diff --git a/src/main/java/com/thealgorithms/slidingwindow/ShortestCoprimeSegment.java b/src/main/java/com/thealgorithms/slidingwindow/ShortestCoprimeSegment.java new file mode 100644 index 000000000000..b99f7ca7d62f --- /dev/null +++ b/src/main/java/com/thealgorithms/slidingwindow/ShortestCoprimeSegment.java @@ -0,0 +1,135 @@ +package com.thealgorithms.slidingwindow; + +import java.util.Arrays; +import java.util.LinkedList; + +/** + * The Sliding Window technique together with 2-stack technique is used to find coprime segment of minimal size in an array. + * Segment a[i],...,a[i+l] is coprime if gcd(a[i], a[i+1], ..., a[i+l]) = 1 + *
+ * Run-time complexity: O(n log n) + * What is special about this 2-stack technique is that it enables us to remove element a[i] and find gcd(a[i+1],...,a[i+l]) in amortized O(1) time. + * For 'remove' worst-case would be O(n) operation, but this happens rarely. + * Main observation is that each element gets processed a constant amount of times, hence complexity will be: + * O(n log n), where log n comes from complexity of gcd. + *
+ * More generally, the 2-stack technique enables us to 'remove' an element fast if it is known how to 'add' an element fast to the set. + * In our case 'adding' is calculating d' = gcd(a[i],...,a[i+l+1]), when d = gcd(a[i],...a[i]) with d' = gcd(d, a[i+l+1]). + * and removing is find gcd(a[i+1],...,a[i+l]). We don't calculate it explicitly, but it is pushed in the stack which we can pop in O(1). + *
+ * One can change methods 'legalSegment' and function 'f' in DoubleStack to adapt this code to other sliding-window type problems. + * I recommend this article for more explanations: "CF Article">Article 1 or USACO Article + *
+ * Another method to solve this problem is through segment trees. Then query operation would have O(log n), not O(1) time, but runtime complexity would still be O(n log n)
+ *
+ * @author DomTr (Github)
+ */
+public final class ShortestCoprimeSegment {
+ // Prevent instantiation
+ private ShortestCoprimeSegment() {
+ }
+
+ /**
+ * @param arr is the input array
+ * @return shortest segment in the array which has gcd equal to 1. If no such segment exists or array is empty, returns empty array
+ */
+ public static long[] shortestCoprimeSegment(long[] arr) {
+ if (arr == null || arr.length == 0) {
+ return new long[] {};
+ }
+ DoubleStack front = new DoubleStack();
+ DoubleStack back = new DoubleStack();
+ int n = arr.length;
+ int l = 0;
+ int shortestLength = n + 1;
+ int beginsAt = -1; // beginning index of the shortest coprime segment
+ for (int i = 0; i < n; i++) {
+ back.push(arr[i]);
+ while (legalSegment(front, back)) {
+ remove(front, back);
+ if (shortestLength > i - l + 1) {
+ beginsAt = l;
+ shortestLength = i - l + 1;
+ }
+ l++;
+ }
+ }
+ if (shortestLength > n) {
+ shortestLength = -1;
+ }
+ if (shortestLength == -1) {
+ return new long[] {};
+ }
+ return Arrays.copyOfRange(arr, beginsAt, beginsAt + shortestLength);
+ }
+
+ private static boolean legalSegment(DoubleStack front, DoubleStack back) {
+ return gcd(front.top(), back.top()) == 1;
+ }
+
+ private static long gcd(long a, long b) {
+ if (a < b) {
+ return gcd(b, a);
+ } else if (b == 0) {
+ return a;
+ } else {
+ return gcd(a % b, b);
+ }
+ }
+
+ /**
+ * This solves the problem of removing elements quickly.
+ * Even though the worst case of 'remove' method is O(n), it is a very pessimistic view.
+ * We will need to empty out 'back', only when 'from' is empty.
+ * Consider element x when it is added to stack 'back'.
+ * After some time 'front' becomes empty and x goes to 'front'. Notice that in the for-loop we proceed further and x will never come back to any stacks 'back' or 'front'.
+ * In other words, every element gets processed by a constant number of operations.
+ * So 'remove' amortized runtime is actually O(n).
+ */
+ private static void remove(DoubleStack front, DoubleStack back) {
+ if (front.isEmpty()) {
+ while (!back.isEmpty()) {
+ front.push(back.pop());
+ }
+ }
+ front.pop();
+ }
+
+ /**
+ * DoubleStack serves as a collection of two stacks. One is a normal stack called 'stack', the other 'values' stores gcd-s up until some index.
+ */
+ private static class DoubleStack {
+ LinkedList