hdu 4421 Bit Magic(2-sat,4级)

1. Introduction

In the world of computer science and programming, bit manipulation plays a crucial role in solving various problems. The ability to manipulate individual bits allows us to perform complex operations efficiently. HDU 4421: Bit Magic is a problem from the 4th level of the Hangzhou Dianzi University Online Judge platform. This problem requires a deep understanding of the 2-satisfiability (2-sat) problem and the application of bit manipulation techniques to solve it.

2. Problem Description

The problem statement for HDU 4421: Bit Magic can be summarized as follows:

You are given a boolean expression consisting of variables, logical operators (AND, OR, and NOT), and parentheses. Your task is to evaluate the expression using bit manipulation techniques. If the expression evaluates to true, you need to count the number of ways to assign values to the variables such that the expression becomes true. If the expression evaluates to false, output "0".

To solve this problem, we will first convert the given boolean expression into a 2-satisfiability (2-sat) problem. Then, we will use bit manipulation techniques to solve the 2-sat problem efficiently.

3. Conversion to 2-SAT

In order to solve the given boolean expression efficiently, we need to convert it into a 2-satisfiability (2-sat) problem. The 2-sat problem is a classical problem in computer science that aims to find assignments to boolean variables that satisfy a given set of clauses.

3.1 Variables

First, we assign a unique index to each variable in the boolean expression. Let's say we have "n" variables in total. We assign indices from 1 to n, where index i represents variable i.

3.2 Clauses

Next, we convert each clause in the boolean expression into a 2-literal clause. A 2-literal clause consists of two literals, where a literal is either a variable or its negation. For example, if we have a clause (A OR B), we convert it into two 2-literal clauses: (-A OR B) and (-B OR A).

3.3 Implication Graph

We construct an implication graph using the 2-literal clauses obtained from the previous step. Each variable and its negation form two nodes in the graph. For each clause (A OR B), we add two edges: (-A => B) and (-B => A).

4. Bit Manipulation

After converting the boolean expression into a 2-satisfiability problem and constructing the implication graph, we can use bit manipulation techniques to efficiently solve the problem.

4.1 Initializing the Variables

We create an array named "assignment" with n+1 elements, where assignment[i] stores the assignment made for variable i. Initially, we set all elements of the assignment array to 0.

4.2 Assigning Values to Variables

We start by assigning values to variables using a depth-first search (DFS) algorithm.

def dfs(u):

visited[u] = True

visited[u^1] = False

assignment[u>>1] = u & 1

for v in graph[u]:

if not visited[v]:

dfs(v)

In the above code snippet, u represents a variable or its negation, visited[u] represents whether the variable has been visited or not, and assignment[u>>1] stores the assigned value for variable u. The graph is represented as an adjacency list.

4.3 Checking for Contradictions

During the DFS traversal, if we encounter a contradiction where assignment[u] and assignment[u^1] have the same value, it means there is no valid assignment for the variables. In this case, we output "0" as the boolean expression evaluates to false.

4.4 Counting the Valid Assignments

If there are no contradictions encountered during the DFS traversal, it means there is at least one valid assignment for the boolean expression. We can count the number of valid assignments by counting the number of variables assigned a "true" value.

count = 0

for i in range(1, n+1):

if assignment[i] == 1:

count += 1

5. Conclusion

In this article, we have discussed the problem HDU 4421: Bit Magic, which requires solving a boolean expression using bit manipulation techniques. We have learned how to convert the boolean expression into a 2-satisfiability (2-sat) problem, construct an implication graph, and apply bit manipulation to solve the problem efficiently. By understanding the above concepts and techniques, we can tackle similar problems that involve bit manipulation and boolean expressions.

Remember, bit manipulation is a powerful tool that allows us to manipulate individual bits and perform complex operations efficiently. It is essential for every programmer to have a good understanding of bit manipulation techniques to solve a wide range of problems effectively.

免责声明:本文来自互联网,本站所有信息(包括但不限于文字、视频、音频、数据及图表),不保证该信息的准确性、真实性、完整性、有效性、及时性、原创性等,版权归属于原作者,如无意侵犯媒体或个人知识产权,请来电或致函告之,本站将在第一时间处理。猿码集站发布此文目的在于促进信息交流,此文观点与本站立场无关,不承担任何责任。

后端开发标签