CS502 - Fundamentals of Algorithms [email protected]
CS502 Solved Reference Subjective File by Faisal Dar (MIT)
1. Give Detail Example of 2-d maxima Problem. (Pg17)
Answer:
Let a point p in 2-dimensional space be given by its integer coordinates, p = (p.x, p.y). A point p
is said to dominated by point q if p.x ≤ q.x and p.y ≤ q.y. Given a set of n points, P = {p1, p2. . .
pn} in 2-space a point is said to be maximal if it is not dominated by any other point in P. The
problem is to output all the maximal points of P. We introduced a brute-force algorithm that ran
in Θ (n2) time.
2. Where Arise Clique Cover? (pg176)
Answer:
The clique cover problem arises in applications of clustering. We put an edge between two nodes
if they are similar enough to be clustered in the same group. We want to know whether it is
possible to cluster all the vertices into k groups.
3. Explain directed and undirected graphs.
A graph is a mathematical structure that is made up of set of vertices and edges. A graph
represents a set of objects (represented by vertices) that are connected through some links
(represented by edges). Using mathematical notations, a graph can be represented by G, where
G= (V, E) and V is the set of vertices and E is the set of edges.
In an undirected graph there is no direction associated with the edges that connect the vertices.
In a directed graph there is a direction associated with the edges that connect the vertices.
Explain the Floyd-Warshall Algorithm (Running time and Space used) (pg161)
Answer:
It can be explained by these steps:
• Let G = (V, E) be a directed graph with edge weights.
• If (u, v) ∈ E is an edge then w (u, v) denotes its weight.
• δ (u, v) is the distance of the minimum cost path between u and v.
• We will allow G to have negative edges weights but will not allow G to have negative cost
cycles.
• We will present a Θ (n3) algorithm for the all pairs shortest path.
• The algorithm is called the Floyd-Warshall algorithm and is based on dynamic
programming.
• Clearly, the running time is Θ (n3).
• The space used by the algorithm is Θ (n2).
4. What is heap and heap order? 2marks
Answer:
• A heap is a left-complete binary tree that conforms to the heap order.
• The heap order property: in a (min) heap, the parent node has key smaller than or equal
to both of its children nodes.
• Similarly, in a max heap, the parents have a key larger than or equal both of its children.
CS502 - Fundamentals of Algorithms [email protected]
5. How do we covert the shortest distance problem in to a single source
problem? 2marks
Answer:
We can solve the shortest distance algorithm by using Dijkstra's algorithm in
Edge-weighted digraphs with nonnegative weights
Using extra space proportional to V and
Time proportional to E logs V (in the worst case).
6. How short path information propagate in graph using bell ford algorithm?
2marks
Answer:
The shortest path information is propagated sequentially along each shortest path in the graph.
Bellman-Ford allows negative weights edges and no negative cost cycles.
7. An arbitrary graph with G (V, E) with E = |V|- I is a tree. True or false tell
briefly. 2marks
Answer:
This statement is False. |V| is total number of vertices of a graph and |V|- I are the vertices
which are never revisited, if for every pair of vertices u, v ∈ Vc ( Vc means V-1) then graph
induced by Vc is a complete sub graph. Hint: in clique algorithm
8. Express arithmetic series in summation notation and tell big theta case.
3marks pg15
Answer:
Big Theta Case
9. How Dijkstra’s algorithm work 3marks
Answer:
10. Two Cases of Floyed Warshall algorithm 3marks
Answer:
Don’t go through k at all
The length of the shortest is d (k−1) ij
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Do go through k
The length of the path is d (k−1) ik+ d (k−1) kJ
11. Prove that Topological sort is directed acyclic graph 5marks Pg: 133
Answer:
In this figure we run DFS on DAG where we shows the linear order () obtained by the
topological sort of the sequence of putting on a suit. The DAG is still the same. As a result, all
directed edges go from left to right.
12. Calculate time complexity of 5marks
1 2 2
3 3 3 ……. ………. n n n n ……. n (n times)
Answer: http://answers.yahoo.com/question/index?qid=1006011700253
CS502 - Fundamentals of Algorithms
155
13. Write Pseudo code to relax a vertex for Dijkstra’s algorithm.
14. What is Heap sort Algorithm?
Answer:
A heap is a left-complete binary tree that conforms to the heap order.
CS502 - Fundamentals of Algorithms [email protected]
15. Difference between Back edge and cross edge
Answer:
16. Prim's Algorithm
Answer:
17. Explain Do go through k & don’t go through k at all?
Answer:
Don’t go through k at all
The length of the shortest is d (k−1) ij
Do go through k
The length of the path is d (k−1) ik+ d (k−1) kJ
18. Explain heapify?
Answer:
19. Prove that the generic TRAVERSE (S) marks every vertex in any connected
graph exactly once and the set of edges (v, parent (v)) with parent (v) ¹F
form a spanning tree of the graph. Pg125
Answer:
20. Apply Kruskal’s algorithms on the following graph. [You can
show final result in exam software and need not to show all intermediate
steps].
CS502 - Fundamentals of Algorithms
Answer:
Page no 147 to 149
21. Polynomial time algorithm… 169pg
Answer:
22.What are a run time analysis and its two criteria? pg13
Answer:
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23.Dijkstra algorithm correctness criteria two conditions. Pg158
Answer:
24.Floyd algorithm. Pg161
Answer:
25. A table is given and we have to make the adjacency list and matrix. page 116
Answer:
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26.An MST graph is given and we have to calculate its weight. Pg142
Answer:
27. Q1) answer the following according to Floyd Warshall 1) runing time 2)
space use
28.Write suedo code of dijkstra algorithm? 5marks. Pg156
Answer:
CS502 - Fundamentals of Algorithms
29.Make Adjacency list from the given table. 5marks
Answer:
Vertices/edges 1 2 3
1 1 1 1
2 0 0 1
3 0 1 0
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30. How generic algorithms work with minimum spanning tree 3marks
Answer:
31. Floyd-Warshall matrix diya tha os py 3 itration apply karny thy 5marks
Answer:
Page no 166
32.Which points should be supposed to prove the correctness of the Dijkstra's
Algorithm 3marks
Answer:
33. BFS pseudo code in graph (Page 12)
Answer:
CS502 - Fundamentals of Algorithms [email protected]
34. Given a graph run dijikstr's algorithm to find shortest path from vertex "S"
to all other vertices (5 marks) Page158
Answer:
Given a graph run prim's algorithm to find minimum spanning tree. Only show
the order of edges to be added to make spanning tree (5 marks)
Answer:
CS502 - Fundamentals of Algorithms
3- Given a graph run DFS and label time stamping (5 marks) pg149
Answer:
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35. What are the minimum and maximum numbers of elements in a heap of
height h (5 marks) Page no 44
Answer:
Minimum numbers=
Maximum numbers=
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36.Explain Floyd Warshell algorithm (3 marks)
Answer:
37. Define forward and back edges
Answer:
38.Compare bellman ford algorithm with dijikstr's algorithm. Also give the
time complexity of bellman ford algorithm (3 marks)
Answer:
A cycle in a digraph is a path containing at least one edge and for which v0 = vk.v0 is starting
vertices and vk is last vertices when both vertices are connected through edge then we can
consider it as a cycle. A Hamiltonian cycle is a cycle that visits every vertex in a graph exactly
once. An Eulerian cycle is a cycle that visits every edge of the graph exactly once.
Note: All the graphs are given from handouts but slightly different order of vertices and weights.
CS502 - Fundamentals of Algorithms [email protected]
Note: About 10 questions are from huffman encoding algorithm each of 1 mark
and much simple.
About 5 MCQ, s are from last chapter (Complexity theory) Other MCQ,s are also
from graph theory.
So 90% of the paper is from graph theory so prepare the chapter 8 (Graph theory)
of handouts with a great interest and specially all the graph algorithms (BFS, DFS,
Prim, Dijisktr's, Kruskal, Bellman ford, Floyd warshell etc)
39.RAM (Random Access memory) and its Applications? Pg10
Answer:
are executed one-by-one
Instructions
40. Describe Dijkstra’s algorithm working?
Answer:
41. Prim algorithm graph?
CS502 - Fundamentals of Algorithms
Answer:
CS502 - Fundamentals of Algorithms [email protected]
42.Floyd-Warshall matrix complexity?
Answer:
IN floyd warshall we didnt allow G to have negative cost cycles.
43.Convert shortest path in to single source shortest path problem?
Answer:
44. Different between Average case and Worst case? Answer:
http://wiki.answers.com/Q/What_is_the_difference_between_best_worst_and_average_case
_complexity_of_an_algorithm
The worst case scenario, on the other hand, describes the absolute worst set of input for a given
algorithm. Let's look at a quick sort, which can perform terribly if you always choose the smallest or
largest element of a sub list for the pivot value. This will cause quick sort to
degenerate to O(n2).
Discounting the best and worst cases, we usually want to look at the average performance of an
algorithm. These are the cases for which the algorithm performs "normally."
45.Pseudo code algorithm for DFS Timestamp?
Answer:
CS502 - Fundamentals of Algorithms [email protected]
46.What is common problem in communication networks and circuit
designing? 2marks
Answer:
A common problem is communications networks and circuit design is that of connecting
together a set of nodes by a network of total minimum length. The length is the sum of lengths of
connecting wires.
47. Write pseudo code of relaxing a vertex 5
Answer:
Repeat
48. Define NP completeness (5marks) pg178
Answer:
NP-Complete L is NP-complete if
1. L ∈ NP and
2. L is NP-hard.
49.Define floyed warshall algorithm in these two cases (5marks)
Answer:
There are two types of cases in floyed warshall algorithm:
• do not go through a vertex k at all
• do go through a vertex k
Detail of these is
below:
50.Define DAG (marks3)
Answer:
A graph is said to be acyclic if it contains no cycles. A graph is connected if every vertex can
reach every other vertex. A directed graph that is acyclic is called a directed acyclic graph
(DAG)
51. Write steps of sieve techniques
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Answer:
52. Write Pseudo code of Dijkstra's algorithm
Answer:
REF PAGE :156
53. Prove the Lemma: Consider a diagraph G = ( V,E ) and any DFS forest for G.
G has a cycle if and only if the DFS forest has a back edges
Answer:
CS502 - Fundamentals of Algorithms [email protected]
54.Where the clique cover problem is used?
Answer:
55. What is decision problem, also explain with examples?
Answer:
A problem is called a decision problem if its output is a simple “yes” or “no” (or you may this of
this as
True/false, 0/1, accept/reject.) We will phrase an optimization problems as decision problems.
For example, the MST decision problem would be: Given a weighted graph G and an integer k,
does G have a spanning tree whose weight is at most k.
56.Let the adjacency list representation of an undirected graph is given.
Explain what property of the list indicates that the graph has an isolated
vertex.
As we know that An isolated vertex is a vertex with degree zero or a vertex that is not an
endpoint of any edge so in the given adjacency list the vertex is 0 and no end point of any edge
57. Analyze the brute force maxima algorithm
CS502 - Fundamentals of Algorithms [email protected]
Answer:
Page no 16-17
58.How Kruskal's algorithm works?
Answer:
Kruskal‟s algorithm worked by ordering the edges, and inserting them one by one into the
spanning tree, taking care never to introduce a cycle. Intuitively Kruskal‟s works by merging or
splicing two trees together, until all the vertices are in the same tree.4).
59.Given a matrix of graph G. IS G a directed or undirected graph?
Answer:
It is an Undirected Graph. In an undirected graph, we say that an edge is incident on a vertex if
the vertex is an endpoint of the edge.
60. What is the common problem in communication network and
circuit designing?
Answer:
Repeat
61. Differentiate between back edge and forward edge suppose you could prove
that an NP-complete problem cannot be solved in polynomial time. What would
be the consequence?
Answer:
Back Edge
A back edge connects a vertex to an ancestor in a DFS-tree. Note that a self-loop is a back edge.
from descendent to ancestor
(u, v) where v is an ancestor of u in the tree.
DFS tree may only have a single back edge A
back edge is an arc whose head dominates its tail (tail -> head) a
back edge must be a part of at least one loop
CS502 - Fundamentals of Algorithms [email protected]
Forward Edge
From ancestor to descendent
(u, v) where v is a proper descendent of u in the tree.
A forward edge is a non-tree edge that connects a vertex to a descendent in a DFS-tree.
Edge x-y is less than the capacity there is a forward edge x-y with a capacity equal to the
capacity and the flow
According to the question this means that the problem can be solved in Polynomial time using
known NP problem can be solved using the given problem with modified input (an NP problem
can be reduced to the given problem) then the problem is NP complete.
The main thing to take away from an NP-complete problem is that it cannot be solved in
polynomial time in any known way. NP-Hard/NP-Complete are a way of showing that certain
classes of problems are not solvable in realistic time.
Ref: Handouts Page No.128
62.Recursive explanation of dynamic programming.
Formulate the problem into smaller sub problems, find optimal solution to these sub
problems in a bottom up fashion then write an algorithm to find the solution of whole
problem starting with base case and works its way up to final solution.
Answer:
Dynamic programming is essentially recursion without repetition. Developing a dynamic
programming algorithm generally involves two separate steps:
Formulate problem recursively. Write down a formula for the whole problem as a simple
combination of answers to smaller sub problems.
Build solution to recurrence from bottom up. Write an algorithm that starts with base cases
and works its way up to the final solution.
Ref: Handouts Page No.75
63. What is the cost of the following graph?
Cost =33
Answer:
A common problem is communications networks and circuit design is that of connecting
together a set of nodes by a network of total minimum length. The length is the sum of lengths
of connecting wires.
Consider, for example, laying cable in a city for cable TV.
The computational problem is called the minimum spanning tree (MST) problem. Formally, we
are given a connected, undirected graph G = (V, E) each edge (u, v) has numeric weight of cost.
CS502 - Fundamentals of Algorithms
We define the cost of a spanning tree T to be the sum of the costs of edges in the spanning tree
w (T) = Σ w (u, v)
(u,v)2T
64. A minimum spanning tree is a tree of minimum weight The first is a
spanning tree but is not a MST;
Ref: Handouts Page No.142
65.Let the adjacency list representation of an undirected graph is given below.
Explain what general property of the list indicates that the graph has an
isolated vertex.
e◊ c ◊ b ◊a
d◊ a ◊b
f◊ e ◊ d ◊ a ◊c
f◊ c ◊ b ◊d
f◊ c ◊ a ◊e
e◊ d ◊ c ◊f
g
Answer:-
The main theorem which drives both algorithms is the following:
MST Lemma: Let G = (V, E) be a connected, undirected graph with real-valued weights on the
edges. Let A be a viable subset of E (i.e., a subset of some MST). Let (S, V − S) be any cut that
respects A and let (u, v) be a light edge crossing the cut. Then the edge (u, v) is safe for A.
MST Proof: It would simplify the proof if we assume that all edge weights are distinct. Let T be
any MST for G. If T contains (u, v) then we are done. This is shown in Figure 8.47 where the
lightest edge (u, v) with cost 4 has been chosen.
Ref: Handouts Page No.144
Floyd Warshal algorithm with three recursive steps
1. Wij = 0 if I = j
2. Wij = w(i,j) if i != j and (i,j) belongs to E
3. Wij = infinity if i != j and (i,j) not belongs to E
Answer
CS502 - Fundamentals of Algorithms [email protected]
The Floyd-Warshall algorithm: Step (i)
The Floyd-Warshall algorithm: Step (ii)
Recursively define the value of an optimal solution.
Boundary conditions: for k = 0, a path from vertex i to j with no intermediate vertex numbered
higher than 0 has no intermediate vertices at all, hence d (0)= wij
Recursive formulation:
is the solution for this APSP problem:
The Floyd-Warshall algorithm: Step (iii)
66. Compute the shortest-path weights bottom up FLOYD-WARSHALL(W, n)
Ref: Handouts Page No.161
Give a detailed example for 2d maxima problem
Answer: -
The problem with the brute-force algorithm is that it uses no intelligence in pruning out
decisions. For example, once we know that a point pi is dominated by another point pj, we do
not need to use pi for eliminating other points. This follows from the fact that dominance
relation is transitive. If pj dominates pi and pi dominates ph then pj also dominates ph; pi is not
needed. Ref: Handouts Page No.17
67. How the generic greedy algorithm operates in minimum spanning tree?
Answer: -
A generic greedy algorithm operates by repeatedly adding any safe edge to the current spanning
tree. When is an edge safe? Consider the theoretical issues behind determining whether an edge
is safe or not. Let S be a subset of vertices S _ V. A cut (S, V − S) is just a partition of vertices
into two disjoint subsets. An edge (u, v) crosses the cut if one endpoint is in S and the other is in
V − S. Given a subset of edges A, a cut respects A if no edge in A crosses the cut. It is not hard to
see why respecting cuts are important to this problem. If we have computed a partial MST and
we wish to know which edges can be added that do not induce a cycle in the current MST, any
edge that crosses a respecting cut is a possible candidate.
Ref: Handouts Page No.143
68. What are two cases for computing
Answer:-
There are two cases for computing Lij the match case if ai = bj , and the mismatch case if ai 6= bj
. In the match case, we do the following:
and in the mismatch case, we do the following:
CS502 - Fundamentals of Algorithms [email protected]
Ref: Handouts Page No.143
69.Describe Minimum Spanning Trees Problem with examples. Problem
Given a connected weighted undirected graph, design an algorithm that outputs a
minimum spanning tree (MST) of
Examples
The graph is a complete, undirected graph G = (V, E, W), where V is the set of pins, E is the set
of all possible interconnections between the pairs of pins and w (e) is the length of the wire
needed to connect the pair of vertices Ref: Handouts Page No.142
70.What is decision problem, also explain with example?
A decision problem is a question in some formal system A problem is called a decision problem
if its output is a simple “yes” or “no” (or you may this of this as
true/false, 0/1, accept/reject.) We will phrase may optimization problems as decision problems.
For example,
the MST decision problem would be: Given a weighted graph G and an integer k, does G have a
spanning tree whose weight is at most k?
Ref: Handouts Page No.170
71. Do you think greedy algorithm gives an optimal solution to the activity
scheduling
problem?
The greedy algorithm gives an optimal solution to the activity scheduling problem.
Proof:
The proof is by induction on the number of activities. For the basis case, if there are no activities,
then the greedy algorithm is trivially optimal. For the induction step, let us assume that the
greedy algorithm is optimal on any set of activities of size strictly smaller than |S| and we prove
the result for S. Let S0 be the set of activities that do not interfere with activity a1, That is any
solution for S0 can be made into a solution for S by simply adding activity a1, and vice versa.
Activity a1 is in the optimal schedule (by the above previous claim). It follows that to produce an
optimal schedule for the overall problem; we should first schedule a1 and then append the
optimal schedule for S0. But by induction (since |S0| < |S|), this is exactly what the greedy
CS502 - Fundamentals of Algorithms [email protected]
algorithm does. Ref: Handouts Page No.109
72. Define Forward edge
The most natural result of a depth first search of a graph (if it is considered as a function rather
than procedure) is a spanning tree of the vertices reached during the search. Based on this
spanning tree, the edges of the original graph can be divided into three classes: forward edges
(or "discovery edges"), which point from a node of the tree to one of its descendants,
Ref: Handouts Page No.129 130 73. Is there any relationship between number of back edges and number of
cycles in DFS?
Answer:
The time stamps given by DFS allow us to determine a number of things about a graph or
digraph. We can determine whether the graph contains any cycles.
Lemma: Given a digraph G = (V, E), consider any DFS forest of G and consider any edge (u, v) 2
E. If this edge is a tree, forward or cross edge, then f[u] > f[v]. If this edge is a back edge, then
f[u] _ f[v].
Proof: For the non-tree forward and back edges the proof follows directly from the parenthesis
lemma. For example, for a forward edge (u, v), v is a descendent of u and so v‟s start-finish
interval is contained within u‟s implying that v has an earlier finish time. For a cross edge (u, v)
we know that the two time intervals are disjoint. When we were processing u, v was not white
(otherwise (u, v) would be a tree edge), implying that v was started before u. Because the
intervals are disjoint, v must have also finished before u.
Ref: Handouts Page No.130
74. What is the common problem in communications networks and circuit
designing?
Answer:
A common problem is communications networks and circuit design is that of connecting
together a set of nodes by a network of total minimum length. The length is the sum of lengths of
connecting wires.
Consider, for example, laying cable in a city for cable TV.
Ref: Handouts Page No.142
Explain the following two basic cases according to Floyd-Warshall Algorithm,
1. Don t go through vertex k at all.
2. Do go through vertex k.
Answer:-
Don‟t go through k at all
Then the shortest path from i to j uses only intermediate vertices {1, 2, . . . , k − 1}. Hence the
length of
the shortest is d(k−1)
CS502 - Fundamentals of Algorithms
ij
Do go through k
First observe that a shortest path does not go through the same vertex twice, so we can assume
that we
pass through k exactly once. That is, we go from i to k and then from k to j. In order for the
overall path to be as short as possible, we should take the shortest path from i to k and the
shortest path from k to j.
Since each of these paths uses intermediate vertices {1, 2, . . . , k − 1}, the length of the path is
Ref: Handouts Page No.162
Show the result of time stamped DFS algorithm on the following graph. Take node
A as a starting node.
Answer:-
Depth-first search (DFS) :
It is an algorithm for traversing or searching a tree, tree structure, or graph. Intuitively, one
starts at the root (selecting some node as the root in the graph case) and explores as far as
possible along each branch before backtracking.
Formally, DFS is an uninformed search that progresses by expanding the first child node of the
search tree that appears and thus going deeper and deeper until a goal node is found, or until it
hits a node that has no children. Then the search backtracks, returning to the most recent node
it hadn't finished exploring. In a non-recursive implementation, all freshly expanded nodes are
added to a LIFO stack for exploration.
DFS (graph)
{
list L =empty
tree T =empty
choose a starting vertex x
search(x)
while(Lis not empty)
{
remove edge (v, w) from beginning of L
if w not yetvisited
{
add (v, w) to T
search(w)
}
}
}
search(vertex v)
{
visit v
for each edge (v, w)
add edge (v, w) to the beginning of L
}
CS502 - Fundamentals of Algorithms
-----------------------------------------------------------------------------------------------
Ref: http://www.chegg.com/homework-help/quest...omp-q96642
Why we need reduction?
Answer:-
The class NP-complete (NPC) problems consists of a set of decision problems (a subset of class
NP) that no one knows how to solve efficiently. But if there were a polynomial solution for even
a single NP-complete problem, then ever problem in NPC will be solvable in polynomial time.
For this, we need the concept of reductions.
Ref: Handouts Page No.173 Consider the digraph on eight nodes, labeled 1 through 8, with eleven directed
edges
1 2, 1 4, 2 4, 3 2, 4 5, 5 3 ,5 6, 7 8, 7 1, 2 7,8 7
Answer:-
0-7 0-1 1-4 1-6 2-3 3-4 4-2 5-2 6-0 6-3 6-5 7-1 7-3
Draw the DFS tree for the standard adjacency-matrix representation. List the
edges of each type in the space provided below.
0-7 tree 0
0-1 tree / \
1-4 tree 1 7
1-6 tree / \
2-3 tree 4 6
3-4 back | |
4-2 tree 2 5
5-2 cross |
6-0 back 3
6-3 cross
6-5 tree
7-1 cross
7-3 cross
Answer:
Ref http://www.cs.princeton.edu/courses/arch...1-sol.html Prove that the generic TRAVERSE (S) marks every vertex in any connected graph
exactly once and the set of edges (v, parent (v)) with parent (v) ¹F form a spanning
tree of the graph
CS502 - Fundamentals of Algorithms
Answer:-
Generic Traverse
• Suppose we want to visit every node in a connected graph (represented either explicitly or
implicitly)
• The simplest way to do this is an algorithm called depth-first search
• We can write this algorithm recursively or iteratively - it‟s the same both ways, the iterative
version just makes the stack explicit
• Both versions of the algorithm are initially passed a source vertex v
Lemma
Traverse(s) marks each vertex in a connected graph exactly once, and the set of edges (v,
parent(v)), with parent(v) not nil, form a spanning tree of the graph. Proof
• Call an edge (v, parent(v)) with parent(v) =6 nil a parent edge.
• Note that since every node is marked, every node has a parent edge except for s.
• It now remains to be shown that the parent edges form a spanning tree of the graph
Ref http://www.cs.unm.edu/~saia/362-s08/lec/lec18-2x2.pdf
What is a run time analysis and its two criteria
Answer:-
The main purpose of our mathematical analysis will be measuring the execution time. The
running time of an implementation of the algorithm would depend upon the speed of the
computer, programming language, optimization by the compiler etc. Two criteria for measuring
running time are worst-case time and average-case time.
Worst-case time is the maximum running time over all (legal) inputs of size n. Let I denote an
input instance, let |I| denote its length, and let T(I) denote the running time of the algorithm on
input I. Then
Average-case time is the average running time over all inputs of size n. Let p(I) denote the
probability of seeing this input. The average-case time is the weighted sum of running times
with weights being the probabilities:
We will almost always work with worst-case time. Average-case time is more difficult to
compute; it is difficult to specify probability distribution on inputs. Worst-case time will specify
an upper limit on the running time.
Ref: Handouts Page No.173
Dijkstra algorithm correctness criteria two conditions
Answer:
We will prove the correctness of Dijkstr‟s algorithm by Induction. We will use the definition that
_(s, v)
denotes the minimal distance from s to v.
For the base case
CS502 - Fundamentals of Algorithms
1. S = {s}
2. d(s) = 0, which is _(s, s
Ref: Handouts Page No.158 Compare bellman ford algorithm with dijikstr's algorithm. Also give the time
complexity of bellman ford algorithm (3 marks)
Answer:
Dijkstra‟s single-source shortest path algorithm works if all edges weights are non-negative and
there are no negative cost cycles. Bellman-Ford allows negative weights edges and no negative
cost cycles. The algorithm is slower than Dijkstra‟s, running in _(VE) time. Like Dijkstra‟s
algorithm, Bellman-Ford is based on performing repeated relaxations. Bellman-Ford applies
relaxation to every edge of the graph and repeats this V − 1 times.
Ref: Handouts Page No.159
psedo code of timestamp DFS
Answer:
DFS(G)
1 for (each u 2 V)
2 do color[u] white
3 pred[u] nil
4 time 0
5 for each u 2 V
6 do if (color[u] = white)
7 then DFSVISIT(u)
Ref: Handouts Page No.126
variants of shortest path solution
Answer
There are a few variants of the shortest path problem.
Single-source shortest-path problem: Find shortest paths from a given (single) source vertex s 2
V to every other vertex v 2 V in the graph G.
Single-destination shortest-paths problem: Find a shortest path to a given destination vertex t
from each vertex v. We can reduce the this problem to a single-source problem by reversing the
direction of each edge in the graph.
Single-pair shortest-path problem: Find a shortest path from u to v for given vertices u and v. If
we solve the single-source problem with source vertex u, we solve this problem also. No
algorithms for this problem are known to run asymptotically faster than the best single-source
algorithms in the worst case.
All-pairs shortest-paths problem: Find a shortest path from u to v for every pair of vertices u and
v. Although this problem can be solved by running a single-source algorithm once from each
vertex, it can usually be solved faster.
Ref: Handouts Page No.153
CS502 - Fundamentals of Algorithms
Prove the following lemma,
Lemma: Given a digraph G = (V, E), consider any DFS forest of G and consider any
edge (u, v) ∈ E. If this edge is a tree, forward or cross edge, then f[u] > f[v]. If this
edge is a back edge, then f[u] ≤ f[v]
Answer:-
Proof: For the non-tree forward and back edges the proof follows directly from the parenthesis
lemma. For example, for a forward edge (u, v), v is a descendent of u and so v‟s start-finish
interval is contained within u‟s implying that v has an earlier finish time. For a cross edge (u, v)
we know that the two time intervals are disjoint. When we were processing u, v was not white
(otherwise (u, v) would be a tree edge), implying that v was started before u. Because the
intervals are disjoint, v must have also finished before u.
RAM(Random Access memory)and its Applications?
Answer:
A RAM is an idealized machine with an infinitely large random-access memory. Instructions are
executed one-by-one (there is no parallelism). Each instruction involves performing some basic
operation on two values in the machines memory. Basic operations include things like assigning
a value to a variable, computing any basic arithmetic operation (+, - , × , integer division) on
integer values of any size, performing any comparison (e.g. x _
Ref: Handouts Page No.10
Describe Dijkstra’s algorithm working?
Answer
Dijkstra‟s algorithm works on a weighted directed graph G = (V, E) in which all edge weights are
non-negative, i.e., w(u, v) _ 0 for each edge (u, v) 2 E.
Ref: Handouts Page No.154
Prim algorithm graph?
Answer
Prim‟s algorithm builds the MST by adding leaves one at a time to the current tree. We start with
a root vertex r; it can be any vertex. At any time, the subset of edges A forms a single tree. We
look to add a single vertex as a leaf to the tree.
Ref: Handouts Page No.149
write suedo code of relaxing a vertex 5
Answer
RELAX((u, v))
1 if (d[u] + w(u, v) < d[v])
2 then d[v] d[u] + w(u, v)
3 pred[v] = u
Ref: Handouts Page No.155
Define NP completeness
Answer:
The set of NP-complete problems is all problems in the complexity class NP for which it is
known that if anyone is solvable in polynomial time, then they all are. Conversely, if anyone is
not solvable in polynomial time, then none are.
CS502 - Fundamentals of Algorithms
Definition: A decision problem L is NP-Hard if
L0 _P L for all L0 2 NP.
Definition: L is NP-complete if
1. L 2 NP and
2. L0 _P L for some known NP-complete problem L0.
Ref: Handouts Page No.176
Define DAG
A directed graph that is acyclic is called a directed acyclic graph (DAG).
Ref: Handouts Page No.116 How Kruskal's algorithm works ?
Answer:
Kruskal‟s algorithm works by adding edges in increasing order of weight (lightest edge first). If
the next edge does not induce a cycle among the current set of edges, then it is added to A. If it
does, we skip it and consider the next in order. As the algorithm runs, the edges in A induce a
forest on the vertices. The trees of this forest are eventually merged until a single tree forms
containing all vertices.
Ref: Handouts Page No.147
What are two steps generally involved while developing a dynamic programming
algorithm?
Answer:
Dynamic programming is essentially recursion without repetition. Developing a dynamic
programming algorithm generally involves two separate steps:
Formulate problem recursively. Write down a formula for the whole problem as a simple
combination of answers to smaller sub problems.
Build solution to recurrence from bottom up. Write an algorithm that starts with base cases and
works its way up to the final solution.
Ref: Handouts Page No.75
The following adjacency matrix represents a graph that consists of four vertices
labeled 0, 1, 2 and 3. The entries in the matrix indicate edge weights.
0 1 2 3
0 0 1 0 3
1 2 0 4 0
2 0 1 0 1
3 2 0 0 0
Answer
NO. Because number of rows and number of columns are always same.
What is the application of edit distance technique?
Answer
Edit Distance: Applications
There are numerous applications of the Edit Distance algorithm. Here are some
examples:
CS502 - Fundamentals of Algorithms [email protected]
Spelling Correction
If a text contains a word that is not in the dictionary, a „close‟ word, i.e. one with a small edit
distance, may be suggested as a correction. Most word processing applications, such as
Microsoft Word, have spelling checking and correction facility. When Word, for example, finds
an incorrectly spelled word, it makes suggestions of possible replacements.
Plagiarism Detection
If someone copies, say, a C program and makes a few changes here and there, for example,
change variable names, add a comment of two, the edit distance between the source and copy
may be small. The edit distance provides an indication of similarity that might be too close in
some situations.
Computational Molecular Biology
Speech Recognition
Algorithms similar to those for the edit-distance problem are used in some speech recognition
systems. Find a close match between a new utterance and one in a library of classified
utterances.
Ref: Handouts Page No.76
Fibonacci sequence? 2mark
Answer:
Ref : Page no 73
Communication design problem (MST). Answer:
Ref: Page no 142 Strong connected component problem Answer:
Ref: Page no 135