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Code Implementation

Hiking Backpack Packer Implementation

Below is the implementation of the hiking backpack packer:

solution.js
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// Recursive Backtracking Approach
function canPackExactly(gearWeights, capacity) {
  return subsetSum(gearWeights, capacity, 0);
}
 
function subsetSum(gearWeights, capacity, index) {
  // Base cases
  if (capacity === 0) {
    return true;
  }
  if (capacity < 0 || index >= gearWeights.length) {
    return false;
  }
  
  // Recursive cases
  
  // Exclude the current item
  if (subsetSum(gearWeights, capacity, index + 1)) {
    return true;
  }
  
  // Include the current item
  return subsetSum(gearWeights, capacity - gearWeights[index], index + 1);
}
 
// Dynamic Programming Approach
function canPackExactlyDP(gearWeights, capacity) {
  const n = gearWeights.length;
  
  // Create a 2D DP table
  const dp = Array(n + 1).fill().map(() => Array(capacity + 1).fill(false));
  
  // Base case: empty subset sums to 0
  for (let i = 0; i <= n; i++) {
    dp[i][0] = true;
  }
  
  // Fill the DP table
  for (let i = 1; i <= n; i++) {
    for (let j = 1; j <= capacity; j++) {
      // If current item is too heavy, exclude it
      if (gearWeights[i - 1] > j) {
        dp[i][j] = dp[i - 1][j];
      } else {
        // Either exclude or include the current item
        dp[i][j] = dp[i - 1][j] || dp[i - 1][j - gearWeights[i - 1]];
      }
    }
  }
  
  return dp[n][capacity];
}
 
// Space-Optimized Dynamic Programming Approach
function canPackExactlyOptimized(gearWeights, capacity) {
  // Create a 1D DP array
  const dp = Array(capacity + 1).fill(false);
  
  // Base case: empty subset sums to 0
  dp[0] = true;
  
  // Fill the DP array
  for (let i = 0; i < gearWeights.length; i++) {
    for (let j = capacity; j >= gearWeights[i]; j--) {
      // Either exclude or include the current item
      dp[j] = dp[j] || dp[j - gearWeights[i]];
    }
  }
  
  return dp[capacity];
}
 
// Test cases
const gearWeights1 = [2, 3, 7, 8, 10];
const capacity1 = 11;
console.log(`Gear weights: [${gearWeights1}], Capacity: ${capacity1}`);
console.log(`Recursive Approach: ${canPackExactly(gearWeights1, capacity1)}`);
console.log(`DP Approach: ${canPackExactlyDP(gearWeights1, capacity1)}`);
console.log(`Optimized DP Approach: ${canPackExactlyOptimized(gearWeights1, capacity1)}`);
 
const gearWeights2 = [1, 3, 5];
const capacity2 = 10;
console.log(`\nGear weights: [${gearWeights2}], Capacity: ${capacity2}`);
console.log(`Recursive Approach: ${canPackExactly(gearWeights2, capacity2)}`);
console.log(`DP Approach: ${canPackExactlyDP(gearWeights2, capacity2)}`);
console.log(`Optimized DP Approach: ${canPackExactlyOptimized(gearWeights2, capacity2)}`);
 
const gearWeights3 = [4, 2, 3, 5, 6];
const capacity3 = 9;
console.log(`\nGear weights: [${gearWeights3}], Capacity: ${capacity3}`);
console.log(`Recursive Approach: ${canPackExactly(gearWeights3, capacity3)}`);
console.log(`DP Approach: ${canPackExactlyDP(gearWeights3, capacity3)}`);
console.log(`Optimized DP Approach: ${canPackExactlyOptimized(gearWeights3, capacity3)}`);

Step-by-Step Explanation

Let's break down the implementation:

  1. Understand the Problem: We need to determine if there's a subset of gear weights that sum exactly to the backpack's capacity.
  2. Identify the Recursive Structure: Recognize that this is a classic Selection Pattern problem where we recursively decide whether to include or exclude each item.
  3. Define the Base Cases: The base cases are when the remaining capacity is 0 (success), or when it's negative or we've processed all items (failure).
  4. Implement the Recursive Solution: Write a function that handles the base cases and recursively explores both options (include/exclude) for each item.
  5. Optimize with Dynamic Programming: Implement a dynamic programming solution to avoid redundant calculations and improve efficiency.
  6. Further Optimize Space Complexity: Implement a space-optimized version of the dynamic programming solution using a 1D array instead of a 2D table.
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