Implemented the Minimum Path Sum algorithm in R.

dynamic_programming/minimum_path_sum.r

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+# Minimum Path Sum Problem
+#
+# The Minimum Path Sum problem finds the minimum sum path from the top-left corner
+# to the bottom-right corner of a grid, moving only right or down at each step.
+# This is a classic dynamic programming problem that appears in various forms.
+#
+# Time Complexity: O(m * n) where m = number of rows, n = number of columns
+# Space Complexity: O(m * n) for DP table, O(min(m, n)) for optimized version
+#
+# Applications:
+# - Grid-based pathfinding algorithms
+# - Resource optimization in 2D grids
+# - Game development (pathfinding with costs)
+# - Network routing optimization
+# - Cost minimization in transportation
+
+# Basic DP solution for Minimum Path Sum
+minimum_path_sum <- function(grid) {
+  #' Find the minimum path sum from top-left to bottom-right corner
+  #' @param grid: 2D matrix of non-negative integers
+  #' @return: List containing minimum sum, path, and DP table
+
+  m <- nrow(grid)
+  n <- ncol(grid)
+
+  # Handle edge case
+  if (m == 0 || n == 0) {
+    return(list(
+      min_sum = 0,
+      path = c(),
+      dp_table = matrix(0, nrow = 1, ncol = 1)
+    ))
+  }
+
+  # Create DP table: dp[i, j] = minimum sum to reach position (i, j)
+  dp <- matrix(0, nrow = m, ncol = n)
+
+  # Initialize first row and column
+  dp[1, 1] <- grid[1, 1]
+
+  # Fill first row (can only move right)
+  if (n > 1) {
+    for (j in 2:n) {
+      dp[1, j] <- dp[1, j - 1] + grid[1, j]
+    }
+  }
+
+  # Fill first column (can only move down)
+  if (m > 1) {
+    for (i in 2:m) {
+      dp[i, 1] <- dp[i - 1, 1] + grid[i, 1]
+    }
+  }
+
+  # Fill remaining cells
+  if (m > 1 && n > 1) {
+    for (i in 2:m) {
+      for (j in 2:n) {
+        dp[i, j] <- min(dp[i - 1, j], dp[i, j - 1]) + grid[i, j]
+      }
+    }
+  }
+
+  # Backtrack to find the path
+  path <- list()
+  i <- m
+  j <- n
+
+  while (i > 1 || j > 1) {
+    path <- c(list(c(i, j)), path)
+
+    if (i == 1) {
+      # Can only move left
+      j <- j - 1
+    } else if (j == 1) {
+      # Can only move up
+      i <- i - 1
+    } else {
+      # Choose direction with minimum sum
+      if (dp[i - 1, j] < dp[i, j - 1]) {
+        i <- i - 1
+      } else {
+        j <- j - 1
+      }
+    }
+  }
+  path <- c(list(c(1, 1)), path)
+
+  return(list(
+    min_sum = dp[m, n],
+    path = path,
+    dp_table = dp
+  ))
+}
+
+# Space-optimized version using only 1D array
+minimum_path_sum_optimized <- function(grid) {
+  #' Space optimized minimum path sum using 1D array
+  #' @param grid: 2D matrix of non-negative integers
+  #' @return: Minimum path sum
+
+  m <- nrow(grid)
+  n <- ncol(grid)
+
+  if (m == 0 || n == 0) return(0)
+
+  # Use the smaller dimension for space optimization
+  if (m <= n) {
+    # Process row by row
+    dp <- rep(0, m)
+    dp[1] <- grid[1, 1]
+
+    # Initialize first row
+    if (m > 1) {
+      for (i in 2:m) {
+        dp[i] <- dp[i - 1] + grid[i, 1]
+      }
+    }
+
+    # Process remaining columns
+    for (j in 2:n) {
+      dp[1] <- dp[1] + grid[1, j]
+      for (i in 2:m) {
+        dp[i] <- min(dp[i - 1], dp[i]) + grid[i, j]
+      }
+    }
+  } else {
+    # Process column by column
+    dp <- rep(0, n)
+    dp[1] <- grid[1, 1]
+
+    # Initialize first column
+    for (j in 2:n) {
+      dp[j] <- dp[j - 1] + grid[1, j]
+    }
+
+    # Process remaining rows
+    for (i in 2:m) {
+      dp[1] <- dp[1] + grid[i, 1]
+      for (j in 2:n) {
+        dp[j] <- min(dp[j - 1], dp[j]) + grid[i, j]
+      }
+    }
+  }
+
+  return(dp[length(dp)])
+}
+
+# Function to find all possible minimum paths
+find_all_minimum_paths <- function(grid) {
+  #' Find all possible paths that achieve the minimum sum
+  #' @param grid: 2D matrix of non-negative integers
+  #' @return: List of all minimum cost paths
+
+  m <- nrow(grid)
+  n <- ncol(grid)
+
+  if (m == 0 || n == 0) return(list())
+
+  # First compute the minimum sum
+  result <- minimum_path_sum(grid)
+  min_sum <- result$min_sum
+
+  all_paths <- list()
+
+  # Use recursive backtracking to find all paths with minimum sum
+  find_paths_recursive <- function(current_path, current_sum, i, j) {
+    current_sum <- current_sum + grid[i, j]
+
+    # If we've reached the bottom-right corner
+    if (i == m && j == n) {
+      if (current_sum == min_sum) {
+        all_paths <<- c(all_paths, list(c(current_path, list(c(i, j)))))
+      }
+      return
+    }
+
+    # If current sum exceeds minimum, prune
+    if (current_sum > min_sum) {
+      return
+    }
+
+    # Move right
+    if (j < n) {
+      find_paths_recursive(c(current_path, list(c(i, j))), current_sum, i, j + 1)
+    }
+
+    # Move down
+    if (i < m) {
+      find_paths_recursive(c(current_path, list(c(i, j))), current_sum, i + 1, j)
+    }
+  }
+
+  find_paths_recursive(list(), 0, 1, 1)
+  return(all_paths)
+}
+
+# Helper function to print DP table
+print_minimum_path_sum_dp <- function(dp_table, grid) {
+  m <- nrow(grid)
+  n <- ncol(grid)
+
+  cat("DP Table for Minimum Path Sum:\n")
+  cat("Grid:\n")
+  for (i in 1:m) {
+    cat("  ")
+    for (j in 1:n) {
+      cat(sprintf("%3d ", grid[i, j]))
+    }
+    cat("\n")
+  }
+  cat("\nDP Table:\n")
+  for (i in 1:m) {
+    cat("  ")
+    for (j in 1:n) {
+      cat(sprintf("%3d ", dp_table[i, j]))
+    }
+    cat("\n")
+  }
+  cat("\n")
+}
+
+# Helper function to visualize path on grid
+visualize_path <- function(grid, path) {
+  m <- nrow(grid)
+  n <- ncol(grid)
+
+  cat("Path Visualization:\n")
+  cat("Grid with path marked (*):\n")
+
+  # Create a matrix to mark the path
+  path_matrix <- matrix(" ", nrow = m, ncol = n)
+
+  for (pos in path) {
+    path_matrix[pos[1], pos[2]] <- "*"
+  }
+
+  for (i in 1:m) {
+    cat("  ")
+    for (j in 1:n) {
+      if (path_matrix[i, j] == "*") {
+        cat(sprintf("%3s ", "*"))
+      } else {
+        cat(sprintf("%3d ", grid[i, j]))
+      }
+    }
+    cat("\n")
+  }
+  cat("\n")
+}
+
+# ===========================
+# Example Usage & Testing
+# ===========================
+cat("=== Minimum Path Sum Problem (Dynamic Programming) ===\n\n")
+
+# Test 1: Basic Example
+cat("Test 1: Basic Example\n")
+grid1 <- matrix(c(1, 3, 1, 1, 5, 1, 4, 2, 1), nrow = 3, ncol = 3, byrow = TRUE)
+cat("Grid:\n")
+print(grid1)
+
+result1 <- minimum_path_sum(grid1)
+print_minimum_path_sum_dp(result1$dp_table, grid1)
+cat("Minimum Path Sum:", result1$min_sum, "\n")
+cat("Path (row, col):", paste(sapply(result1$path, function(x) paste("(", x[1], ",", x[2], ")", sep="")), collapse = " -> "), "\n")
+visualize_path(grid1, result1$path)
+cat("\n")
+
+# Test 2: Optimized Version
+cat("Test 2: Space Optimized Version\n")
+min_sum_opt <- minimum_path_sum_optimized(grid1)
+cat("Minimum Path Sum (Optimized):", min_sum_opt, "\n")
+cat("Verification: Both methods match:", result1$min_sum == min_sum_opt, "\n\n")
+
+# Test 3: Single Row/Column Cases
+cat("Test 3: Edge Cases\n")
+cat("Single row grid:\n")
+grid_row <- matrix(c(1, 2, 3, 4, 5), nrow = 1)
+print(grid_row)
+result_row <- minimum_path_sum(grid_row)
+cat("Minimum sum:", result_row$min_sum, "\n\n")
+
+cat("Single column grid:\n")
+grid_col <- matrix(c(1, 2, 3, 4, 5), ncol = 1)
+print(grid_col)
+result_col <- minimum_path_sum(grid_col)
+cat("Minimum sum:", result_col$min_sum, "\n\n")
+
+# Test 4: Larger Grid
+cat("Test 4: Larger Grid (4x5)\n")
+# Set random seed for reproducibility in tests. The value 42 is chosen arbitrarily.
+SEED <- 42
+set.seed(SEED)
+grid_large <- matrix(sample(1:9, 20, replace = TRUE), nrow = 4, ncol = 5)
+cat("Grid:\n")
+print(grid_large)
+
+result_large <- minimum_path_sum(grid_large)
+cat("Minimum Path Sum:", result_large$min_sum, "\n")
+cat("Path length:", length(result_large$path), "steps\n")
+visualize_path(grid_large, result_large$path)
+cat("\n")
+
+# Test 5: Performance Comparison
+cat("Test 5: Performance Comparison (6x8 grid)\n")
+grid_perf <- matrix(sample(1:20, 48, replace = TRUE), nrow = 6, ncol = 8)
+
+library(microbenchmark)
+
+mbm <- microbenchmark(
+  std = minimum_path_sum(grid_perf),
+  opt = minimum_path_sum_optimized(grid_perf),
+  times = 100L
+)
+
+result_std <- minimum_path_sum(grid_perf)
+result_opt <- minimum_path_sum_optimized(grid_perf)
+
+cat("Standard DP result:", result_std$min_sum, "\n")
+cat("Optimized DP result:", result_opt, "\n")
+cat("Standard DP median time:", sprintf("%.6f sec", median(mbm$time[mbm$expr == "std"])/1e9), "\n")
+cat("Optimized DP median time:", sprintf("%.6f sec", median(mbm$time[mbm$expr == "opt"])/1e9), "\n")
+cat("Results match:", result_std$min_sum == result_opt, "\n\n")
+
+# Test 6: Multiple Minimum Paths
+cat("Test 6: Multiple Minimum Paths\n")
+grid_multiple <- matrix(c(1, 2, 1, 1, 1, 1, 1, 1, 1), nrow = 3, ncol = 3, byrow = TRUE)
+cat("Grid:\n")
+print(grid_multiple)
+
+result_multiple <- minimum_path_sum(grid_multiple)
+cat("Minimum Path Sum:", result_multiple$min_sum, "\n")
+cat("One possible path:", paste(sapply(result_multiple$path, function(x) paste("(", x[1], ",", x[2], ")", sep="")), collapse = " -> "), "\n")
+
+# Find all minimum paths
+all_paths <- find_all_minimum_paths(grid_multiple)
+cat("Total number of minimum paths:", length(all_paths), "\n")
+for (i in seq_along(all_paths)) {
+  path_str <- paste(sapply(all_paths[[i]], function(x) paste("(", x[1], ",", x[2], ")", sep="")), collapse = " -> ")
+  path_sum <- sum(sapply(all_paths[[i]], function(x) grid_multiple[x[1], x[2]]))
+  cat("Path", i, ":", path_str, "(sum =", path_sum, ")\n")
+}
+cat("\n")
+
+# Test 7: Real-world Example - Cost Optimization
+cat("Test 7: Real-world Example - Transportation Cost Optimization\n")
+# Grid representing transportation costs between cities
+transport_grid <- matrix(c(
+  2, 3, 4, 2, 1,
+  1, 2, 1, 3, 2,
+  3, 1, 2, 1, 4,
+  2, 4, 1, 2, 3
+), nrow = 4, ncol = 5, byrow = TRUE)
+
+cat("Transportation Cost Grid:\n")
+print(transport_grid)
+
+transport_result <- minimum_path_sum(transport_grid)
+cat("Minimum transportation cost:", transport_result$min_sum, "\n")
+cat("Optimal route:", paste(sapply(transport_result$path, function(x) paste("City(", x[1], ",", x[2], ")", sep="")), collapse = " -> "), "\n")
+visualize_path(transport_grid, transport_result$path)
+
+# Calculate cost breakdown
+cat("Cost breakdown:\n")
+total_cost <- 0
+for (i in seq_along(transport_result$path)) {
+  pos <- transport_result$path[[i]]
+  cost <- transport_grid[pos[1], pos[2]]
+  total_cost <- total_cost + cost
+  cat("  Step", i, ": City(", pos[1], ",", pos[2], ") =", cost, "\n")
+}
+cat("Total cost verification:", total_cost, "\n")