grid47 Exploring patterns and algorithms
Aug 23, 2024
5 min read
Solution to LeetCode 754: Reach a Number Problem
You are standing at position 0 on an infinite number line. There is a destination at position target. You can make some number of moves where, on the i-th move, you take exactly i steps either left or right. The goal is to determine the minimum number of moves required to reach the target.
Problem
Approach
Steps
Complexity
Input: You are given an integer target, representing the position on the number line that you want to reach.
Example: target = 4
Constraints:
• -10^9 <= target <= 10^9
• target != 0
Output: Return the minimum number of moves required to reach the target, or -1 if it's not possible.
Example: Output: 3
Constraints:
• The solution must be efficient for large inputs
Goal: To determine the minimum number of moves to reach the target by analyzing the required steps.
Steps:
• 1. Convert the target to its absolute value to simplify the problem.
• 2. Calculate the smallest number of moves needed to reach or exceed the target distance.
• 3. Check if the difference between the total steps taken and the target is even, in which case the solution is valid.
• 4. Return the number of moves required to reach the target.
Goal: This problem requires efficient computation of steps and consideration of large input values.
Steps:
• The solution must work within the constraints of large numbers, with target ranging from -10^9 to 10^9.
Assumptions:
• The target value is always non-zero.
• Input: Example 1: target = 4
• Explanation: Start at position 0. Move 1 step to the right, then 2 steps to the left, and finally 3 steps to the right to reach position 4 in 3 moves.
• Input: Example 2: target = 5
• Explanation: Start at position 0. Move 1 step to the right, then 2 steps to the right, and finally 3 steps to the right to reach position 5 in 3 moves.
Approach: The goal is to compute the smallest number of moves that allow us to reach the target position, by carefully analyzing the total steps taken at each stage.
Observations:
• We need to compute the minimum number of moves efficiently for large target values.
• We can calculate the smallest n such that the sum of the first n natural numbers exceeds or equals the target distance.
Steps:
• 1. Use the formula for the sum of the first n natural numbers to compute the total number of steps.
• 2. If the total steps is exactly equal to the target, return n.
• 3. If not, calculate the difference and check if it's possible to adjust the steps to reach the target by flipping some of the steps' directions.
Empty Inputs:
• There are no empty inputs as the target is always non-zero.
Large Inputs:
• Ensure that the solution handles cases where target is at the extreme ends of the range (-10^9 to 10^9).
Special Values:
• No special cases besides the minimum and maximum target values.
Constraints:
• The solution must handle large target values efficiently.
intreachNumber(int target) {
target = abs(target);
longlong n = ceil((-1.0+ sqrt(1+8.0*target)) /2);
longlong sum = n * (n +1) /2;
if (sum == target) return n;
longlong res = sum - target;
if((res &1) ==0) return n;
elsereturn n + ((n&1)?2:1);
}
1 : Function Definition
intreachNumber(int target) {
Defines the function 'reachNumber', which takes the target number and calculates the minimum steps required to reach it starting from 0.
2 : Absolute Value
target = abs(target);
Ensures the target number is positive by converting it to its absolute value, as the problem is symmetric with respect to the direction of movement.
3 : Mathematical Calculation
longlong n = ceil((-1.0+ sqrt(1+8.0*target)) /2);
Calculates the smallest integer 'n' such that the sum of the first 'n' integers is greater than or equal to the target using the formula derived from the quadratic equation.
4 : Sum Calculation
longlong sum = n * (n +1) /2;
Calculates the sum of the first 'n' integers (i.e., 1 + 2 + ... + n). This represents the total distance after 'n' steps.
5 : Exact Match Check
if (sum == target) return n;
Checks if the sum of the first 'n' integers equals the target. If they are equal, the function returns 'n' because the target is reached exactly after 'n' steps.
6 : Difference Calculation
longlong res = sum - target;
Calculates the difference between the sum of the first 'n' integers and the target, to determine how far the sum overshoots the target.
7 : Parity Check
if((res &1) ==0) return n;
Checks if the difference 'res' is even. If it is, then it's possible to adjust the sum to match the target by flipping the direction of one of the steps, so the function returns 'n'.
8 : Adjust for Odd Difference
elsereturn n + ((n&1)?2:1);
If the difference 'res' is odd, it means an additional step is needed to make the sum match the target. If 'n' is odd, add 2 steps; otherwise, add 1 step.
Best Case: O(1)
Average Case: O(1)
Worst Case: O(1)
Description: The solution involves simple mathematical calculations, which are constant time operations.
Best Case: O(1)
Worst Case: O(1)
Description: The space complexity is constant as only a few variables are used to store intermediate values.
