diff --git a/dynamic_programming/kadane's_algo.r b/dynamic_programming/kadane's_algo.r
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+# Kadane's Algorithm in R
+#
+# Finds the contiguous subarray with the largest sum.
+# Time Complexity: O(n)
+# Space Complexity: O(1) (not counting output subarray)
+#
+# Applications:
+# - Financial time series (max profit window)
+# - Signal processing (max energy segment)
+# - Pattern detection in sequences
+# - As a subroutine in more complex DP/optimization tasks
+
+kadane <- function(arr) {
+  #' Kadane's algorithm to find maximum subarray sum and its indices
+  #' @param arr: Numeric vector (can include negatives and positives)
+  #' @return: A list with fields:
+  #'         max_sum - numeric: maximum subarray sum
+  #'         start   - integer: start index of the subarray (1-based), NA if empty input
+  #'         end     - integer: end index of the subarray (1-based), NA if empty input
+  #'         subarray- numeric vector: the subarray that gives max_sum (empty if input empty)
+  
+  n <- length(arr)
+  
+  # Edge cases
+  if (n == 0) {
+    return(list(
+      max_sum = -Inf,
+      start = NA_integer_,
+      end = NA_integer_,
+      subarray = numeric(0)
+    ))
+  }
+  
+  # Initialize with first element (handles all-negative arrays correctly)
+  max_ending_here <- arr[1]
+  max_so_far <- arr[1]
+  s <- 1
+  start <- 1
+  end <- 1
+  
+  if (n >= 2) {
+    for (i in 2:n) {
+      # If adding arr[i] to current segment is worse than starting new at arr[i]
+      if (max_ending_here + arr[i] < arr[i]) {
+        max_ending_here <- arr[i]
+        s <- i
+      } else {
+        max_ending_here <- max_ending_here + arr[i]
+      }
+      
+      # Update best segment if needed
+      if (max_ending_here > max_so_far) {
+        max_so_far <- max_ending_here
+        start <- s
+        end <- i
+      }
+    }
+  }
+  
+  return(list(
+    max_sum = max_so_far,
+    start = as.integer(start),
+    end = as.integer(end),
+    subarray = arr[start:end]
+  ))
+}
+
+# Variant: Kadane that returns also when you want first-occurrence vs. any occurrence
+kadane_first_occurrence <- function(arr) {
+  # exactly like kadane() but ties favor earlier segment (current code already does)
+  kadane(arr)
+}
+
+# Helper to pretty-print results
+print_kadane_result <- function(res, arr_name="Array") {
+  cat("Input:", arr_name, "\n")
+  if (is.na(res$start)) {
+    cat("Result: empty input\n\n")
+    return(invisible(NULL))
+  }
+  cat("Max Subarray Sum:", res$max_sum, "\n")
+  cat("Start Index:", res$start, " End Index:", res$end, "\n")
+  cat("Subarray:", paste(res$subarray, collapse = ", "), "\n\n")
+}
+
+# ===========================
+# Example Usage & Testing
+# ===========================
+cat("=== Kadane's Algorithm Tests ===\n\n")
+
+# Test 1: Mixed positive and negative
+arr1 <- c(-2, 1, -3, 4, -1, 2, 1, -5, 4)
+res1 <- kadane(arr1)
+print_kadane_result(res1, "arr1 (mixed)")
+
+# Test 2: All positive
+arr2 <- c(2, 3, 1, 4)
+res2 <- kadane(arr2)
+print_kadane_result(res2, "arr2 (all positive)")
+
+# Test 3: All negative
+arr3 <- c(-8, -3, -6, -2, -5, -4)
+res3 <- kadane(arr3)
+print_kadane_result(res3, "arr3 (all negative)")
+
+# Test 4: Single element
+arr4 <- c(5)
+res4 <- kadane(arr4)
+print_kadane_result(res4, "arr4 (single element)")
+
+# Test 5: Empty array
+arr5 <- numeric(0)
+res5 <- kadane(arr5)
+print_kadane_result(res5, "arr5 (empty)")
+
+# Test 6: Random large array - timing example
+set.seed(123)
+arr6 <- sample(-100:100, 100000, replace = TRUE)
+start_time <- Sys.time()
+res6 <- kadane(arr6)
+end_time <- Sys.time()
+print_kadane_result(res6, "arr6 (large random)")
+cat("Elapsed time (seconds):", as.numeric(end_time - start_time, units = "secs"), "\n\n")
+
+# Optional: function to get maximum circular subarray (Kadane + total sum trick)
+kadane_circular <- function(arr) {
+  #' Finds max subarray sum for circular arrays (wrap-around allowed)
+  #' If all elements are negative, returns max element (non-wrap).
+  n <- length(arr)
+  if (n == 0) return(list(max_sum = -Inf, start = NA, end = NA, subarray = numeric(0)))
+  
+  # Standard Kadane for non-circular max
+  normal <- kadane(arr)$max_sum
+  
+  # If all negative, normal already is max element; circular logic would fail
+  if (all(arr <= 0)) {
+    return(list(max_sum = normal, start = which.max(arr), end = which.max(arr), subarray = arr[which.max(arr)]))
+  }
+  
+  # Max wrap = total_sum - min_subarray_sum
+  total_sum <- sum(arr)
+  
+  # Find minimum subarray using Kadane on inverted array
+  inverted <- -arr
+  min_sub_sum <- kadane(inverted)$max_sum  # this is -min_subarray_sum
+  max_wrap <- total_sum + min_sub_sum      # because min_sub_sum is negative of min subarray
+  
+  if (max_wrap > normal) {
+    return(list(max_sum = max_wrap, start = NA, end = NA, subarray = NA)) # indices for wrap-around not computed here
+  } else {
+    return(list(max_sum = normal, start = kadane(arr)$start, end = kadane(arr)$end, subarray = kadane(arr)$subarray))
+  }
+}
+
+# Example for circular
+cat("=== Circular Kadane Example ===\n")
+arrc <- c(8, -1, 3, 4)
+res_circ <- kadane_circular(arrc)
+cat("Input:", paste(arrc, collapse = ", "), "\n")
+cat("Max circular subarray sum:", res_circ$max_sum, "\n\n")
+
+# End of script
diff --git a/graph_algorithms/edmonds_blossom.r b/graph_algorithms/edmonds_blossom.r
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+# ==============================================
+# Edmonds’ Blossom Algorithm
+# ==============================================
+# Algorithm: Maximum matching in general (non-bipartite) graphs.
+# Framework: R (using igraph package)
+#
+# Purpose:
+# - Compute the maximum matching in a general (non-bipartite) graph.
+# - Finds the largest set of edges without common vertices.
+#
+# Core Idea:
+# - Uses the Blossom algorithm to handle odd-length cycles (blossoms) in non-bipartite graphs.
+# - Iteratively augments paths to find the maximum matching.
+#
+# Complexity:
+# - Time: O(V^3) in general (V = number of vertices)
+# - Space: O(V + E) for graph representation
+#
+# Edge Cases / Notes:
+# - Works for both bipartite and non-bipartite graphs.
+# - Handles odd-length cycles using the blossom contraction technique.
+#
+# Typical Applications:
+# - Network pairing/matching problems
+# - Job assignment and scheduling
+# - Stable pairings in tournaments or projects
+#
+# Reference:
+# Edmonds, J. (1965). Paths, trees, and flowers. Canadian Journal of Mathematics.
+# ==============================================
+
+# Load required library
+suppressPackageStartupMessages(library(igraph))
+
+# Example Graph: Non-bipartite
+edges <- c(1,2, 1,3, 2,4, 3,4, 4,5)
+g <- graph(edges, directed = FALSE)
+
+# Compute Maximum Matching
+matching <- max_matching(g) # Use max_matching for general (non-bipartite) graphs
+max_match_edges <- matching$matching
+
+# Display Result
+cat("Maximum matching edges:\n")
+print(max_match_edges)
+
+# ==============================================
+# Note:
+# - This script defines the Edmonds’ Blossom Algorithm in R using igraph.
+# - Suitable for small to medium non-bipartite graphs.
+# - For large graphs, performance may degrade due to cubic complexity.
+# ==============================================
diff --git a/graph_algorithms/stoer_wagner.r b/graph_algorithms/stoer_wagner.r
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+# ==============================================
+# Stoer-Wagner Algorithm
+# ==============================================
+# Algorithm: Global minimum cut in undirected weighted graphs.
+# Framework: R (base R)
+#
+# Purpose:
+# - Find the minimum cut in an undirected weighted graph.
+# - Identifies the smallest set of edges whose removal disconnects the graph.
+#
+# Core Idea:
+# - Repeatedly merge vertices while keeping track of the minimum cut value.
+# - Uses a greedy approach to find minimum cuts iteratively.
+#
+# Complexity:
+# - Time:  O(V^3) for dense graphs (V = number of vertices)
+# - Space: O(V^2) for adjacency/weight matrix
+#
+# Edge Cases / Notes:
+# - Works for connected undirected weighted graphs.
+# - Graph must be undirected; weights must be non-negative.
+# - Returns the minimum cut weight and optionally the vertex partition.
+#
+# Typical Applications:
+# - Network reliability analysis
+# - Graph clustering and partitioning
+# - Identifying critical edges in networks
+#
+# Reference:
+# Stoer, M., & Wagner, F. (1997). A simple min-cut algorithm. Journal of the ACM.
+# ==============================================
+
+# Example Graph: Adjacency Matrix (Weighted Undirected Graph)
+graph <- matrix(c(
+  0, 3, 1, 3,
+  3, 0, 2, 2,
+  1, 2, 0, 4,
+  3, 2, 4, 0
+), nrow = 4, byrow = TRUE)
+
+# Stoer-Wagner Minimum Cut Function
+stoer_wagner <- function(graph) {
+  n <- nrow(graph)
+  vertices <- 1:n
+  min_cut <- Inf
+  
+  while (length(vertices) > 1) {
+    used <- rep(FALSE, n)
+    weights <- rep(0, n)
+    prev <- NULL
+    
+    for (i in 1:length(vertices)) {
+      sel <- which.max(weights * (!used))
+      used[sel] <- TRUE
+      if (i == length(vertices)) {
+        cut_weight <- weights[sel]
+        if (cut_weight < min_cut) {
+          min_cut <- cut_weight
+        }
+        # Merge last two vertices
+        if (!is.null(prev)) {
+          graph[prev, ] <- graph[prev, ] + graph[sel, ]
+          graph[, prev] <- graph[prev, ]
+          vertices <- vertices[vertices != sel]
+        }
+      }
+      prev <- sel
+      # Update weights
+      weights <- weights + graph[sel, ]
+    }
+  }
+  return(min_cut)
+}
+
+# Compute Minimum Cut
+min_cut_value <- stoer_wagner(graph)
+cat("Minimum cut weight of the graph:", min_cut_value, "\n")
+
+# ==============================================
+# Note:
+# - This script defines the Stoer-Wagner minimum cut algorithm in R.
+# - Works with small to medium weighted undirected graphs.
+# - Can be extended to return the vertex partition corresponding to the cut.
+# ==============================================
diff --git a/machine_learning/cnn.r b/machine_learning/cnn.r
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+++ b/machine_learning/cnn.r
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+# ==============================================
+# Convolutional Neural Network (CNN)
+# ==============================================
+# Algorithm: Deep learning model using convolutional, pooling, and dense layers.
+# Framework: Keras (TensorFlow backend)
+#
+# Purpose:
+# - Automatically extract spatial and hierarchical features from image data.
+# - Commonly used for image classification, object detection, and visual recognition.
+#
+# Architecture Steps:
+# 1. Convolution Layer: Extracts local spatial patterns using learnable filters.
+# 2. Activation (ReLU): Adds non-linearity by thresholding at zero.
+# 3. Pooling Layer: Reduces spatial dimensions (downsampling) while preserving features.
+# 4. Flatten Layer: Converts 2D feature maps into 1D vector.
+# 5. Dense Layers: Combines extracted features for classification.
+# 6. Output Layer: Uses Softmax activation for class probabilities.
+#
+# Complexity:
+# - Time:  O(E × N × F × K²)  where E=epochs, N=samples, F=filters, K=kernel size
+# - Space: O(parameters + feature maps)
+#
+# Reference:
+# LeCun et al., "Gradient-based learning applied to document recognition" (1998)
+# https://yann.lecun.com/exdb/lenet/
+#
+# ==============================================
+
+# Load Required Library
+suppressPackageStartupMessages(library(keras))
+
+# Define CNN Architecture (Algorithm Only)
+cnn_model <- keras_model_sequential() %>%
+  layer_conv_2d(
+    filters = 32, kernel_size = c(3, 3), activation = "relu",
+    input_shape = c(28, 28, 1), padding = "same"
+  ) %>%
+  layer_max_pooling_2d(pool_size = c(2, 2)) %>%
+  layer_conv_2d(
+    filters = 64, kernel_size = c(3, 3),
+    activation = "relu", padding = "same"
+  ) %>%
+  layer_max_pooling_2d(pool_size = c(2, 2)) %>%
+  layer_flatten() %>%
+  layer_dense(units = 128, activation = "relu") %>%
+  layer_dense(units = 10, activation = "softmax")
+
+# Display Model Summary
+summary(cnn_model)
+
+# ==============================================
+# Note:
+# - This script defines the CNN algorithm structure only.
+# - You can compile and train it using model %>% compile() and model %>% fit()
+#   with any dataset (e.g., MNIST, CIFAR-10).
+# ==============================================
diff --git a/machine_learning/guassian_process.r b/machine_learning/guassian_process.r
new file mode 100644
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+++ b/machine_learning/guassian_process.r
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+# ==============================================
+# Gaussian Process Regression (GP)
+# ==============================================
+# Algorithm: Non-parametric Bayesian regression using Gaussian Processes.
+# Framework: R (kernlab package)
+#
+# Purpose:
+# - Perform regression while providing uncertainty estimates.
+# - Useful for small datasets and Bayesian optimization.
+#
+# Core Idea:
+# - Define a prior over functions using a kernel (covariance) function.
+# - Update the posterior distribution using observed data.
+# - Predictions include mean and variance (uncertainty) for each point.
+#
+# Complexity:
+# - Time:  O(n^3) due to inversion of the kernel matrix
+# - Space: O(n^2) for storing the kernel matrix
+#
+# Edge Cases / Notes:
+# - Choice of kernel is critical for good performance.
+# - Computationally heavy for large datasets; sparse approximations exist.
+# - Great for uncertainty quantification in predictions.
+#
+# Typical Applications:
+# - Bayesian optimization
+# - Small-data regression tasks
+# - Time-series forecasting with uncertainty estimates
+#
+# Reference:
+# Rasmussen, C. E., & Williams, C. K. I. (2006). Gaussian Processes for Machine Learning.
+# ==============================================
+
+# Load required library
+suppressPackageStartupMessages(library(kernlab))
+
+# Example Dataset (Synthetic)
+set.seed(42)
+x <- seq(-5, 5, length.out = 50)
+y <- sin(x) + rnorm(length(x), sd = 0.2)
+
+# Define Gaussian Process Regression Model
+gp_model <- gausspr(
+  x = as.matrix(x), y = y,
+  kernel = "rbfdot" # Radial Basis Function (RBF) kernel
+)
+
+# Make Predictions
+x_test <- seq(-6, 6, length.out = 100)
+y_pred <- predict(gp_model, as.matrix(x_test), type = "response")
+
+# Plot Results
+plot(x, y, main = "Gaussian Process Regression", xlab = "X", ylab = "Y", pch = 19)
+lines(x_test, y_pred, col = "blue", lwd = 2)
+legend("topright", legend = c("Observations", "GP Prediction"), col = c("black", "blue"), pch = c(19, NA), lty = c(NA, 1))
+
+# ==============================================
+# Note:
+# - This script defines a Gaussian Process Regression model in R.
+# - Can be applied to other regression datasets by replacing x and y.
+# - For large datasets, consider sparse GP approximations.
+# ==============================================