# Integer Replacement

LeetCode problem 397.

This is a problem I brought along to the WICSxACM Technical Interview Prep workshop on January 13, 2020.

## Problem Description ¶

Source: LeetCode 397

Given a positive integer $n$, you can perform the following operations:

1. If $n$ is even, replace $n$ with $n/2$
2. If $n$ is odd, replace $n$ with either $n + 1$ or $n - 1$

What is the minimum number of replacements needed for $n$ to become $1$?

## Hints ¶

1. The initial temptation is to go for a fancy, math-based $O(1)$ solution, but there are more general approaches that will increase your chances of actually coming up with a solution. What are some other approaches you can think of?
2. How can you think of the problem as a graph? (For this workshop, I know there have been quite a few other graph problems, so this will be fresh in your mind.)

## Solution ¶

I have implemented one possible solution to the problem in C++.

### Approach 1: BFS ¶

This solution was suggested in the Hints. The idea is to think of the numbers as a graph, where each number is a node and there are edges connecting each node $x$ to $x/2$ if $x$ is even, or to $x + 1$ and $x - 1$ if $x$ is odd. We can then perform a BFS (Breadth-First Search) on the graph to find the length of the shortest path from $n$ to $1$.

• Time complexity: $O(n)$ – we can only encounter numbers in the range $[1, 2n]$ since if you reach $2n$, you wasted a bunch of steps because $2n$ can just be divided by $2$ to get $n$. Since we never visit the same number twice, the time complexity is $O(2n) = O(n)$
• Space complexity: $O(n)$ – same reason as above.
class Solution { public:  int integerReplacement(int n) {    // Queue for BFS -- each pair holds a number x and the number    // of steps it is from n. We use long long to account for the    // edge case when n == INT_MAX. Note that as we pop nodes from    // and add nodes to this queue, the number of steps gradually    // increases. Furthermore, note that we are guaranteed to visit all nodes    // that are k steps from n before any nodes that are k + 1 steps    // from n.    queue<pair<long long, int>> q;    // Visited set -- holds numbers that have already been encountered.    // Numbers enter this set when they are first encountered by the    // algorithm. We do not want to revisit these numbers because we only    // care about the minimum number of steps to visit them.    unordered_set<long long> visited;    // The queue starts with n at 0 steps, and n is visited to begin with.    q.push({n, 0});    visited.insert(n);    // BFS loop.    while (!q.empty()) {      // Retrieve the next x and steps from the queue.      long long x;      int steps;      tie(x, steps) = q.front();  // See std::tie      q.pop();      // We solved the problem -- return the number of steps.      if (x == 1) return steps;      if (x % 2 == 0) {        // If x is even, the only node connected to it will be x/2        if (visited.find(x / 2) == visited.end()) {          // Make sure to record that x/2 is visited.          visited.insert(x / 2);          // Add x/2 with one more step than x.          q.push({x / 2, steps + 1});        }      } else {        // Same as above, except we now have x + 1 and x - 1        if (visited.find(x + 1) == visited.end()) {          visited.insert(x + 1);          q.push({x + 1, steps + 1});        }        if (visited.find(x - 1) == visited.end()) {          visited.insert(x - 1);          q.push({x - 1, steps + 1});        }      }    }    // If for some reason no solution was found.    return -1;  }};

### Alternate Approaches ¶

If you look at the discussions on the original LeetCode problem, you will see references to dynamic programming (DP) and recursion.