《算法》(1)--动态连接与算法分析
来源:互联网 发布:dean fujioka 知乎 编辑:程序博客网 时间:2024/06/05 14:21
First Chapter of <Algorithms>(a)--Dynamic Connectivity and Algorithm Analysis
Basic Statement
This poster of algorithms is based on the book and the lecture of algorithms on coursera which is taught by Robert Sedgewick and Kevin Wayne, writers of the book mentioned above.
First chapter of “Algorithm” is the fundamental of java programming language and a brief introduction to algorithms and data structure, including their concepts. some interesting problems and the basic implementations of them.
Introduction
Why we study algorithms
- Their impact is broad and far-reaching.
- Old roots, new opportunities.
- To solve problems that could not otherwise be addressed.
- For intellectual stimulation.
- To become a proficient programmer.
- They may unlock the secrets of life and of the universe.
- For fun and profit.
After summarizing some reason why we study algorithms, we may draw a conclusion that algorithms are widely used and we can get much fun from it.
Steps to developing a usable algorithm
- Model the problem
- Find an algorithm to solve it
- Fast enough? Fits in memory?
- If not, figure out why
- Find a way to address the problem
- Iterate until satisfied
Developing a usable algorithm is the same as mathematical modeling which using iteration until the performance is satisfied.
Scientific method:
Develop hypotheses about performance, create mathematical models. and run experiments to test them, repeating the process as necessary.
Algorithm
The term algorithm is used in computer science to describe a finite, deterministic, and effective problem-solving method suitable for implementation as a computer program.
Dynamic Connectivity-Union Find
Form of Union Find problem
Given a set of N objects.
- Union command: connect two objects
- Find/connected query: is there a path connecting the two objects?
QuickFind
Implementation of Quick-Find
QuickFind.java
/** * Created by WilliamYi * 3/11/2017 * the direct implementation of Union-Find * the connected two points have the same index */package quickfind;public class QuickFind { private int[] id; //set id of each object to itself public void QuickFindUF (int N) { id = new int[N]; for(int i = 0; i < N; i++) id[i] = i; } //check whether p and q are connected public boolean connected(int p, int q) { return id[p] == id[q]; } //change all entries with id[p] to id[q] public void union(int p, int q) { int pid = id[p]; int qid = id[q]; for(int i = 0; i < id.length; i++) { if(id[i] == pid) id[i] = qid; } }}
QuickFindTest.java
package quickfind;public class QuickFindTest { public static void main(String [] args) { QuickFind qf = new QuickFind(); qf.QuickFindUF(6); qf.union(0, 3); qf.union(1, 4); qf.union(4, 5); qf.union(2, 5); System.out.println(qf.connected(1, 3)); System.out.println(qf.connected(1, 2)); }}
QuickUnion
Implementation of QuickUnion
QuickUnion.java
/** * Created by WilliamYi * 3/11/2017 * the quick union implementation of Union-Find */package quickunion;public class QuickUnion { private int[] id; public void QuickUnionUF(int N) { id = new int[N]; for(int i = 0; i < N; i++) id[i] = i; } public int root(int i) { while(i != id[i]) i = id[i]; return i; } public boolean connected(int p, int q) { return root(p) == root(q); } public void union(int p, int q) { int i = root(p); int j = root(q); id[i] = j; }}
QuickUnionTest.java
package quickunion;public class QuickUnionTest { public static void main(String[] args) { QuickUnion qu = new QuickUnion(); qu.QuickUnionUF(6); qu.union(0, 3); qu.union(1, 4); qu.union(4, 5); qu.union(2, 5); System.out.println(qu.connected(1, 3)); System.out.println(qu.connected(1, 2)); }}
Weighted Quick Union
The main idea of weighted quick union is to create a balance treee.
Implementation of WeightedQuickUnion
WeightedQuickUnion.java
package weightedquickunion;public class WeightedQuickUnion { private int[] id; private int[] sz; public void WeightedQU(int N) { id = new int[N]; sz = new int[N]; for(int i = 0; i < N; i++) { id[i] = i; sz[i] = 1; } } public int root(int i) { while(i != id[i]) i = id[i]; return i; } public void union(int p, int q) { int i = root(p); int j = root(q); if(i == j) return; if(sz[i] < sz[j]) {id[i] = j; sz[j] += sz[i];} else {id[j] = i; sz[i] += sz[j];} } public boolean connected(int p, int q) { return root(p) == root(q); }}
WeightedQUTest.java
package weightedquickunion;public class WeightedQUTest { public static void main(String[] args) { WeightedQuickUnion wqu = new WeightedQuickUnion(); wqu.WeightedQU(6); wqu.union(0, 3); wqu.union(1, 4); wqu.union(4, 5); wqu.union(2, 5); System.out.println(wqu.connected(1, 3)); System.out.println(wqu.connected(1, 2));// wqu.WeightedQU(6);// wqu.union(0, 1);// System.out.println(wqu.connected(1, 0)); }}
Analysis of algorithm
Scientific method applied to analysis of algorithms
Scientific Method
- Observe some feature of the natural world
- Hypothesize a model that is consistent with the observations
- Predict events using the hypothesis
- Verify the predictions by making further observations
- Validate by repeating until the hypothesis and observations agree
Principles
- Experiments must be reporducible
- Hypotheses must be falsifiable
Three-Sum
Description
Given N distinct integers, how many triples sum to exactly zero?
BruteForceMethod
BruteForceMethod.java
package threesum;public class BruteForceMethod { public int NumOfThreeSum(int[] a) { int count = 0; for(int i = 0; i < a.length; i++) for(int j = i+1; j < a.length; j++) for(int k = j+1; k < a.length; k++) if(a[i] + a[j] + a[k] == 0) { System.out.println(a[i] + " " + a[j] + " " + " " + a[k]); count++; } return count; }}
ThreeSumTest.java
package threesum;public class ThreeSumTest { public static void main(String[] args) { BruteForceMethod bfm = new BruteForceMethod(); int a[] = {10, 20, 0, -10, 40, -40, 30}; System.out.println(bfm.NumOfThreeSum(a)); }}
Time Calculator and Random Size Three Sum
Stopwatch.java
package threesum;public class Stopwatch { private final long start; public Stopwatch() { start = System.currentTimeMillis(); } public double elapsedTime() { long now = System.currentTimeMillis(); return (now - start) / 1000.0; }}
BruteForceMethod.java
package threesum;public class BruteForceMethod { public int NumOfThreeSum(int[] a) { int count = 0; for(int i = 0; i < a.length; i++) for(int j = i+1; j < a.length; j++) for(int k = j+1; k < a.length; k++) if(a[i] + a[j] + a[k] == 0) { System.out.println(a[i] + " " + a[j] + " " + " " + a[k]); count++; } return count; }}
ThreeSumTest.java
package threesum;import java.util.*;public class ThreeSumTest { public static void main(String[] args) { Scanner sc = new Scanner(System.in); System.out.print("Please input the size: "); int N = sc.nextInt(); int a[] = new int[N]; BruteForceMethod bfm = new BruteForceMethod(); Stopwatch timer = new Stopwatch(); //int a[] = {10, 20, 0, -10, 40, -40, 30}; for(int i = 0; i < N; i++) { //generate random number ranging from -10000 to 10000 a[i] = (int)(Math.random() * 10000 * Math.pow(-1, (int)(10 * Math.random()))); } System.out.println(bfm.NumOfThreeSum(a)); double time = timer.elapsedTime(); System.out.println("Running time is" + time + "seconds"); }}
Binary Search
BinarySearch.java
package binarysearch;public class BinarySearch { public int rank(int key, int[] a) { int lo = 0; int hi = a.length - 1; //lo represent low, hi represent high while (lo <= hi) { int mid = lo + (hi - lo) / 2; //use this form instead of (hi + lo)/2 because it can prevent overflow if(key < a[mid]) hi = mid - 1; else if(key > a[mid]) lo = mid + 1; else return mid; } return -1; }}
BinarySearchTest.java
package binarysearch;import java.util.Arrays;import java.util.Scanner;public class BinarySearchTest { public static void main(String[] args) { Scanner sc = new Scanner(System.in); BinarySearch bs = new BinarySearch(); int a[] = {11,23,54,68,15,18,19,56,35}; Arrays.sort(a); System.out.println("The sorted array is: "); for(int i = 0; i < a.length; i++) { System.out.print(a[i] + " "); } System.out.println(); System.out.print("Please input the searching number: "); int key = sc.nextInt(); System.out.print("The index of the it is: " + bs.rank(key, a)); }}
An N2logN algorithm for Three-Sum
SortingBase3Sum.java
package sortingbased3sum;import java.util.Arrays;public class SortingBased3Sum { public int SortingBsedThreeSum(int[] a) { int count = 0; Arrays.sort(a); for(int i = 0; i < a.length; i++) { for(int j = 1 + 1; j < a.length; j++) { if(Arrays.binarySearch(a, -(a[i]+a[j])) >= 0) { System.out.println(a[i] + " " + a[j] + " " + (-a[i]-a[j])); count++; } } } return count; }}
SortingBased3SumTest.java
package sortingbased3sum;import java.util.Scanner;public class SortingBased3SumTest { public static void main(String [] args) {Scanner sc = new Scanner(System.in); System.out.print("Please input the size: "); int N = sc.nextInt(); int a[] = new int[N]; SortingBased3Sum sortingbased3sum = new SortingBased3Sum(); //int a[] = {10, 20, 0, -10, 40, -40, 30}; for(int i = 0; i < N; i++) { //generate random number ranging from -10000 to 10000 a[i] = (int)(Math.random() * 10000 * Math.pow(-1, (int)(10 * Math.random()))); } System.out.println(sortingbased3sum.SortingBsedThreeSum(a)); }}
END
- 《算法》(1)--动态连接与算法分析
- 算法设计与分析入门篇----动态规划 1
- 动态规划算法分析与探究
- 动态规划(算法分析与设计)
- 算法分析与设计-动态规划0
- 算法设计与分析 (1)
- 算法分析与设计之五大常用算法 (II)—— 动态规划算法
- 单连接算法与全连接算法
- 算法设计与分析 递归与分治 回溯算法 分支限界 动态规划 贪心算法 近拟算法
- 算法设计与分析 递归与分治 回溯算法 分支限界 动态规划 贪心算法 近拟算法
- 1.4 算法与算法分析
- 南邮《算法设计与分析》动态规划源码
- 算法设计与分析复习-动态规划例子代码
- 算法设计与分析之动态规划法
- 算法设计与分析入门篇----动态规划 2
- 算法设计与分析(二)动态规划
- 字符串相似度算法 递归与动态规划求解分析
- 算法设计与分析——第六篇,动态规划
- XMLUtil工具封装
- Linux五种IO模型
- Plug-in a UITextView Placeholder Label Extension
- eclipse 从开辟空间到web项目
- UVALive
- 《算法》(1)--动态连接与算法分析
- Linux常用26个命令
- GYM 100182 H.Robot Challenge(dp)
- 2014第五届蓝桥杯预赛试题本科c++啤酒和饮料
- Apache POI TXT转成EXCEL(XLSX)
- Django项目关闭debug模式后,静态文件无法加载的解决办法
- [编程题] 买苹果
- ListView的基础学习
- js 创建对象与原型链继承