@@ -16,6 +16,7 @@ public static long Fib(int N) {
1616 return Fib (N - 2 ) + Fib (N - 1 );
1717 }
1818
19+ 1920 // dynamic programming and memoization
2021 public static long FastFib (int N , HashMap <Integer , Long > cache ) {
2122 if (N == 0 ) {
@@ -38,6 +39,7 @@ public static boolean isBetweenZeroAndOne(double x, double y) {
3839 return (0 < x && x < 1 ) && (0 < y && y < 1 );
3940 }
4041
42+ 4143 /**
4244 * static method to return the binary string representation of an integer
4345 * */
@@ -48,6 +50,7 @@ public static String toBinaryString(int x) {
4850 return binaryString .toString ();
4951 }
5052
53+ 5154 /**
5255 * static recursive method that "builds" the binary string
5356 * */
@@ -57,6 +60,7 @@ private static String binaryFormString(int x, String binString) {
5760 return binaryFormString (x / 2 , binString ) + (x % 2 );
5861 }
5962
63+ 6064 /**
6165 * static method to print a "*" or " " depending on the contents in a two-dimensional array
6266 * */
@@ -73,6 +77,7 @@ public static void printTwoDmBooleanArray(boolean[][] theArray) {
7377 }
7478 }
7579
80+ 7681 /**
7782 * static method to print the formatted contents of a two-dimensional int array
7883 * */
@@ -86,6 +91,7 @@ public static void printTwoDmIntArray(int[][] theArray) {
8691 }
8792 }
8893
94+ 8995 /**
9096 * static method that returns the transposition of an input matrix
9197 * */
@@ -104,6 +110,7 @@ public static int[][] matrixTransposition(int[][] theMatrix) {
104110 return transposedMatrix ;
105111 }
106112
113+ 107114 /**
108115 * recursive method that returns the greatest integer not greater than log(N)
109116 * to base two
@@ -114,6 +121,7 @@ public static int lg(int N) {
114121 return 1 + lg (N / 2 );
115122 }
116123
124+ 117125 public static void main (String [] args ) {
118126 Random generator = new Random ();
119127 HashMap <Integer , Long > cache = new HashMap <>();
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