From 8b30acf58c4636541e76801c4f9f22dfcd53d85b Mon Sep 17 00:00:00 2001
From: Prathamesh Kalshetti <prathameshkalshetti450@gmail.com>
Date: Sat, 18 Oct 2025 11:44:39 +0530
Subject: [PATCH 1/6] bidirectional_bfs

---
 graph_algorithms/bidirectional_bfs.r | 153 +++++++++++++++++++++++++++
 1 file changed, 153 insertions(+)
 create mode 100644 graph_algorithms/bidirectional_bfs.r

diff --git a/graph_algorithms/bidirectional_bfs.r b/graph_algorithms/bidirectional_bfs.r
new file mode 100644
index 00000000..8daf6f38
--- /dev/null
+++ b/graph_algorithms/bidirectional_bfs.r
@@ -0,0 +1,153 @@
+# ==============================================================
+# Bidirectional Breadth-First Search (BFS) Shortest Path Algorithm
+# ==============================================================
+#
+# Description:
+#   Finds the shortest path between a source and target in an
+#   unweighted graph using Bidirectional BFS.
+#
+# Time Complexity: O(b^(d/2)) — much faster than normal BFS O(b^d)
+# Space Complexity: O(V)
+#
+# Input:
+#   graph  - adjacency list (list of integer vectors)
+#   source - integer (starting vertex)
+#   target - integer (destination vertex)
+#
+# Output:
+#   A list containing:
+#       path       - vector of vertices representing the path
+#       distance   - number of edges in the shortest path
+#       found      - logical flag (TRUE if path found, else FALSE)
+#
+# Example usage at bottom of file.
+# ==============================================================
+
+bidirectional_bfs <- function(graph, source, target) {
+  if (source == target) {
+    return(list(path = c(source), distance = 0, found = TRUE))
+  }
+
+  # Initialize BFS from both ends
+  visited_from_source <- setNames(rep(FALSE, length(graph)), names(graph))
+  visited_from_target <- setNames(rep(FALSE, length(graph)), names(graph))
+  
+  parent_from_source <- rep(NA, length(graph))
+  parent_from_target <- rep(NA, length(graph))
+  
+  queue_source <- c(source)
+  queue_target <- c(target)
+  
+  visited_from_source[source] <- TRUE
+  visited_from_target[target] <- TRUE
+  
+  meeting_node <- NA
+  
+  # Function to check intersection
+  get_intersection <- function() {
+    common <- which(visited_from_source & visited_from_target)
+    if (length(common) > 0) return(common[1])
+    return(NA)
+  }
+  
+  # Main loop
+  while (length(queue_source) > 0 && length(queue_target) > 0) {
+    # Expand one level from source side
+    next_queue <- c()
+    for (u in queue_source) {
+      for (v in graph[[as.character(u)]]) {
+        if (!visited_from_source[v]) {
+          visited_from_source[v] <- TRUE
+          parent_from_source[v] <- u
+          next_queue <- c(next_queue, v)
+        }
+      }
+    }
+    queue_source <- next_queue
+    
+    # Check intersection
+    meeting_node <- get_intersection()
+    if (!is.na(meeting_node)) break
+    
+    # Expand one level from target side
+    next_queue <- c()
+    for (u in queue_target) {
+      for (v in graph[[as.character(u)]]) {
+        if (!visited_from_target[v]) {
+          visited_from_target[v] <- TRUE
+          parent_from_target[v] <- u
+          next_queue <- c(next_queue, v)
+        }
+      }
+    }
+    queue_target <- next_queue
+    
+    # Check intersection again
+    meeting_node <- get_intersection()
+    if (!is.na(meeting_node)) break
+  }
+  
+  if (is.na(meeting_node)) {
+    return(list(path = NULL, distance = Inf, found = FALSE))
+  }
+  
+  # Reconstruct path from source → meeting_node
+  path1 <- c()
+  node <- meeting_node
+  while (!is.na(node)) {
+    path1 <- c(node, path1)
+    node <- parent_from_source[node]
+  }
+  
+  # Reconstruct path from meeting_node → target
+  path2 <- c()
+  node <- parent_from_target[meeting_node]
+  while (!is.na(node)) {
+    path2 <- c(path2, node)
+    node <- parent_from_target[node]
+  }
+  
+  full_path <- c(path1, path2)
+  return(list(path = full_path, distance = length(full_path) - 1, found = TRUE))
+}
+
+# ==============================================================
+# Example Usage and Test
+# ==============================================================
+
+cat("=== Bidirectional BFS Shortest Path ===\n")
+
+# Example Graph (Unweighted)
+# 1 -- 2 -- 3
+#  |    |
+#  4 -- 5 -- 6
+
+graph <- list(
+  "1" = c(2, 4),
+  "2" = c(1, 3, 5),
+  "3" = c(2, 6),
+  "4" = c(1, 5),
+  "5" = c(2, 4, 6),
+  "6" = c(3, 5)
+)
+
+cat("Graph adjacency list:\n")
+for (v in names(graph)) {
+  cat("Vertex", v, "-> [", paste(graph[[v]], collapse = ", "), "]\n")
+}
+
+cat("\nRunning Bidirectional BFS from 1 to 6...\n")
+result <- bidirectional_bfs(graph, 1, 6)
+
+if (result$found) {
+  cat("Shortest Path Found!\n")
+  cat("Path:", paste(result$path, collapse = " -> "), "\n")
+  cat("Distance:", result$distance, "\n")
+} else {
+  cat("No path found between source and target.\n")
+}
+    return(list(
+        distances = distances,
+        predecessor = predecessor,
+        found = found
+    ))
\ No newline at end of file

From 4921341b2921457245427d4b465e0eb1478f28e6 Mon Sep 17 00:00:00 2001
From: Prathamesh Kalshetti <prathameshkalshetti450@gmail.com>
Date: Sat, 18 Oct 2025 12:35:09 +0530
Subject: [PATCH 2/6] viterbi

---
 dynamic_programming/viterbi.r | 113 ++++++++++++++++++++++++++++++++++
 1 file changed, 113 insertions(+)
 create mode 100644 dynamic_programming/viterbi.r

diff --git a/dynamic_programming/viterbi.r b/dynamic_programming/viterbi.r
new file mode 100644
index 00000000..7d656562
--- /dev/null
+++ b/dynamic_programming/viterbi.r
@@ -0,0 +1,113 @@
+# ==============================================================
+# Viterbi Algorithm — Hidden Markov Model (HMM) Decoding
+# ==============================================================
+#
+# Description:
+#   The Viterbi algorithm finds the most probable sequence of
+#   hidden states (state path) that results in a given sequence of
+#   observed events in a Hidden Markov Model.
+#
+# Time Complexity: O(N * T)
+#   - N = number of hidden states
+#   - T = length of observation sequence
+#
+# Space Complexity: O(N * T)
+#
+# Input:
+#   states        - vector of hidden states
+#   observations  - vector of observed symbols
+#   start_prob    - named vector of initial probabilities (state → prob)
+#   trans_prob    - matrix of transition probabilities (from_state → to_state)
+#   emit_prob     - matrix of emission probabilities (state → observation)
+#
+# Output:
+#   A list containing:
+#       best_path     - most probable state sequence
+#       best_prob      - probability of the best path
+#
+# Example usage provided at bottom of file.
+# ==============================================================
+
+viterbi <- function(states, observations, start_prob, trans_prob, emit_prob) {
+  N <- length(states)
+  T_len <- length(observations)
+  
+  # Initialize matrices
+  V <- matrix(0, nrow = N, ncol = T_len)  # probability table
+  path <- matrix(NA, nrow = N, ncol = T_len)  # backpointer table
+  
+  # Initialization step
+  for (i in 1:N) {
+    V[i, 1] <- start_prob[states[i]] * emit_prob[states[i], observations[1]]
+    path[i, 1] <- 0
+  }
+  
+  # Recursion step
+  for (t in 2:T_len) {
+    for (j in 1:N) {
+      probs <- V[, t - 1] * trans_prob[, states[j]] * emit_prob[states[j], observations[t]]
+      V[j, t] <- max(probs)
+      path[j, t] <- which.max(probs)
+    }
+  }
+  
+  # Termination step
+  best_last_state <- which.max(V[, T_len])
+  best_prob <- V[best_last_state, T_len]
+  
+  # Backtrack the best path
+  best_path <- rep(NA, T_len)
+  best_path[T_len] <- best_last_state
+  
+  for (t in (T_len - 1):1) {
+    best_path[t] <- path[best_path[t + 1], t + 1]
+  }
+  
+  best_state_sequence <- states[best_path]
+  
+  return(list(
+    best_path = best_state_sequence,
+    best_prob = best_prob
+  ))
+}
+
+# ==============================================================
+# Example Usage and Test
+# ==============================================================
+
+cat("=== Viterbi Algorithm — Hidden Markov Model ===\n")
+
+# Example: Weather HMM
+# States: Rainy, Sunny
+# Observations: walk, shop, clean
+states <- c("Rainy", "Sunny")
+observations <- c("walk", "shop", "clean")
+
+# Start probabilities
+start_prob <- c(Rainy = 0.6, Sunny = 0.4)
+
+# Transition probabilities
+trans_prob <- matrix(c(
+  0.7, 0.3,   # from Rainy to (Rainy, Sunny)
+  0.4, 0.6    # from Sunny to (Rainy, Sunny)
+), nrow = 2, byrow = TRUE)
+rownames(trans_prob) <- states
+colnames(trans_prob) <- states
+
+# Emission probabilities
+emit_prob <- matrix(c(
+  0.1, 0.4, 0.5,  # Rainy emits (walk, shop, clean)
+  0.6, 0.3, 0.1   # Sunny emits (walk, shop, clean)
+), nrow = 2, byrow = TRUE)
+rownames(emit_prob) <- states
+colnames(emit_prob) <- observations
+
+# Observed sequence
+obs_seq <- c("walk", "shop", "clean")
+
+cat("Observation sequence:", paste(obs_seq, collapse = ", "), "\n")
+result <- viterbi(states, obs_seq, start_prob, trans_prob, emit_prob)
+
+cat("Most probable state sequence:\n")
+cat(paste(result$best_path, collapse = " -> "), "\n")
+cat("Probability of this sequence:", result$best_prob, "\n")

From a0236f8c81c6356eee2f535bca6ed95a19bc8b4a Mon Sep 17 00:00:00 2001
From: Prathamesh Kalshetti <prathameshkalshetti450@gmail.com>
Date: Mon, 20 Oct 2025 23:26:05 +0530
Subject: [PATCH 3/6] Update dynamic_programming/viterbi.r

Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com>
---
 dynamic_programming/viterbi.r | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/dynamic_programming/viterbi.r b/dynamic_programming/viterbi.r
index 7d656562..b51294af 100644
--- a/dynamic_programming/viterbi.r
+++ b/dynamic_programming/viterbi.r
@@ -23,7 +23,7 @@
 # Output:
 #   A list containing:
 #       best_path     - most probable state sequence
-#       best_prob      - probability of the best path
+#      best_prob      - probability of the best path
 #
 # Example usage provided at bottom of file.
 # ==============================================================

From e73a1b8dcee3ac3ad75d5e68005fbb3e39d23264 Mon Sep 17 00:00:00 2001
From: Prathamesh Kalshetti <prathameshkalshetti450@gmail.com>
Date: Mon, 20 Oct 2025 23:41:17 +0530
Subject: [PATCH 4/6] Update dynamic_programming/viterbi.r

Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com>
---
 dynamic_programming/viterbi.r | 60 ++++++++++++++++++-----------------
 1 file changed, 31 insertions(+), 29 deletions(-)

diff --git a/dynamic_programming/viterbi.r b/dynamic_programming/viterbi.r
index b51294af..88b7a948 100644
--- a/dynamic_programming/viterbi.r
+++ b/dynamic_programming/viterbi.r
@@ -75,39 +75,41 @@ viterbi <- function(states, observations, start_prob, trans_prob, emit_prob) {
 # Example Usage and Test
 # ==============================================================
 
-cat("=== Viterbi Algorithm — Hidden Markov Model ===\n")
+if (!exists(".test_mode")) {
+  cat("=== Viterbi Algorithm — Hidden Markov Model ===\n")
 
-# Example: Weather HMM
-# States: Rainy, Sunny
-# Observations: walk, shop, clean
-states <- c("Rainy", "Sunny")
-observations <- c("walk", "shop", "clean")
+  # Example: Weather HMM
+  # States: Rainy, Sunny
+  # Observations: walk, shop, clean
+  states <- c("Rainy", "Sunny")
+  observations <- c("walk", "shop", "clean")
 
-# Start probabilities
-start_prob <- c(Rainy = 0.6, Sunny = 0.4)
+  # Start probabilities
+  start_prob <- c(Rainy = 0.6, Sunny = 0.4)
 
-# Transition probabilities
-trans_prob <- matrix(c(
-  0.7, 0.3,   # from Rainy to (Rainy, Sunny)
-  0.4, 0.6    # from Sunny to (Rainy, Sunny)
-), nrow = 2, byrow = TRUE)
-rownames(trans_prob) <- states
-colnames(trans_prob) <- states
+  # Transition probabilities
+  trans_prob <- matrix(c(
+    0.7, 0.3,   # from Rainy to (Rainy, Sunny)
+    0.4, 0.6    # from Sunny to (Rainy, Sunny)
+  ), nrow = 2, byrow = TRUE)
+  rownames(trans_prob) <- states
+  colnames(trans_prob) <- states
 
-# Emission probabilities
-emit_prob <- matrix(c(
-  0.1, 0.4, 0.5,  # Rainy emits (walk, shop, clean)
-  0.6, 0.3, 0.1   # Sunny emits (walk, shop, clean)
-), nrow = 2, byrow = TRUE)
-rownames(emit_prob) <- states
-colnames(emit_prob) <- observations
+  # Emission probabilities
+  emit_prob <- matrix(c(
+    0.1, 0.4, 0.5,  # Rainy emits (walk, shop, clean)
+    0.6, 0.3, 0.1   # Sunny emits (walk, shop, clean)
+  ), nrow = 2, byrow = TRUE)
+  rownames(emit_prob) <- states
+  colnames(emit_prob) <- observations
 
-# Observed sequence
-obs_seq <- c("walk", "shop", "clean")
+  # Observed sequence
+  obs_seq <- c("walk", "shop", "clean")
 
-cat("Observation sequence:", paste(obs_seq, collapse = ", "), "\n")
-result <- viterbi(states, obs_seq, start_prob, trans_prob, emit_prob)
+  cat("Observation sequence:", paste(obs_seq, collapse = ", "), "\n")
+  result <- viterbi(states, obs_seq, start_prob, trans_prob, emit_prob)
 
-cat("Most probable state sequence:\n")
-cat(paste(result$best_path, collapse = " -> "), "\n")
-cat("Probability of this sequence:", result$best_prob, "\n")
+  cat("Most probable state sequence:\n")
+  cat(paste(result$best_path, collapse = " -> "), "\n")
+  cat("Probability of this sequence:", result$best_prob, "\n")
+}

From 9d8d9d3ced22ad1bf389a240378b69aa80609e25 Mon Sep 17 00:00:00 2001
From: Prathamesh Kalshetti <prathameshkalshetti450@gmail.com>
Date: Mon, 20 Oct 2025 23:41:59 +0530
Subject: [PATCH 5/6] Update dynamic_programming/viterbi.r

Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com>
---
 dynamic_programming/viterbi.r | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

diff --git a/dynamic_programming/viterbi.r b/dynamic_programming/viterbi.r
index 88b7a948..89958005 100644
--- a/dynamic_programming/viterbi.r
+++ b/dynamic_programming/viterbi.r
@@ -59,8 +59,10 @@ viterbi <- function(states, observations, start_prob, trans_prob, emit_prob) {
   best_path <- rep(NA, T_len)
   best_path[T_len] <- best_last_state
   
-  for (t in (T_len - 1):1) {
-    best_path[t] <- path[best_path[t + 1], t + 1]
+  if (T_len > 1) {
+    for (t in (T_len - 1):1) {
+      best_path[t] <- path[best_path[t + 1], t + 1]
+    }
   }
   
   best_state_sequence <- states[best_path]

From 411216bf076c6cc08b00d5f22cb4c676468fd8b3 Mon Sep 17 00:00:00 2001
From: Prathamesh Kalshetti <prathameshkalshetti450@gmail.com>
Date: Mon, 20 Oct 2025 23:42:08 +0530
Subject: [PATCH 6/6] Update dynamic_programming/viterbi.r

Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com>
---
 dynamic_programming/viterbi.r | 12 +++++++-----
 1 file changed, 7 insertions(+), 5 deletions(-)

diff --git a/dynamic_programming/viterbi.r b/dynamic_programming/viterbi.r
index 89958005..56105392 100644
--- a/dynamic_programming/viterbi.r
+++ b/dynamic_programming/viterbi.r
@@ -43,11 +43,13 @@ viterbi <- function(states, observations, start_prob, trans_prob, emit_prob) {
   }
   
   # Recursion step
-  for (t in 2:T_len) {
-    for (j in 1:N) {
-      probs <- V[, t - 1] * trans_prob[, states[j]] * emit_prob[states[j], observations[t]]
-      V[j, t] <- max(probs)
-      path[j, t] <- which.max(probs)
+  if (T_len > 1) {
+    for (t in 2:T_len) {
+      for (j in 1:N) {
+        probs <- V[, t - 1] * trans_prob[, states[j]] * emit_prob[states[j], observations[t]]
+        V[j, t] <- max(probs)
+        path[j, t] <- which.max(probs)
+      }
     }
   }