|
| 1 | + |
| 2 | +Given the array: |
| 3 | +``` |
| 4 | +[38, 27, 43, 3, 9, 82, 10] |
| 5 | +``` |
| 6 | + |
| 7 | +## Step-by-Step Simulation on Ones Place Value |
| 8 | + |
| 9 | +### Counting Occurrences by Ones Place |
| 10 | + |
| 11 | +1. **Count Array Initialization:** |
| 12 | + ``` |
| 13 | + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0] |
| 14 | + ``` |
| 15 | + |
| 16 | +2. **Number: 38, Ones Digit: 8 -> Count[8]++** |
| 17 | + ``` |
| 18 | + [0, 0, 0, 0, 0, 0, 0, 0, 1, 0] |
| 19 | + ``` |
| 20 | + |
| 21 | +3. **Number: 27, Ones Digit: 7 -> Count[7]++** |
| 22 | + ``` |
| 23 | + [0, 0, 0, 0, 0, 0, 0, 1, 1, 0] |
| 24 | + ``` |
| 25 | + |
| 26 | +4. **Number: 43, Ones Digit: 3 -> Count[3]++** |
| 27 | + ``` |
| 28 | + [0, 0, 0, 1, 0, 0, 0, 1, 1, 0] |
| 29 | + ``` |
| 30 | + |
| 31 | +5. **Number: 3, Ones Digit: 3 -> Count[3]++** |
| 32 | + ``` |
| 33 | + [0, 0, 0, 2, 0, 0, 0, 1, 1, 0] |
| 34 | + ``` |
| 35 | + |
| 36 | +6. **Number: 9, Ones Digit: 9 -> Count[9]++** |
| 37 | + ``` |
| 38 | + [0, 0, 0, 2, 0, 0, 0, 1, 1, 1] |
| 39 | + ``` |
| 40 | + |
| 41 | +7. **Number: 82, Ones Digit: 2 -> Count[2]++** |
| 42 | + ``` |
| 43 | + [0, 0, 1, 2, 0, 0, 0, 1, 1, 1] |
| 44 | + ``` |
| 45 | + |
| 46 | +8. **Number: 10, Ones Digit: 0 -> Count[0]++** |
| 47 | + ``` |
| 48 | + [1, 0, 1, 2, 0, 0, 0, 1, 1, 1] |
| 49 | + ``` |
| 50 | + |
| 51 | +### Transforming to Cumulative Count Array |
| 52 | + |
| 53 | +1. **Count[1] = Count[1] + Count[0] = 0 + 1 = 1** |
| 54 | + ``` |
| 55 | + [1, 1, 1, 2, 0, 0, 0, 1, 1, 1] |
| 56 | + ``` |
| 57 | + |
| 58 | +2. **Count[2] = Count[2] + Count[1] = 1 + 1 = 2** |
| 59 | + ``` |
| 60 | + [1, 1, 2, 2, 0, 0, 0, 1, 1, 1] |
| 61 | + ``` |
| 62 | + |
| 63 | +3. **Count[3] = Count[3] + Count[2] = 2 + 2 = 4** |
| 64 | + ``` |
| 65 | + [1, 1, 2, 4, 0, 0, 0, 1, 1, 1] |
| 66 | + ``` |
| 67 | + |
| 68 | +4. **Count[4] = Count[4] + Count[3] = 0 + 4 = 4** |
| 69 | + ``` |
| 70 | + [1, 1, 2, 4, 4, 0, 0, 1, 1, 1] |
| 71 | + ``` |
| 72 | + |
| 73 | +5. **Count[5] = Count[5] + Count[4] = 0 + 4 = 4** |
| 74 | + ``` |
| 75 | + [1, 1, 2, 4, 4, 4, 0, 1, 1, 1] |
| 76 | + ``` |
| 77 | + |
| 78 | +6. **Count[6] = Count[6] + Count[5] = 0 + 4 = 4** |
| 79 | + ``` |
| 80 | + [1, 1, 2, 4, 4, 4, 4, 1, 1, 1] |
| 81 | + ``` |
| 82 | + |
| 83 | +7. **Count[7] = Count[7] + Count[6] = 1 + 4 = 5** |
| 84 | + ``` |
| 85 | + [1, 1, 2, 4, 4, 4, 4, 5, 1, 1] |
| 86 | + ``` |
| 87 | + |
| 88 | +8. **Count[8] = Count[8] + Count[7] = 1 + 5 = 6** |
| 89 | + ``` |
| 90 | + [1, 1, 2, 4, 4, 4, 4, 5, 6, 1] |
| 91 | + ``` |
| 92 | + |
| 93 | +9. **Count[9] = Count[9] + Count[8] = 1 + 6 = 7** |
| 94 | + ``` |
| 95 | + [1, 1, 2, 4, 4, 4, 4, 5, 6, 7] |
| 96 | + ``` |
| 97 | + |
| 98 | +### Building the Output Array |
| 99 | + |
| 100 | +1. **Output Array:** |
| 101 | + ``` |
| 102 | + [0, 0, 0, 0, 0, 0, 0] |
| 103 | + ``` |
| 104 | + |
| 105 | +2. **Place Number: 10, Ones Digit: 0** |
| 106 | + - Place 10 at index `count[0] - 1 = 0` |
| 107 | + - Decrement `count[0]` |
| 108 | + ``` |
| 109 | + Output Array: [10, 0, 0, 0, 0, 0, 0] |
| 110 | + Count Array: [0, 1, 2, 4, 4, 4, 4, 5, 6, 7] |
| 111 | + ``` |
| 112 | + |
| 113 | +3. **Place Number: 82, Ones Digit: 2** |
| 114 | + - Place 82 at index `count[2] - 1 = 1` |
| 115 | + - Decrement `count[2]` |
| 116 | + ``` |
| 117 | + Output Array: [10, 82, 0, 0, 0, 0, 0] |
| 118 | + Count Array: [0, 1, 1, 4, 4, 4, 4, 5, 6, 7] |
| 119 | + ``` |
| 120 | + |
| 121 | +4. **Place Number: 9, Ones Digit: 9** |
| 122 | + - Place 9 at index `count[9] - 1 = 6` |
| 123 | + - Decrement `count[9]` |
| 124 | + ``` |
| 125 | + Output Array: [10, 82, 0, 0, 0, 0, 9] |
| 126 | + Count Array: [0, 1, 1, 4, 4, 4, 4, 5, 6, 6] |
| 127 | + ``` |
| 128 | + |
| 129 | +5. **Place Number: 3, Ones Digit: 3** |
| 130 | + - Place 3 at index `count[3] - 1 = 3` |
| 131 | + - Decrement `count[3]` |
| 132 | + ``` |
| 133 | + Output Array: [10, 82, 0, 3, 0, 0, 9] |
| 134 | + Count Array: [0, 1, 1, 3, 4, 4, 4, 5, 6, 6] |
| 135 | + ``` |
| 136 | + |
| 137 | +6. **Place Number: 43, Ones Digit: 3** |
| 138 | + - Place 43 at index `count[3] - 1 = 2` |
| 139 | + - Decrement `count[3]` |
| 140 | + ``` |
| 141 | + Output Array: [10, 82, 43, 3, 0, 0, 9] |
| 142 | + Count Array: [0, 1, 1, 2, 4, 4, 4, 5, 6, 6] |
| 143 | + ``` |
| 144 | + |
| 145 | +7. **Place Number: 27, Ones Digit: 7** |
| 146 | + - Place 27 at index `count[7] - 1 = 4` |
| 147 | + - Decrement `count[7]` |
| 148 | + ``` |
| 149 | + Output Array: [10, 82, 43, 3, 27, 0, 9] |
| 150 | + Count Array: [0, 1, 1, 2, 4, 4, 4, 4, 6, 6] |
| 151 | + ``` |
| 152 | + |
| 153 | +8. **Place Number: 38, Ones Digit: 8** |
| 154 | + - Place 38 at index `count[8] - 1 = 5` |
| 155 | + - Decrement `count[8]` |
| 156 | + ``` |
| 157 | + Output Array: [10, 82, 43, 3, 27, 38, 9] |
| 158 | + Count Array: [0, 1, 1, 2, 4, 4, 4, 4, 5, 6] |
| 159 | + ``` |
| 160 | + |
| 161 | +### Sorted by Ones Place |
| 162 | + |
| 163 | +The array sorted by the ones place is: |
| 164 | + |
| 165 | +``` |
| 166 | +[10, 82, 43, 3, 27, 38, 9] |
| 167 | +``` |
| 168 | + |
| 169 | +### Recap of Radix Sort Steps |
| 170 | + |
| 171 | +1. **Sort by Least Significant Digit (Ones Place):** |
| 172 | + ``` |
| 173 | + [10, 82, 43, 3, 27, 38, 9] |
| 174 | + ``` |
| 175 | + |
| 176 | +2. **Sort by Next Significant Digit (Tens Place):** |
| 177 | + ``` |
| 178 | + [3, 9, 10, 27, 38, 43, 82] |
| 179 | + ``` |
| 180 | + |
| 181 | +### Overview of Each Step and Keyword |
| 182 | + |
| 183 | +- **Digit Position:** Each iteration focuses on sorting elements based on a specific digit position, starting from the least significant to the most significant. |
| 184 | +- ** |
| 185 | + |
| 186 | +**Counting Sort:** Counting sort is applied at each digit level, ensuring the algorithm is stable and correctly sorts elements at each step. |
| 187 | +- **Cumulative Count:** The cumulative count array helps place elements in the correct position in the output array. |
| 188 | +- **Stability:** Ensures that elements with the same digit values appear in the same order as they were in the input array. |
| 189 | + |
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