Answer any FIVE questions 5 x 10 = 50. Graph traversal algorithms process all the vertices of a graph in a systematic fashion.

Transcription

1 PES Institute of Technology, Bangalore South Campus (Hosur Road, 1KM before Electronic City, Bangalore ) Solution Set Test III Subject & Code: Design and Analysis of Algorithms(10MCA44) Name of faculty: Ms. A. D. Ashwini Department: MCA IV SEM. Max Marks: 50 Answer any FIVE questions 5 x 10 = Write algorithm for DFS and BFS. Graph Traversal Graph traversal algorithms process all the vertices of a graph in a systematic fashion. They are useful for many graph problems such as checking connectivity, checking acyclicity, connected components, finding articulation points, and topological sorting. First all the vertices are marked as unvisited. Then an unvisited vertex is selected, marked as visited, and all unvisited vertices reachable from that vertex are marked as visited. Repeat above step until all vertices are visited. Graph Marking Algorithm algorithm MarkGraph(G) // Marks vertices and calls traversal algorithm // Input: Graph G = (V, E) // Output: Vertices marked in traversal order mark each vertex in V with 0 // means unvisited count 0 // count is global for each vertex v in v do if v is marked with 0 call traversal algorithm with v

2 BFS: algorithm BFS(v) // Visits unvisited vertices from v // Input: Vertex v // Output: Unvisited vertices from v are marked count count + 1; mark v with count initialize a queue with v while the queue is not empty do u remove vertex from the queue for each vertex w adjacent from u do if w is marked with 0 count count + 1 mark w with count add w to the queue DFS: algorithm DFS(v) // Recursively visits unvisited vertices from v // Input: Vertex v // Output: Unvisited vertices from v are marked count count + 1 mark v with count for each vertex u adjacent from v do if v is marked with 0 DF S(u)

3 2. a. Write briefly about Decrease and Conquer. Decrease and conquer is an approach to solving a problem by: Change an instance into one smaller instance of the problem. Solve the smaller instance. Convert the solution of the smaller instance into a solution for the larger instance. b. Write a program for Topological sorting. #include<stdio.h> #definemax20 int n,adj[max][max]; int front= 1,rear= 1,queue[MAX]; void main() int i,j=0,k; int topsort[max],indeg[max]; create_graph(); printf( The adjacency matrix is:\n ); display(); for(i=1;i<+n;i++) indeg[i]=indegree(i); if(indeg[i]==0) insert_queue(i); while(front<=rear) k=delete_queue(); topsort[j++]=k; for(i=1;i<=n;i++) if(adj[k][i]==1) adj[k][i]=0; indeg[i]=indeg[i] 1; if(indeg[i]==0) insert_queue(i);

4 printf("nodes after topological sorting are:\n"); for(i=0;i<=n;i++) printf("%d",topsort[i]); printf("\n"); create_graph() int i,max_edges,origin,destin; printf("\n Enter number of vertices:"); scamf("%d",&n); max_edges=n*(n 1); for(i=1;i<=max_edges;i++) printf("\n Enter edge %d (00 to quit):",i); scanf("%d%d",&origin,&destin); if((origin==0)&&(destin==0)) printf("invalid edge!!\n"); i ; else adj[origin][destin]=1; return; display() int i,j; for(i=0;i<=n;i++) for(j=1;jrear) printf( Queue Underflow ); return; else del_item=queue[front]; front=front+1; return del_item; int indegree(int node) int i,in_deg=0; for(i=1;i<=n;i++) if(adj[i][node]==1) in_deg++; returnin_deg;

5 3. a. Explain the below with proper examples. P, NP, and NP Complete P, NP, and NP Complete If there s an algorithm to solve a problem that runs in polynomial time, the problem is said to be in the set P If the outcome of an algorithm to solve a problem can be verified in polynomial time, the problem is said to be in the set NP (non deterministic polynomial, the non determinism refers to the outcome of the algorithm, not the verification) There is a set of problems in NP for which if there s a polynomial solution to one there will be a polynomial solution to all the set is called NP Complete b. Define the following with example: Trivial Lower bound Trivial lower bounds are based on counting the number of items that must be processed in input and generated as output to solve a problem. Some examples: Generating all permutations of a set of n elements has a trivial lower bound of Ω(n!) since all n! permutations must be generated. This lower bound is tight since we have algorithms to do this that operate in Θ(n!) 4. a. What is Backtracking approach? Write generic algorithm for backtracking. Backtracking is a refinement of the brute force approach, which systematically searches for a solution to a problem among all available options. It does so by assuming that the solutions are represented by vectors (v 1,..., v m ) of values and by traversing, in a depth first manner, the domains of the vectors until the solutions are found. When invoked, the algorithm starts with an empty vector. At each stage it extends the partial vector with a new value. Upon reaching a partial vector (v 1,..., v i ) which can t represent a partial solution, the algorithm backtracks by removing the trailing value from the vector, and then proceeds by trying to extend the vector with alternative values. ALGORITHM try(v1,...,vi) IF (v1,...,vi) is a solution THEN RETURN (v1,...,vi) FOR each v DO IF (v1,...,vi,v) is acceptable vector THEN sol = try(v1,...,vi,v) IF sol!= () THEN RETURN sol END END RETURN () If S i is the domain of v i, then S 1... S m is the solution space of the problem. The validity criteria used in checking for acceptable vectors determines what portion of that space needs to be searched, and so it also determines the resources required by the algorithm.

6 b. Define Branch and Bound. Compare Backtracking, Branch and Bound approaches. Backtracking is a problem solving method according to which one systematically searches for one or all solutions to a problem by repeatedly trying to extend an approximate solution in all possible ways. Whenever it turns out that such a solution fails, one backtracks to the last point of choice where there are still alternatives available. For most problems it is of the utmost importance to spot early on that an approximate solution can not be extended to a full solution, so that a huge amount of failing trials can be saved. This is called cutting down the search space. It may diminish the running time of the algorithm by several orders of magnitude. Now suppose that it is required to find not just any one or all solutions, but an optimal one. In this case one can apply the same method, but every time a solution is encountered the search space can be reduced even further: from then onwards one need not try to extend approximate solutions if it is sure that their extensions can not be as good as the currently optimal one. In this case we speak of Branch and Bound 5. a. Constract Hamiltonian circuit. A hamiltonian path in a graph is a path that passes through every vertex in the graph exactly once. A hamiltonain path does not necessarily pass through all the edges of the graph, however. A hamiltonian path which ends in the same place in which it began is called a hamiltonian circuit or a hamiltonain cycle. 6. Explain about Approximation algorithm for NP hard problem.

7 An approximate algorithm is a way of dealing with NP completeness for optimization problem. This technique does not guarantee the best solution. The goal of an approximation algorithm is to come as close as possible to the optimum value in a reasonable amount of time which is at most polynomial time. Suppose we have some optimization problem instance i, which has a large number of feasible solutions. Also let c(i) be the cost of solution produced by approximate algorithm and c*(i) be the cost of optimal solution. For minimization problem, we are interested in finding a solution of a given instance i in the set of feasible solutions, such that c(i)/c*(i) be as small as possible. On the other hand, for maximization problem, we are interested in finding a solution in the feasible solution set such that c*(i)/c(i) be as small as possible. We say that an approximation algorithm for the given problem instance i, has a ratio bound of p(n) if for any input of sign n, the cost c of the solution produced by the approximation algorithm is within a factor of p(n) of the cost c* of an optimal solution. That is max(c(i)/c*(i), c*(i)/c(i)) p(n) This definition applies for both minimization and maximization problems. Note that p(n) is always greater than or equal to 1. If solution produced by approximation algorithm is true optimal solution then clearly we have p(n) = 1. For a minimization problem, 0 < c*(i) c(i), and the ratio c(i)/c*(i) gives the factor by which the cost of the approximate solution is larger than the cost of an optimal solution. Similarly, for a maximization problem, 0 < c(i) c*(i), and the ratio c*(i)/c(i) gives the factor by which the cost of an optimal solution is larger than the cost of the approximate solution. Relative Error We define the relative error of the approximate algorithm for any input size as mod[c(i) c*(i)/ c*(i)] We say that an approximate algorithm has a relative bound of ε( n) if mod[c(i) c*(i)/ c*(i)] ε( n) 7. Briefly write about Horspool's algorithm with an example. The Boyer Moore algorithm uses two heuristics in order to determine the shift distance of the pattern in case of a mismatch: the bad character and the good suffix heuristics. Since the good suffix heuristics is rather complicated to implement there is a need for a simple algorithm that is based merely on the bad character heuristics. Due to an idea of Horspool [Hor 80], instead of the "bad character" that caused the mismatch, in each case the rightmost character of the current text window is used for determining the shift distance. Example: a b c a b d a a c b a b c a a b b c a a b a b c a b d a a c b a b c a a b b c a a b

8 (a) Boyer Moore (b) Horspool In this example, t 0,..., t 4 = a b c a b is the current text window that is compared with the pattern. Its suffix a b has matched, but the comparison c a causes a mismatch. The badcharacter heuristics of the Boyer Moore algorithm (a) uses the "bad" text character c to determine the shift distance. The Horspool algorithm (b) uses the rightmost character b of the current text window. The pattern can be shifted until the rightmost occurrence of b in the pattern matches the text character b, where the occurence at the last position of the pattern does not count. Like the Boyer Moore algorithm, the Horspool algorithm assumes its best case if every time in the first comparison a text symbol is found that does not occur at all in the pattern. Then the algorithm performs just O(n/m) comparisons. Preprocessing The function occ required for the bad character heuristics is computed slightly different as in the Boyer Moore algorithm. For every alphabet symbol a, the function value occ(p, a) is equal to the rightmost position of a in p 0... p m 2, or 1, if a does not occur at all. Observe that the last symbol p m 1 of the pattern is not taken into account. Example: occ(text, x) = 2 occ(text, t) = 0 occ(next, t) = 1 The occurrence function for a certain pattern p is stored in an array occ that is indexed by the alphabet symbols. For every symbol a A the entry occ[a] holds the corresponding function value occ(p, a). Given a pattern p, the following function horspoolinitocc computes the occurrence function. void horspoolinitocc() int j; char a; for (a=0; a<alphabetsize; a++) occ[a]= 1; for (j=0; j<m 1; j++) a=p[j]; occ[a]=j;

9 Searching algorithm As in the Boyer Moore algorithm, the pattern is compared from right to left with the text. After a complete match or in case of a mismatch, the pattern is shifted according to the precomputed function occ. void horspoolsearch() int i=0, j; while (i<=n m) j=m 1; while (j>=0 && p[j]==t[i+j]) j ; if (j<0) report(i); i+=m 1; i =occ[t[i]]; 8. Explain List ranking and Graph problems. List ranking Input: A linked list L of n elements. L is given in an array S such that the entry S(i) contains the index of the node which is the successor of the node i in L. Output: The distance of each node i from the end of the list. Sequential list ranking algorithm List ranking can be solved in O(n) time sequentially for a list of length n. A simple parallel list ranking algorithm Output: For each 1 i n, the distance R(i) of node i from the end of the list. begin for 1 i n do in parallel if S(i) 0 then R(i) := 1 else R(i) := 0 endfor while S(i) 0 and S(S(i)) 0 do Set R(i) := R(i) + R(S(i)) Set S(i) := S(S(i)) end