Algorithm - one page

 

edge case 

mid = l+h/2 and l+ h-l/2

public class Main {

    public static void main(String[] args) {

        int lower = Integer.MAX_VALUE - 2;

        int high = Integer.MAX_VALUE;

        

        // Potential overflow scenario

        int middle1 = (lower + high) / 2;

        System.out.println("Middle index using (lower + high) / 2: " + middle1);

        // Safe calculation to avoid overflow

        int middle2 = lower + (high - lower) / 2;

        System.out.println("Middle index using lower + (high - lower) / 2: " + middle2);

    }

}

NP-complete problems are a class of computational problems that are believed to be difficult to solve efficiently. The term "NP-complete" stands for "nondeterministic polynomial-time complete." It means that if a problem is NP-complete, any other problem in the class NP (nondeterministic polynomial-time) can be reduced to it in polynomial time.

Here are some examples of well-known NP-complete problems:

1. Traveling Salesman Problem (TSP): Given a list of cities and the distances between them, find the shortest possible route that visits each city exactly once and returns to the starting city.

2. Knapsack Problem: Given a set of items, each with a weight and value, determine the most valuable combination of items that can be included in a knapsack of a given weight capacity.

3. Boolean Satisfiability Problem (SAT): Given a boolean formula in conjunctive normal form (CNF), determine if there exists an assignment of truth values to the variables that satisfies the formula.

4. Graph Coloring Problem: Given an undirected graph, assign colors to the vertices in such a way that no two adjacent vertices have the same color, using the fewest possible colors.

5. Subset Sum Problem: Given a set of integers and a target sum, determine whether there exists a subset of the integers that adds up to the target sum.

6. 3-SAT: A special case of the SAT problem where each clause in the boolean formula consists of exactly three literals combined using logical OR and logical AND operations.

How are the shortest-path and traveling-salesman problems given above similar? How are they different?

The shortest-path problem and the traveling salesman problem (TSP) are both fundamental problems in graph theory and optimization. While they share some similarities, they have distinct characteristics that set them apart. Let's examine the similarities and differences between these two problems:

Similarities:

1. Graph Representation: Both problems involve finding optimal paths in a graph. The graph can be represented as a set of vertices (nodes) connected by edges (links) with associated weights or costs.

2. Optimization Objective: The objective in both problems is to minimize a certain metric. In the shortest-path problem, the goal is to find the path with the minimum total weight or cost. In the TSP, the objective is to find the shortest possible closed tour that visits all nodes exactly once and returns to the starting node.

3. Graph Traversal: Both problems require traversing the graph to find the optimal solution. Various algorithms and techniques can be employed to explore different paths and evaluate their costs.

Differences:

1. Problem Statement: The main difference lies in the specific problem statements. The shortest-path problem focuses on finding the minimum-cost path between two specific nodes in a graph. It seeks a single path that minimizes the cost between a source node and a target node.

2. Constraints: The TSP introduces additional constraints compared to the shortest-path problem. In the TSP, the objective is to find a closed tour that visits all nodes exactly once and returns to the starting node. This requirement of visiting all nodes and returning to the starting node creates a combinatorial optimization challenge.

3. Complexity: The TSP is known to be an NP-hard problem, meaning that no efficient algorithm exists to solve it exactly for large problem sizes. On the other hand, various efficient algorithms, such as Dijkstra's algorithm or A* search, can solve the shortest-path problem optimally or approximately in polynomial time.

4. Solution Space: The number of possible solutions grows exponentially with the number of nodes in the TSP. The brute-force approach to solve the TSP requires evaluating all possible permutations, making it computationally expensive. In contrast, the shortest-path problem typically involves finding a single path, making the solution space more manageable.

In summary, both the shortest-path problem and the traveling salesman problem involve finding optimal paths in a graph, but the TSP introduces additional constraints and complexities related to visiting all nodes and returning to the starting node. The TSP is a more challenging problem in terms of computation and solution space, often requiring approximate algorithms or heuristics to find near-optimal solutions.

1.1-2 Other than speed, what other measures of efficiency might one use in a real-world setting?

Algorithms as a technology:

computers may be fast, but they are not infinitely fast. And memory may be inexpensive, but it is not free. Computing time is therefore a bounded resource, and so is space in memory. You should use these resources wisely, and algorithms that are efficient in terms of time or space will help you do so