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+# Subset Sum Problem
+#
+# The Subset Sum problem determines whether there exists a subset of a given set
+# of positive integers that sums to a target value. This is a classic NP-complete
+# problem solved using dynamic programming.
+#
+# Time Complexity: O(n * sum) where n = number of elements, sum = target sum
+# Space Complexity: O(n * sum) for DP table, O(sum) for optimized version
+#
+# Applications:
+# - Partition problems in computer science
+# - Knapsack problem variations
+# - Resource allocation and optimization
+# - Cryptography and number theory
+# - Game theory and decision making
+
+# Basic DP solution for Subset Sum Problem
+subset_sum <- function(arr, target) {
+  #' Check if there exists a subset that sums to the target value
+  #' @param arr: Numeric vector of positive integers
+  #' @param target: Target sum value
+  #' @return: Boolean indicating if subset exists, along with DP table
+  
+  n <- length(arr)
+  
+  # Handle edge cases
+  if (n == 0) {
+    return(list(
+      exists = (target == 0),
+      dp_table = matrix(FALSE, nrow = 1, ncol = 1),
+      subset = c()
+    ))
+  }
+  
+  if (target == 0) {
+    return(list(
+      exists = TRUE,
+      dp_table = matrix(TRUE, nrow = n + 1, ncol = 1),
+      subset = c()
+    ))
+  }
+  
+  # Create DP table: dp[i, j] = TRUE if sum j can be achieved using first i elements
+  dp <- matrix(FALSE, nrow = n + 1, ncol = target + 1)
+  
+  # Base case: sum 0 can always be achieved with empty subset
+  for (i in 1:(n + 1)) {
+    dp[i, 1] <- TRUE
+  }
+  
+  # Fill DP table
+  for (i in 2:(n + 1)) {
+    for (j in 1:(target + 1)) {
+      current_sum <- j - 1  # Convert to 0-based indexing
+      
+      # Don't include current element
+      dp[i, j] <- dp[i - 1, j]
+      
+      # Include current element (if it doesn't exceed current sum)
+      if (arr[i - 1] <= current_sum) {
+        dp[i, j] <- dp[i, j] || dp[i - 1, j - arr[i - 1]]
+      }
+    }
+  }
+  
+  # Backtrack to find one possible subset
+  subset <- c()
+  if (dp[n + 1, target + 1]) {
+    i <- n + 1
+    j <- target + 1
+    
+    while (i > 1 && j > 1) {
+      # If current sum was achieved without including arr[i-1]
+      if (dp[i - 1, j]) {
+        i <- i - 1
+      } else {
+        # Current element was included
+        subset <- c(arr[i - 1], subset)
+        j <- j - arr[i - 1]
+        i <- i - 1
+      }
+    }
+  }
+  
+  return(list(
+    exists = dp[n + 1, target + 1],
+    dp_table = dp,
+    subset = subset
+  ))
+}
+
+# Space-optimized version using only 1D array
+subset_sum_optimized <- function(arr, target) {
+  #' Space optimized subset sum using 1D array
+  #' @param arr: Numeric vector of positive integers
+  #' @param target: Target sum value
+  #' @return: Boolean indicating if subset exists
+  
+  n <- length(arr)
+  
+  if (n == 0) return(target == 0)
+  if (target == 0) return(TRUE)
+  
+  dp <- rep(FALSE, target + 1)
+  dp[1] <- TRUE  # sum 0 is always possible
+  
+  for (i in 1:n) {
+    # Traverse from right to left to avoid overwriting needed values
+    for (j in target:1) {
+      if (arr[i] <= j) {
+        dp[j + 1] <- dp[j + 1] || dp[j - arr[i] + 1]
+      }
+    }
+  }
+  
+  return(dp[target + 1])
+}
+
+# Function to find all subsets that sum to target
+find_all_subsets <- function(arr, target) {
+  #' Find all subsets that sum to the target value
+  #' @param arr: Numeric vector of positive integers
+  #' @param target: Target sum value
+  #' @return: List of subsets (each subset is a numeric vector) that sum to target
+  
+  n <- length(arr)
+  results <- list()
+  
+  # Helper recursive function
+  find_subsets_rec <- function(idx, current_subset, current_sum) {
+    if (current_sum == target) {
+      results <<- c(results, list(current_subset))
+      return()
+    }
+    if (idx > n || current_sum > target) {
+      return()
+    }
+    # Include arr[idx]
+    find_subsets_rec(idx + 1, c(current_subset, arr[idx]), current_sum + arr[idx])
+    # Exclude arr[idx]
+    find_subsets_rec(idx + 1, current_subset, current_sum)
+  }
+  
+  find_subsets_rec(1, c(), 0)
+  return(results)
+}
+
+# Helper function to print DP table
+print_subset_sum_dp <- function(dp_table, arr, target) {
+  cat("DP Table for Subset Sum Problem:\n")
+  cat("Array:", paste(arr, collapse = ", "), "\n")
+  cat("Target Sum:", target, "\n\n")
+  
+  # Print column headers (sums)
+  cat("        ")
+  cat(paste(sprintf("%4d", 0:target), collapse = " "))
+  cat("\n")
+  cat(paste(rep("-", 8 + 5 * (target + 1)), collapse = ""), "\n")
+  
+  for (i in 1:nrow(dp_table)) {
+    if (i == 1) {
+      cat("Empty  | ")
+    } else {
+      cat(sprintf("Elem%2d| ", i - 1))
+    }
+    
+    for (j in 1:ncol(dp_table)) {
+      cat(sprintf("%4s", ifelse(dp_table[i, j], " T", " F")))
+    }
+    cat("\n")
+  }
+  cat("\n")
+}
+
+# ===========================
+# Example Usage & Testing
+# ===========================
+cat("=== Subset Sum Problem (Dynamic Programming) ===\n\n")
+
+# Test 1: Basic Example
+arr1 <- c(3, 34, 4, 12, 5, 2)
+target1 <- 9
+cat("Test 1: Basic Example\n")
+cat("Array:", paste(arr1, collapse = ", "), "\n")
+cat("Target Sum:", target1, "\n\n")
+
+result1 <- subset_sum(arr1, target1)
+print_subset_sum_dp(result1$dp_table, arr1, target1)
+cat("Subset exists:", result1$exists, "\n")
+if (result1$exists) {
+  cat("One possible subset:", paste(result1$subset, collapse = ", "), "\n")
+  cat("Sum verification:", sum(result1$subset), "\n")
+}
+cat("\n")
+
+# Test 2: Optimized Version
+cat("Test 2: Space Optimized Version\n")
+exists_opt <- subset_sum_optimized(arr1, target1)
+cat("Subset exists (Optimized):", exists_opt, "\n")
+cat("Verification: Both methods match:", result1$exists == exists_opt, "\n\n")
+
+# Test 3: No Solution Case
+cat("Test 3: No Solution Case\n")
+arr3 <- c(3, 34, 4, 12, 5, 2)
+target3 <- 30
+cat("Array:", paste(arr3, collapse = ", "), "\n")
+cat("Target Sum:", target3, "\n")
+
+result3 <- subset_sum(arr3, target3)
+cat("Subset exists:", result3$exists, "\n\n")
+
+# Test 4: Multiple Solutions
+cat("Test 4: Multiple Solutions\n")
+arr4 <- c(1, 2, 3, 4, 5)
+target4 <- 6
+cat("Array:", paste(arr4, collapse = ", "), "\n")
+cat("Target Sum:", target4, "\n")
+
+result4 <- subset_sum(arr4, target4)
+cat("Subset exists:", result4$exists, "\n")
+if (result4$exists) {
+  cat("One possible subset:", paste(result4$subset, collapse = ", "), "\n")
+  
+  # Find all possible subsets
+  all_subsets <- find_all_subsets(arr4, target4)
+  cat("Total number of subsets:", length(all_subsets), "\n")
+  for (i in seq_along(all_subsets)) {
+    cat("Subset", i, ":", paste(all_subsets[[i]], collapse = ", "), 
+        "(sum =", sum(all_subsets[[i]]), ")\n")
+  }
+}
+cat("\n")
+
+# Test 5: Edge Cases
+cat("Test 5: Edge Cases\n")
+cat("Empty array, target 0:", subset_sum(c(), 0)$exists, "\n")
+cat("Empty array, target 5:", subset_sum(c(), 5)$exists, "\n")
+cat("Array [1,2,3], target 0:", subset_sum(c(1, 2, 3), 0)$exists, "\n")
+cat("Array [5], target 5:", subset_sum(c(5), 5)$exists, "\n")
+cat("Array [5], target 3:", subset_sum(c(5), 3)$exists, "\n\n")
+
+# Test 6: Larger Dataset
+cat("Test 6: Larger Dataset (n=15)\n")
+set.seed(42)
+arr_large <- sample(1:20, 15)
+target_large <- 50
+cat("Array:", paste(arr_large, collapse = ", "), "\n")
+cat("Target Sum:", target_large, "\n")
+
+start_time <- Sys.time()
+result_large <- subset_sum(arr_large, target_large)
+dp_time <- as.numeric(Sys.time() - start_time, units = "secs")
+
+start_time <- Sys.time()
+exists_large_opt <- subset_sum_optimized(arr_large, target_large)
+opt_time <- as.numeric(Sys.time() - start_time, units = "secs")
+
+cat("Subset exists:", result_large$exists, "\n")
+cat("DP method time:", sprintf("%.4f sec", dp_time), "\n")
+cat("Optimized method time:", sprintf("%.4f sec", opt_time), "\n")
+cat("Results match:", result_large$exists == exists_large_opt, "\n")
+
+if (result_large$exists) {
+  cat("One possible subset:", paste(result_large$subset, collapse = ", "), "\n")
+  cat("Sum verification:", sum(result_large$subset), "\n")
+}
+cat("\n")
+
+# Test 7: Real-world Example - Budget Allocation
+cat("Test 7: Real-world Example - Budget Allocation\n")
+project_costs <- c(10, 20, 30, 40, 50, 60, 70, 80, 90, 100)
+budget <- 150
+cat("Project costs:", paste(project_costs, collapse = ", "), "\n")
+cat("Available budget:", budget, "\n")
+
+budget_result <- subset_sum(project_costs, budget)
+cat("Exact budget allocation possible:", budget_result$exists, "\n")
+
+if (budget_result$exists) {
+  selected_projects <- budget_result$subset
+  cat("Selected projects (costs):", paste(selected_projects, collapse = ", "), "\n")
+  cat("Total cost:", sum(selected_projects), "\n")
+  cat("Remaining budget:", budget - sum(selected_projects), "\n")
+} else {
+  # Find closest possible sum (≤ budget) in a single pass
+  closest_sum <- max_subset_sum_leq(project_costs, budget)
+  cat("Closest possible sum:", closest_sum, "\n")
+}