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/**
 * Form the largest integer with digits that add up to target.
 * Dynamic Programming approach.
 *
 * @param {number[]} cost - The cost of printing each digit (1-indexed)
 * @param {number} target - The target budget
 * @return {string} - The largest possible integer
 */
function largestNumber(cost, target) {
  // Initialize DP array
  const dp = new Array(target + 1).fill(-1);
  dp[0] = 0;
 
  // Fill DP array
  for (let i = 1; i <= target; i++) {
    for (let j = 1; j <= 9; j++) {
      if (i >= cost[j - 1] && dp[i - cost[j - 1]] !== -1) {
        dp[i] = Math.max(dp[i], dp[i - cost[j - 1]] + 1);
      }
    }
  }
 
  // Check if it's possible to form any integer
  if (dp[target] === -1) {
    return "";
  }
 
  // Construct the largest possible integer
  let result = "";
  let remaining = target;
 
  for (let pos = 1; pos <= dp[target]; pos++) {
    for (let digit = 9; digit >= 1; digit--) {
      if (remaining >= cost[digit - 1] && dp[remaining - cost[digit - 1]] === dp[target] - pos) {
        result += digit;
        remaining -= cost[digit - 1];
        break;
      }
    }
  }
 
  return result;
}
 
/**
 * Form the largest integer with digits that add up to target.
 * Greedy approach.
 *
 * @param {number[]} cost - The cost of printing each digit (1-indexed)
 * @param {number} target - The target budget
 * @return {string} - The largest possible integer
 */
function largestNumberGreedy(cost, target) {
  // Find the digit with the minimum cost
  let minCost = Infinity;
  let minDigit = -1;
 
  for (let i = 0; i < 9; i++) {
    if (cost[i] < minCost) {
      minCost = cost[i];
      minDigit = i + 1;
    }
  }
 
  // Check if it's possible to form any integer
  if (target < minCost) {
    return "";
  }
 
  // Construct the largest possible integer
  let result = "";
 
  // Try to add as many high digits as possible
  for (let digit = 9; digit >= 1; digit--) {
    // Calculate how many of this digit we can add
    while (target >= cost[digit - 1] && (target - cost[digit - 1]) >= 0) {
      result += digit;
      target -= cost[digit - 1];
    }
  }
 
  // If we couldn't form any integer, return an empty string
  if (result === "") {
    return "";
  }
 
  return result;
}
 
/**
 * Form the largest integer with digits that add up to target.
 * Recursive approach with memoization.
 *
 * @param {number[]} cost - The cost of printing each digit (1-indexed)
 * @param {number} target - The target budget
 * @return {string} - The largest possible integer
 */
function largestNumberRecursive(cost, target) {
  // Memoization table
  const memo = new Map();
 
  // Recursive function to find the maximum number of digits
  function maxDigits(budget) {
    // Base cases
    if (budget < 0) return -1;
    if (budget === 0) return 0;
 
    // Check if we've already computed this
    if (memo.has(budget)) {
      return memo.get(budget);
    }
 
    // Try each digit
    let result = -1;
 
    for (let digit = 1; digit <= 9; digit++) {
      const subResult = maxDigits(budget - cost[digit - 1]);
 
      if (subResult !== -1) {
        result = Math.max(result, subResult + 1);
      }
    }
 
    // Memoize and return
    memo.set(budget, result);
    return result;
  }
 
  // Check if it's possible to form any integer
  const maxLen = maxDigits(target);
 
  if (maxLen === -1) {
    return "";
  }
 
  // Construct the largest possible integer
  let result = "";
  let remaining = target;
 
  for (let pos = 1; pos <= maxLen; pos++) {
    for (let digit = 9; digit >= 1; digit--) {
      if (remaining >= cost[digit - 1] && maxDigits(remaining - cost[digit - 1]) === maxLen - pos) {
        result += digit;
        remaining -= cost[digit - 1];
        break;
      }
    }
  }
 
  return result;
}
 
// Test cases
console.log(largestNumber([4, 3, 2, 5, 6, 7, 2, 5, 5], 9));  // "7721"
console.log(largestNumber([7, 6, 5, 5, 5, 6, 8, 7, 8], 12));  // "85"
console.log(largestNumber([2, 4, 6, 2, 4, 6, 4, 4, 4], 5));  // "4"
console.log(largestNumber([6, 10, 15, 40, 40, 40, 40, 40, 40], 47));  // ""
 
console.log(largestNumberGreedy([4, 3, 2, 5, 6, 7, 2, 5, 5], 9));  // "7721"
console.log(largestNumberGreedy([7, 6, 5, 5, 5, 6, 8, 7, 8], 12));  // "85"
console.log(largestNumberGreedy([2, 4, 6, 2, 4, 6, 4, 4, 4], 5));  // "4"
console.log(largestNumberGreedy([6, 10, 15, 40, 40, 40, 40, 40, 40], 47));  // ""
 
console.log(largestNumberRecursive([4, 3, 2, 5, 6, 7, 2, 5, 5], 9));  // "7721"
console.log(largestNumberRecursive([7, 6, 5, 5, 5, 6, 8, 7, 8], 12));  // "85"
console.log(largestNumberRecursive([2, 4, 6, 2, 4, 6, 4, 4, 4], 5));  // "4"
console.log(largestNumberRecursive([6, 10, 15, 40, 40, 40, 40, 40, 40], 47));  // ""