From 294c59220e6bb0e9e3aa35bb4556b2bc27e103c2 Mon Sep 17 00:00:00 2001
From: juan <juancloudcomputing@protonmail.com>
Date: Mon, 5 Sep 2022 14:48:29 +0800
Subject: [PATCH] vault backup: 2022-09-05 14:48:29

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+# Leetcode Product-of-Array-Except-Self
+
+2022-09-05 14:27
+
+> ##### Algorithms:
+>
+> #algorithm #prefix_sum 
+>
+> ##### Data structures:
+>
+> #DS
+>
+> ##### Difficulty:
+>
+> #coding_problem #difficulty-medium
+>
+> ##### Additional tags:
+>
+> #leetcode #CS_list_need_understanding 
+>
+> ##### Revisions:
+>
+> N/A
+
+##### Links:
+
+- [Link to problem](https://leetcode.com/problems/product-of-array-except-self/)
+- [Solution with explanation](https://leetcode.com/problems/product-of-array-except-self/discuss/1597994/C%2B%2BPython-4-Simple-Solutions-w-Explanation-or-Prefix-and-Suffix-product-O(1)-space-approach)
+
+---
+
+### Problem
+
+Given an integer array `nums`, return _an array_ `answer` _such that_ `answer[i]` _is equal to the product of all the elements of_ `nums` _except_ `nums[i]`.
+
+The product of any prefix or suffix of `nums` is **guaranteed** to fit in a **32-bit** integer.
+
+You must write an algorithm that runs in `O(n)` time and without using the division operation.
+
+#### Examples
+
+**Example 1:**
+
+**Input:** nums = [1,2,3,4]
+**Output:** [24,12,8,6]
+
+**Example 2:**
+
+**Input:** nums = [-1,1,0,-3,3]
+**Output:** [0,0,9,0,0]
+
+#### Constraints
+
+-   `2 <= nums.length <= 105`
+-   `-30 <= nums[i] <= 30`
+-   The product of any prefix or suffix of `nums` is **guaranteed** to fit in a **32-bit** integer.
+
+### Thoughts
+
+> [!summary]
+> This is a variation of #prefix_sum
+
+I thought of using prefix product, a variation of 
+prefix_sum, which perfectly fits the requirement.
+
+#### First iteration:
+
+keep two arrays: the suffix product and the prefix product
+of the `nums` array.
+
+Then, `ans = prefix[i] * suffix[i]`
+
+#### Second iteration:
+
+> It's like rolling snowball, the suffix sum variable
+> gets reused
+
+We can use only one output array:
+
+write the prefix to the `ans` array, and update it from
+end using suffix sum variable.
+
+This achieves the `O(1) space except output` requirement.
+
+#### Third iteration:
+
+Since the multiplying process is independent, and can be
+reversed, We can do that in one loop.
+
+`1 * A * B == 1 * B * A`
+
+If we initialize the ans array as one, we can do that in one
+loop.
+
+### Solution
+
+Third version:
+
+Second version:
+
+```cpp
+class Solution {
+public:
+  vector<int> productExceptSelf(vector<int> &nums) {
+    // Optimized prefix sum & suffix sum
+    int size = nums.size();
+    vector<int> ans(size);
+
+    ans[0] = 1;
+    for (int i = 1; i < size; i++) {
+      ans[i] = ans[i - 1] * nums[i - 1];
+    }
+
+    int suffix = 1;
+
+    for (int i = size - 1; i >= 0; i--) {
+      ans[i] = ans[i] * suffix;
+      suffix = suffix * nums[i];
+    }
+
+    return ans;
+  }
+};
+```
+
+First iteration:
+```cpp
+class Solution {
+public:
+  vector<int> productExceptSelf(vector<int> &nums) {
+    // method 1: prefix sum & suffix sum
+    vector<int> ans;
+    vector<int> suffix;
+    int size = nums.size();
+    ans.resize(size);
+    suffix.resize(size);
+
+    ans[0] = 1;
+    for (int i = 1; i < size; i++) {
+      ans[i] = ans[i - 1] * nums[i - 1];
+    }
+
+    suffix[size - 1] = 1;
+    for (int i = size - 2; i >= 0; i--) {
+      suffix[i] = suffix[i + 1] * nums[i + 1];
+    }
+
+    for (int i = 0; i < size; i++) {
+      ans[i] = ans[i] * suffix[i];
+    }
+
+    return ans;
+  }
+};
+```
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