiedges, multiple edges between the same pair of vertices. A graph is said to be simple if it contains no loops and multiple edges. lec15-3.au
A path is a sequence of edges connecting two vertices. Since Mel Brooks is my father's-sister's-husband's cousin, there is a path between me and him!
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If I were trying to impress you with how tight I am with Mel Brooks, I would be much better off saying that Uncle Lenny knows him than to go into the details of how connected I am to Uncle Lenny. Thus we are often interested in the shortest path between two nodes.
A graph is connected if there is a path between any two vertices. A directed graph is strongly connected if there is a directed path between any two vertices.
The degree of a vertex is the number of edges adjacent to it. lec15-4.au
A social clique is a group of mutual friends who all hang around together. A graph theoretic clique is a complete subgraph, where each vertex pair has an edge between them. Cliques are the densest possible subgraphs. Within the friendship graph, we would expect that large cliques correspond to workplaces, neighborhoods, religious organizations, schools, and the like.
A cycle is a path where the last vertex is adjacent to the first. A cycle in which no vertex repeats (such as 1-2-3-1 verus 1-2-3-2-1) is said to be simple. The shortest cycle in the graph defines its girth, while a simple cycle which passes through each vertex is said to be a Hamiltonian cycle.
Data Structures for Graphs
There are two main data structures used to represent graphs.
Adjacency Matrices
An adjacency matrix is an 
matrix,
where M[i,j] = 0 iff there is no edge from vertex i to vertex j
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It takes 
time to test if (i,j) is in a graph represented
by an adjacency matrix.
Can we save space if (1) the graph is undirected? (2) if the graph is sparse? lec15-6.au
Adjacency Lists
An adjacency list consists of a 
array of pointers,
where the ith element points to a linked list of the
edges incident on vertex i.
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To test if edge (i,j) is in the graph, we search the ith list for
j,
which takes 
, where 
is the degree of the ith vertex.
Note that can be much less than n when the graph is sparse.
If necessary, the two copies of each edge can be linked by a pointer
to facilitate deletions.
lec15-7.au
Tradeoffs Between Adjacency Lists and Adjacency Matrices
Both representations are very useful and have different properties, although adjacency lists are probably better for most problems. lec16-2.au Traversing a Graph
One of the most fundamental graph problems is to traverse every edge and vertex in a graph. Applications include:
For efficiency, we must make sure we visit each edge at most twice.
For correctness, we must do the traversal in a systematic way so that we don't miss anything.
Since a maze is just a graph, such an algorithm must be powerful enough to enable us to get out of an arbitrary maze. lec16-3.au Marking Vertices
The idea in graph traversal is that we must mark each vertex when we first visit it, and keep track of what have not yet completely explored.
For each vertex, we can maintain two flags:
We must also maintain a structure containing all the vertices we have discovered but not yet completely explored.
Initially, only a single start vertex is considered to be discovered.
To completely explore a vertex, we look at each edge going out of it. For each edge which goes to an undiscovered vertex, we mark it discovered and add it to the list of work to do.
Note that regardless of what order we fetch the next vertex to explore, each edge is considered exactly twice, when each of its endpoints are explored. lec16-4.au Correctness of Graph Traversal
Every edge and vertex in the connected component is eventually visited.
Suppose not, ie. there exists a vertex which was unvisited whose neighbor was visited. This neighbor will eventually be explored so we would visit it:
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