Merge pull request #1189 from mah-shamim/001404-number-of-steps-to-re…

…duce-a-number-in-binary-representation-to-one

001404 number of steps to reduce a number in binary representation to one

algorithms/001404-number-of-steps-to-reduce-a-number-in-binary-representation-to-one/README.md

@@ -1,6 +1,8 @@
 1404\. Number of Steps to Reduce a Number in Binary Representation to One
 
-Medium
+**Difficulty:** Medium
+
+**Topics:** `String`, `Bit Manipulation`
 
 Given the binary representation of an integer as a string `s`, return _the number of steps to reduce it to `1` under the following rules:_
 
@@ -39,3 +41,112 @@ It is guaranteed that you can always reach one for all test cases.
 - <code>s</code> consists of characters '0' or '1'
 - <code>s[0] == '1'</code>
 
+
+**Hint:**
+1. Read the string from right to left, if the string ends in '0' then the number is even otherwise it is odd.
+2. Simulate the steps described in the binary string.
+
+
+
+**Solution:**
+
+We are given a binary string `s` that represents a number. The goal is to reduce this number to `1` by performing the following operations:
+- If the number is even, divide it by `2`.
+- If the number is odd, add `1` to it.
+
+We need to determine how many steps it takes to achieve this.
+
+### Observations
+1. A binary number is **even** if its last digit is `0` and **odd** if its last digit is `1`.
+2. Dividing a binary number by `2` is equivalent to removing its last digit (`s = substr($s, 0, -1)`).
+3. Adding `1` to a binary number can sometimes cause a carry that affects the preceding digits.
+
+### Approach
+Simulate the process step by step:
+1. Start from the least significant bit (rightmost character of the string).
+2. Check if the number is odd or even:
+  - If odd: Add `1`. This might propagate a carry, modifying the preceding bits.
+  - If even: Divide by `2`, which removes the last digit.
+3. Continue the simulation until the number is reduced to `1`.
+4. Keep a counter to track the number of steps.
+
+Let's implement this solution in PHP: **[1404. Number of Steps to Reduce a Number in Binary Representation to One](https://github.com/mah-shamim/leet-code-in-php/tree/main/algorithms/001404-number-of-steps-to-reduce-a-number-in-binary-representation-to-one/solution.php)**
+
+```php
+<?php
+/**
+ * @param String $s
+ * @return Integer
+ */
+function numSteps($s) {
+    ...
+    ...
+    ...
+    /**
+     * go to ./solution.php
+     */
+}
+
+// Example usage:
+$s1 = "1101";
+$s2 = "10";
+$s3 = s = "1";
+
+echo numSteps($s1) . "\n"; // Output: 6
+echo numSteps($s2) . "\n"; // Output: 1
+echo numSteps($s3) . "\n"; // Output: 0
+?>
+```
+
+### Explanation:
+
+1. **Variables**:
+  - `steps`: Keeps track of the total number of operations performed.
+  - `carry`: Tracks if there is a carry after adding `1` to a binary number.
+
+2. **Iteration**:
+  - Start from the rightmost bit and process each bit up to the second bit (skip the first bit for now).
+  - For each bit:
+    - If it’s `1`: Adding `1` makes it `0`, but it propagates a carry to the next bit. This requires two operations (add and divide).
+    - If it’s `0`: Division directly removes the bit. However, if there is a carry from the previous bit, the `0` effectively acts as a `1`, requiring extra operations.
+
+3. **Final Step**:
+  - After processing all bits except the first, check if there’s a carry for the most significant bit. If so, add one final step.
+
+### Complexity Analysis
+1. **Time Complexity**: _**O(n)**_, where _**n**_ is the length of the binary string `s`. This is because we iterate through each bit once.
+2. **Space Complexity**: _**O(1)**_, as we use only a few variables and do not allocate additional data structures.
+
+### Example Walkthrough
+
+#### Input: `"1101"` (Binary for 13)
+1. Start processing from the rightmost bit:
+  - Bit 1: Odd → Add `1` → Becomes `1110`. Steps: `2`.
+  - Bit 0: Even → Divide by `2` → Becomes `111`. Steps: `3`.
+  - Bit 1: Odd → Add `1` → Becomes `1000`. Steps: `5`.
+  - Bit 0: Even → Divide by `2` → Becomes `100`. Steps: `6`.
+  - Bit 0: Even → Divide by `2` → Becomes `10`. Steps: `7`.
+  - Bit 0: Even → Divide by `2` → Becomes `1`. Steps: `8`.
+
+#### Final Steps: Total = `6`.
+
+### Edge Cases
+1. **Input `"1"`**:
+  - Already `1`, no steps needed. Output: `0`.
+
+2. **Input `"10"`**:
+  - `"10"` → `"1"` in one step. Output: `1`.
+
+3. **Large Binary Strings**:
+  - The algorithm handles large inputs efficiently due to its linear time complexity.
+
+This solution is robust, optimized, and adheres to the problem constraints.
+
+**Contact Links**
+
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+
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