算法Sedgewick第四版-第1章基础-1.4 Analysis of Algorithms-002如何改进算法

1.

package algorithms.analysis14;

import algorithms.util.In;
import algorithms.util.StdOut;

/******************************************************************************
 *  Compilation:  javac TwoSum.java
 *  Execution:    java TwoSum input.txt
 *  Dependencies: StdOut.java In.java Stopwatch.java
 *  Data files:   http://algs4.cs.princeton.edu/14analysis/1Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/2Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/4Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/8Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/16Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/32Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/1Mints.txt
 *
 *  A program with N^2 running time. Read in N integers
 *  and counts the number of pairs that sum to exactly 0.
 *
 *
 *  Limitations
 *  -----------
 *     - we ignore integer overflow
 *
 *
 *  % java TwoSum 2Kints.txt
 *  2
 *
 *  % java TwoSum 1Kints.txt
 *  1
 *
 *  % java TwoSum 2Kints.txt
 *  2
 *
 *  % java TwoSum 4Kints.txt
 *  3
 *
 *  % java TwoSum 8Kints.txt
 *  19
 *
 *  % java TwoSum 16Kints.txt
 *  66
 *
 *  % java TwoSum 32Kints.txt
 *  273
 *
 ******************************************************************************/

public class TwoSum {

    // print distinct pairs (i, j) such that a[i] + a[j]  = 0
    public static void printAll(int[] a) {
        int N = a.length;
        for (int i = 0; i < N; i++) {
            for (int j = i+1; j < N; j++) {
                if (a[i] + a[j] == 0) {
                    StdOut.println(a[i] + " " + a[j]);
                }
            }
        }
    } 

    // return number of distinct triples (i, j) such that a[i] + a[j] = 0
    public static int count(int[] a) {
        int N = a.length;
        int cnt = 0;
        for (int i = 0; i < N; i++) {
            for (int j = i+1; j < N; j++) {
                if (a[i] + a[j] == 0) {
                    cnt++;
                }
            }
        }
        return cnt;
    } 

    public static void main(String[] args)  { 
        In in = new In(args[0]);
        int[] a = in.readAllInts();
        Stopwatch timer = new Stopwatch();
        int cnt = count(a);
        StdOut.println("elapsed time = " + timer.elapsedTime());
        StdOut.println(cnt);
    } 
} 

The answer to this question is that we have discussed and used two classic algorithms,
mergesort and binary search, have introduced the facts that the mergesort is linearith-
mic and binary search is logarithmic.

2.

package algorithms.analysis14;

/******************************************************************************
 *  Compilation:  javac TwoSumFast.java
 *  Execution:    java TwoSumFast input.txt
 *  Dependencies: In.java Stopwatch.java
 *  Data files:   http://algs4.cs.princeton.edu/14analysis/1Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/2Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/4Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/8Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/16Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/32Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/1Mints.txt
 *
 *  A program with N log N running time. Read in N integers
 *  and counts the number of pairs that sum to exactly 0.
 *
 *  Limitations
 *  -----------
 *     - we ignore integer overflow
 *
 *
 *  % java TwoSumFast 2Kints.txt
 *  2
 *
 *  % java TwoSumFast 1Kints.txt
 *  1
 *
 *  % java TwoSumFast 2Kints.txt
 *  2
 *
 *  % java TwoSumFast 4Kints.txt
 *  3
 *
 *  % java TwoSumFast 8Kints.txt
 *  19
 *
 *  % java TwoSumFast 16Kints.txt
 *  66
 *
 *  % java TwoSumFast 32Kints.txt
 *  273
 *
 ******************************************************************************/

import java.util.Arrays;

import algorithms.util.In;
import algorithms.util.StdOut;

public class TwoSumFast {

    // print distinct pairs (i, j) such that a[i] + a[j] = 0
    public static void printAll(int[] a) {
        int N = a.length;
        Arrays.sort(a);
        for (int i = 0; i < N; i++) {
            int j = Arrays.binarySearch(a, -a[i]);
            if (j > i) StdOut.println(a[i] + " " + a[j]);
        }
    } 

    // return number of distinct pairs (i, j) such that a[i] + a[j] = 0
    public static int count(int[] a) {
        int N = a.length;
        Arrays.sort(a);
        int cnt = 0;
        for (int i = 0; i < N; i++) {
            int j = Arrays.binarySearch(a, -a[i]);
            if (j > i) cnt++;
        }
        return cnt;
    } 

    public static void main(String[] args)  { 
        In in = new In(args[0]);
        int[] a = in.readAllInts();
        int cnt = count(a);
        StdOut.println(cnt);
    } 
} 

　　

3.

package algorithms.analysis14;

import algorithms.util.In;
import algorithms.util.StdOut;

/******************************************************************************
 *  Compilation:  javac ThreeSum.java
 *  Execution:    java ThreeSum input.txt
 *  Dependencies: In.java StdOut.java Stopwatch.java
 *  Data files:   http://algs4.cs.princeton.edu/14analysis/1Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/2Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/4Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/8Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/16Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/32Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/1Mints.txt
 *
 *  A program with cubic running time. Read in N integers
 *  and counts the number of triples that sum to exactly 0
 *  (ignoring integer overflow).
 *
 *  % java ThreeSum 1Kints.txt 
 *  70
 *
 *  % java ThreeSum 2Kints.txt 
 *  528
 *
 *  % java ThreeSum 4Kints.txt 
 *  4039
 *
 ******************************************************************************/

/**
 *  The <tt>ThreeSum</tt> class provides static methods for counting
 *  and printing the number of triples in an array of integers that sum to 0
 *  (ignoring integer overflow).
 *  <p>
 *  This implementation uses a triply nested loop and takes proportional to N^3,
 *  where N is the number of integers.
 *  <p>
 *  For additional documentation, see <a href="http://algs4.cs.princeton.edu/14analysis">Section 1.4</a> of
 *  <i>Algorithms, 4th Edition</i> by Robert Sedgewick and Kevin Wayne.
 *
 *  @author Robert Sedgewick
 *  @author Kevin Wayne
 */
public class ThreeSum {

    // Do not instantiate.
    private ThreeSum() { }

    /**
     * Prints to standard output the (i, j, k) with i < j < k such that a[i] + a[j] + a[k] == 0.
     * @param a the array of integers
     */
    public static void printAll(int[] a) {
        int N = a.length;
        for (int i = 0; i < N; i++) {
            for (int j = i+1; j < N; j++) {
                for (int k = j+1; k < N; k++) {
                    if (a[i] + a[j] + a[k] == 0) {
                        StdOut.println(a[i] + " " + a[j] + " " + a[k]);
                    }
                }
            }
        }
    } 

    /**
     * Returns the number of triples (i, j, k) with i < j < k such that a[i] + a[j] + a[k] == 0.
     * @param a the array of integers
     * @return the number of triples (i, j, k) with i < j < k such that a[i] + a[j] + a[k] == 0
     */
    public static int count(int[] a) {
        int N = a.length;
        int cnt = 0;
        for (int i = 0; i < N; i++) {
            for (int j = i+1; j < N; j++) {
                for (int k = j+1; k < N; k++) {
                    if (a[i] + a[j] + a[k] == 0) {
                        cnt++;
                    }
                }
            }
        }
        return cnt;
    } 

    /**
     * Reads in a sequence of integers from a file, specified as a command-line argument;
     * counts the number of triples sum to exactly zero; prints out the time to perform
     * the computation.
     */
    public static void main(String[] args)  { 
        In in = new In(args[0]);
        int[] a = in.readAllInts();

        Stopwatch timer = new Stopwatch();
        int cnt = count(a);
        StdOut.println("elapsed time = " + timer.elapsedTime());
        StdOut.println(cnt);
    } 
} 

 

4.

package algorithms.analysis14;

/******************************************************************************
 *  Compilation:  javac ThreeSumFast.java
 *  Execution:    java ThreeSumFast input.txt
 *  Dependencies: StdOut.java In.java Stopwatch.java
 *  Data files:   http://algs4.cs.princeton.edu/14analysis/1Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/2Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/4Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/8Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/16Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/32Kints.txt
 *                http://algs4.cs.princeton.edu/14analysis/1Mints.txt
 *
 *  A program with N^2 log N running time. Read in N integers
 *  and counts the number of triples that sum to exactly 0.
 *
 *  Limitations
 *  -----------
 *     - we ignore integer overflow
 *     - doesn't handle case when input has duplicates
 *
 *
 *  % java ThreeSumFast 1Kints.txt
 *  70
 *  
 *  % java ThreeSumFast 2Kints.txt
 *  528
 *                
 *  % java ThreeSumFast 4Kints.txt
 *  4039
 * 
 *  % java ThreeSumFast 8Kints.txt
 *  32074
 *
 *  % java ThreeSumFast 16Kints.txt
 *  255181
 *
 *  % java ThreeSumFast 32Kints.txt
 *  2052358
 *
 ******************************************************************************/

import java.util.Arrays;

import algorithms.util.In;
import algorithms.util.StdOut;

/**
 *  The <tt>ThreeSumFast</tt> class provides static methods for counting
 *  and printing the number of triples in an array of distinct integers that
 *  sum to 0 (ignoring integer overflow).
 *  <p>
 *  This implementation uses sorting and binary search and takes time 
 *  proportional to N^2 log N, where N is the number of integers.
 *  <p>
 *  For additional documentation, see <a href="http://algs4.cs.princeton.edu/14analysis">Section 1.4</a> of
 *  <i>Algorithms, 4th Edition</i> by Robert Sedgewick and Kevin Wayne.
 *
 *  @author Robert Sedgewick
 *  @author Kevin Wayne
 */
public class ThreeSumFast {

    // Do not instantiate.
    private ThreeSumFast() { }

    // returns true if the sorted array a[] contains any duplicated integers
    private static boolean containsDuplicates(int[] a) {
        for (int i = 1; i < a.length; i++)
            if (a[i] == a[i-1]) return true;
        return false;
    }

    /**
     * Prints to standard output the (i, j, k) with i < j < k such that a[i] + a[j] + a[k] == 0.
     * @param a the array of integers
     * @throws IllegalArgumentException if the array contains duplicate integers
     */
    public static void printAll(int[] a) {
        int N = a.length;
        Arrays.sort(a);
        if (containsDuplicates(a)) throw new IllegalArgumentException("array contains duplicate integers");
        for (int i = 0; i < N; i++) {
            for (int j = i+1; j < N; j++) {
                int k = Arrays.binarySearch(a, -(a[i] + a[j]));
                if (k > j) StdOut.println(a[i] + " " + a[j] + " " + a[k]);
            }
        }
    } 

    /**
     * Returns the number of triples (i, j, k) with i < j < k such that a[i] + a[j] + a[k] == 0.
     * @param a the array of integers
     * @return the number of triples (i, j, k) with i < j < k such that a[i] + a[j] + a[k] == 0
     */
    public static int count(int[] a) {
        int N = a.length;
        Arrays.sort(a);
        if (containsDuplicates(a)) throw new IllegalArgumentException("array contains duplicate integers");
        int cnt = 0;
        for (int i = 0; i < N; i++) {
            for (int j = i+1; j < N; j++) {
                int k = Arrays.binarySearch(a, -(a[i] + a[j]));
                if (k > j) cnt++;
            }
        }
        return cnt;
    } 

    /**
     * Reads in a sequence of distinct integers from a file, specified as a command-line argument;
     * counts the number of triples sum to exactly zero; prints out the time to perform
     * the computation.
     */
    public static void main(String[] args)  { 
        In in = new In(args[0]);
        int[] a = in.readAllInts();
        int cnt = count(a);
        StdOut.println(cnt);
    } 
}