Problem Description

Alice has n balloons tied in a row and each balloon is colored some color. This sequence of colors is given as a string colors where colors[i] is the color of the ith balloon. Alice desires the string of balloons to be "colorful," which means no two adjacent balloons should share the same color. To help achieve this, Bob can remove balloons, but it takes time. The time it takes to remove each balloon is specified in an integer array neededTime, where neededTime[i] represents the seconds required to remove the ith balloon.

The goal is to find the minimum amount of time Bob needs to spend to turn the string of balloons into one where no two consecutive balloons are of the same color.

Intuition

Given that we want to make the rope of balloons colorful with the least amount of time needed, we can intuit that when we encounter a sequence of balloons of the same color, we should remove all but one of them. Instead of randomly choosing which balloons to remove, we should aim to remove the balloons with the least time cost. The optimal balloon to keep is the one with the highest neededTime value because removing it would contribute the most to the total time.

Throughout the string of balloons, we iteratively look for sequences of consecutive balloons with the same color. For each of these sequences, we calculate the total time needed to remove all balloons (s) and the maximum time needed to remove a single balloon within that sequence (mx). To make the sequence colorful, we will keep the balloon with the maximum neededTime (the most costly to remove) and remove all others. Hence, we subtract the mx from the total s and add this to our answer (ans).

The algorithm is efficient because it goes through the string once, using a while loop, checking each sequence of same-colored balloons and calculating the time cost on the fly. It has a linear time complexity relative to the length of the colors string, making it suitable even for a large number of balloons.

Learn more about Greedy and Dynamic Programming patterns.

Solution Approach

The solution implementation follows a greedy approach, which entails iterating through the colors string and grouping balloons with the same color. During this grouping, the algorithm also keeps track of the sum of neededTimes of these balloons (s) as well as the maximum neededTime for a single balloon in the group (mx). To do this, two pointers are used. The first pointer i marks the beginning of a group of same-colored balloons, and a second pointer j is used to find the end of this group.

The approach can be broken down into the following steps:

1. Initialize ans as zero, which will store the minimum total time required to make the rope colorful.

2. Loop through the string colors using the index i, which serves as the starting point of each group of same-colored balloons:

   • Within this loop, start a nested loop with index j, beginning at the same position as i.

   • Initialize s as zero, which will hold the sum of the neededTime for all balloons currently in the sequence.

   • Initialize mx as zero, which will keep track of the maximum neededTime of a balloon in the current sequence.

   • For each balloon that has the same color as the one at position i (i.e., colors[j] == colors[i]), accumulate its neededTime in s and update mx if the current balloon's neededTime[j] is greater than the current mx.

   • Increment j for as long as it does not exceed the length of the colors string and the color of the jth balloon is the same as the ith balloon.

   • If the sequence length (j - i) is greater than 1, it means there are duplicate balloons. In this case, subtract the maximum neededTime (mx) from the sum of neededTime (s) to get the time required to make the current sequence colorful and add this value to ans.

   • Set i to the value of j to begin processing the next sequence.

3. Once the loop is complete, all the neededTime for removing necessary balloons to avoid consecutive same colors has been added to ans. Return ans as the answer.

It is important to note that because this approach always chooses the most expensive balloon to keep in each consecutive sequence of same-colored balloons, the accumulated removal time is as small as possible. Essentially, you're selecting which balloons to remove based on the criteria of the lowest time cost, which aligns with our greedy strategy.

Example Walkthrough

Let's consider an example where colors = "abcccbad" and neededTime = [1, 2, 3, 4, 5, 6, 7, 8]. Our goal is to make the sequence of balloons colorful by removing consecutive balloons of the same color in the least total time possible.

Using the solution approach outlined earlier, we can work through this step by step:

1. We initialize ans as zero. This will keep track of the minimum total time needed.

2. We will loop through the string colors using index i. For simplicity, we consider each character as a group of same-colored balloons, even if it's a group of one.

   • Start with i = 0; this is the first balloon with color 'a'.

   • There are no consecutive balloons with the same color, so i is incremented to 1.

   • Now at i = 1; this balloon has color 'b'.

   • There are no consecutive balloons with the same color, so i is incremented to 2.

   • At i = 2; this balloon has the color 'c'.

   • We find there is a sequence of balloons with the same color 'c' starting from index 2 to 4.

   • We set j = 2 and start the nested loop.

   • Initialize s = 0 and mx = 0.

   • We go through the sequence, updating s and mx as follows:

     • For j = 2, colors[j] = 'c', so we update s += neededTime[2] (s becomes 3) and mx = max(mx, neededTime[2]) (mx becomes 3).
     • For j = 3, colors[j] = 'c', s += neededTime[3] (s becomes 7) and mx = max(mx, neededTime[3]) (mx remains 3).
     • For j = 4, colors[j] = 'c', s += neededTime[4] (s becomes 12) and mx = max(mx, neededTime[4]) (mx becomes 5).

   • Since j - i (4 - 2) is greater than 1, we have duplicate balloons and we calculate the time to make the sequence colorful: ans += s - mx (ans becomes 12 - 5 = 7).

   • We set i to 5 as the next starting point.

   • Continuing with i = 5, there are no more consecutive balloons with the same color, so nothing gets added to ans.

3. Moving on, loop continues, but no more sequences of consecutive colors are found.

The final ans represents the minimum total time to remove all the consecutive same-colored balloons and is equal to 7 in this example.

Solution Implementation

Time and Space Complexity

Time Complexity

The provided code consists of a single while loop that iterates through the string colors. Inside the loop, there's another while loop that also iterates through the string but only when it finds consecutive characters that are the same. Despite this nested structure, the inner loop does not lead to a quadratic time complexity, because each character in the string colors is visited exactly once. After the inner loop, the index i jumps to the position j, skipping all the characters that have already been accounted for.

Thus, each character in the string causes an iteration of the outer loop, and the inner loop only runs for characters of a sequence of identical colors once. This leads to a linear time complexity with respect to the length of the string colors.

Hence, the time complexity is O(n), where n is the length of colors.

Space Complexity

The code uses a constant amount of extra space: variables ans, i, s, mx, and j. When iterating over the input, no additional space that scales with the size of the input is used. The input itself (colors and neededTime) is not counted towards the space complexity.

Therefore, the space complexity is O(1) since it does not allocate any additional space that grows with the input size.

Learn more about how to find time and space complexity quickly using problem constraints.