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3-SUM · binary-search approach
Combine sorting, pair enumeration and binary search to avoid the third linear loop.
Overview
Combine sorting, pair enumeration and binary search to avoid the third linear loop. There are Θ(n²) pairs and each binary search costs O(log n). Sorting is performed first and does not dominate that bound. The method modifies the input by sorting it.
Step-by-step walkthrough
- Sort the input array.
- Reject duplicate integers as the supplied code requires.
- For each i < j, binary-search for the negated pair sum.
- Count only a result index greater than j.
Complexity analysis
| Best case | Ω(n²) |
|---|---|
| Average / aggregate | O(n² log n) |
| Worst case | O(n² log n) |
| Space | Sorting-dependent; pair loop uses Θ(1) |
There are Θ(n²) pairs and each binary search costs O(log n). Sorting is performed first and does not dominate that bound. The method modifies the input by sorting it.
A worked example
For [−4, −1, 0, 1, 4], the pair (−4, 0) finds 4 after j. A result at or before j is not counted.
Common mistake / implementation note
Without the k > j check the same triple can be counted more than once. This code is not a general duplicate-aware 3-SUM implementation.
Without the k > j check the same triple can be counted more than once. This code is not a general duplicate-aware 3-SUM implementation.
Original course code
Source: code/ThreeSumFast.java. Original logic, comments, variable names and attribution are retained.
Download ThreeSumFast.javaThreeSumFast.javaJAVA
/****************************************************************************** * Compilation: javac ThreeSumFast.java * Execution: java ThreeSumFast input.txt * Dependencies: StdOut.java In.java Stopwatch.java * Data files: https://algs4.cs.princeton.edu/14analysis/1Kints.txt * https://algs4.cs.princeton.edu/14analysis/2Kints.txt * https://algs4.cs.princeton.edu/14analysis/4Kints.txt * https://algs4.cs.princeton.edu/14analysis/8Kints.txt * https://algs4.cs.princeton.edu/14analysis/16Kints.txt * https://algs4.cs.princeton.edu/14analysis/32Kints.txt * https://algs4.cs.princeton.edu/14analysis/1Mints.txt * * A program with n^2 log n running time. Reads 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 * ******************************************************************************/ // package edu.princeton.cs.algs4; import java.util.Arrays; /** * The {@code ThreeSumFast} 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="https://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 {@code i < j < k} * such that {@code 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 {@code i < j < k} * such that {@code a[i] + a[j] + a[k] == 0}. * * @param a the array of integers * @return the number of triples (i, j, k) with {@code i < j < k} * such that {@code 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 count = 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) count++; } } return count; } /** * 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. * * @param args the command-line arguments */ public static void main(String[] args) { In in = new In(args[0]); int[] a = in.readAllInts(); int count = count(a); StdOut.println(count); } } /****************************************************************************** * Copyright 2002-2020, Robert Sedgewick and Kevin Wayne. * * This file is part of algs4.jar, which accompanies the textbook * * Algorithms, 4th edition by Robert Sedgewick and Kevin Wayne, * Addison-Wesley Professional, 2011, ISBN 0-321-57351-X. * http://algs4.cs.princeton.edu * * * algs4.jar is free software: you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation, either version 3 of the License, or * (at your option) any later version. * * algs4.jar is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with algs4.jar. If not, see http://www.gnu.org/licenses. ******************************************************************************/