Algorithm

Getting Started with AlgorithmWhat is an Algorithm?

Characteristics of Algorithm1 Topic

Analysis Framework

Performance Analysis3 Topics

Mathematical Analysis2 Topics

Sorting AlgorithmSorting Algorithm10 Topics

Searching Algorithm6 Topics

Fundamental of Data StructuresStacks

Queues

Graphs

Trees

Sets

Dictionaries

Divide and ConquerGeneral Method

Binary Search

Recurrence Equation for Divide and Conquer

Finding the Maximum and Minimum

Merge Sort

Quick Sort

Stassen’s Matrix Multiplication

Advantages and Disadvantages of Divide and Conquer

Decrease and ConquerInsertion Sort

Topological Sort

Greedy MethodGeneral Method

Coin Change Problem

Knapsack Problem

Job Sequencing with Deadlines

Minimum Cost Spanning Trees2 Topics

Single Source Shortest Paths1 Topic

Optimal Tree Problem1 Topic

Transform and Conquer Approach1 Topic

Dynamic ProgrammingGeneral Method with Examples

Multistage Graphs

Transitive Closure1 Topic

All Pairs Shortest Paths6 Topics

BacktrackingGeneral Method

NQueens Problem

Sum of Subsets problem

Graph Coloring

Hamiltonian Cycles

Branch and Bound2 Topics

0/1 Knapsack problem2 Topics

NPComplete and NPHard Problems1 Topic
Participants2253
Travelling Sales Person Problem
We will be able to apply the branchandbound technique to instances of the traveling salesman problem if we come up with a reasonable lower bound on tour lengths. One very simple lower bound can be obtained by finding the smallest element in the intercity distance matrix D and multiplying it by the number of cities n. But there is a less obvious and more informative lower bound for instances with symmetric matrix D, which does not require a lot of work to compute. It is not difficult to show that we can compute a lower bound on the length l of any tour as follows. For each city i, 1 ≤ i ≤ n, find the sum si of the distances from city i to the two nearest cities; compute the sum s of these n numbers, divide the result by 2, and, if all the distances are integers, round up the result to the nearest integer:
lb = [s/2] —formula (a)
For example, for the instance in Figure a, formula (a) yields
lb = [[(1 + 3) + (3 + 6) + (1 + 2) + (3 + 4) + (2 + 3)]/2] = 14.
Moreover, for any subset of tours that must include particular edges of a given graph, we can modify lower bound (a) accordingly. For example, for all the Hamiltonian circuits of the graph in Figure a that must include edge (a, d), we get the following lower bound by summing up the lengths of the two shortest edges incident with each of the vertices, with the required inclusion of edges(a, d) and (d, a):
[[(1 + 5) + (3 + 6) + (1 + 2) + (3 + 5) + (2 + 3)]/2] = 16.
We now apply the branchandbound algorithm, with the bounding function given by formula (a), to find the shortest Hamiltonian circuit for the graph in Figure a. To reduce the amount of potential work, we take advantage of two observations. First, without loss of generality, we can consider only tours that start at a. Second, because our graph is undirected, we can generate only tours in which b is visited before c. In addition, after visiting n − 1 = 4 cities, a tour has no choice but to visit the remaining unvisited city and return to the starting one. The statespace tree tracing the algorithm’s application is given in Figure b.
To reiterate the main point: these statespace tree techniques enable us to solve many large instances of difficult combinatorial problems. As a rule, however, it is virtually impossible to predict which instances will be solvable in a realistic amount of time and which will not.