#1449

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Form Largest Integer With Digits That Add up to Target

Maximize the integer you can paint with given digit costs under a target sum, using dynamic programming to optimize the solution.

array dynamic programming

Problem Statement

You are given an array cost of integers where cost[i] represents the cost to paint the digit i+1 (1-indexed). You are also given a target integer. Your task is to return the largest integer that can be painted under the condition that the sum of the costs of the digits painted must not exceed the target value.

Return the largest possible integer as a string. If no integer can be formed within the target cost, return "0". The result can be very large, so return it as a string. The answer can be optimized by using dynamic programming to find the maximum number of digits that can be painted given the total cost.

Examples

Example 1

Input: cost = [4,3,2,5,6,7,2,5,5], target = 9

Output: "7772"

The cost to paint the digit '7' is 2, and the digit '2' is 3. Then cost("7772") = 2 3+ 3 1 = 9. You could also paint "977", but "7772" is the largest number. Digit cost 1 -> 4 2 -> 3 3 -> 2 4 -> 5 5 -> 6 6 -> 7 7 -> 2 8 -> 5 9 -> 5

Example 2

Input: cost = [7,6,5,5,5,6,8,7,8], target = 12

Output: "85"

The cost to paint the digit '8' is 7, and the digit '5' is 5. Then cost("85") = 7 + 5 = 12.

Example 3

Input: cost = [2,4,6,2,4,6,4,4,4], target = 5

Output: "0"

It is impossible to paint any integer with total cost equal to target.

Constraints

cost.length == 9
1 <= cost[i], target <= 5000

Solution Approach

Dynamic Programming Approach

The problem boils down to finding the most digits that can be painted for a given cost. Using dynamic programming, we maintain a dp array where each index represents the maximum number of digits that can be formed for a specific cost. Starting from the maximum target, we transition through the digits using their respective costs and update the possible integer value accordingly.

State Transition Maximization

By iterating through each possible digit and its associated cost, we update the possible states, ensuring we choose the largest digits first. We utilize a greedy strategy within the DP approach, maximizing each state by transitioning to higher digits when possible, making sure the final integer is as large as possible.

Edge Case Handling

If no integer can be formed within the target cost, the algorithm should return '0'. Special cases like when target is too small for any digit should be handled correctly, returning a result without further computation. These cases often appear early in the DP array transitions.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity depends on the number of digits (9 in this case) and the target cost. The space complexity is dependent on the size of the dp array, which is proportional to the target value. Given the constraints, both complexities should scale reasonably well, especially with optimized state transitions.

What Interviewers Usually Probe

Look for understanding of dynamic programming and its efficient use in state transitions.
Check for clear explanation of greedy strategies within the DP solution to maximize the integer value.
Gauge familiarity with handling edge cases where no valid number can be formed.

Common Pitfalls or Variants

Common pitfalls

Not properly updating the DP array for each state transition can lead to incorrect results.
Overcomplicating the solution by using inefficient data structures or failing to optimize the transitions.
Ignoring edge cases where it's impossible to form any valid integer, resulting in incorrect output.

Follow-up variants

How would you modify the solution if there were more digits or different constraints?
Can you optimize the space complexity further by reducing the size of the DP array?
What changes would you make if the costs were dynamic or changed during computation?

FAQ

What is the primary approach to solving the 'Form Largest Integer With Digits That Add up to Target' problem?

The problem is typically solved using dynamic programming with state transitions to maximize the number of digits within a target cost.

How can I optimize the time complexity of this problem?

The time complexity can be optimized by ensuring efficient state transitions within the DP array, reducing unnecessary recalculations.

What happens if no valid integer can be formed under the target cost?

In that case, the function should return '0', indicating that it is impossible to paint any integer under the given constraints.

How do I handle the edge case where the target cost is too small?

For edge cases where the target cost is too small to paint any digit, the result should immediately be '0'.

What would happen if the array length or the cost values were larger?

The dynamic programming approach still applies, but larger inputs may require optimizations in both time and space complexity to handle larger target values.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def largestNumber(self, cost: List[int], target: int) -> str:
        f = [[-inf] * (target + 1) for _ in range(10)]
        f[0][0] = 0
        g = [[0] * (target + 1) for _ in range(10)]
        for i, c in enumerate(cost, 1):
            for j in range(target + 1):
                if j < c or f[i][j - c] + 1 < f[i - 1][j]:
                    f[i][j] = f[i - 1][j]
                    g[i][j] = j
                else:
                    f[i][j] = f[i][j - c] + 1
                    g[i][j] = j - c
        if f[9][target] < 0:
            return "0"
        ans = []
        i, j = 9, target
        while i:
            if j == g[i][j]:
                i -= 1
            else:
                ans.append(str(i))
                j = g[i][j]
        return "".join(ans)

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