Dinics algo

dynamic_programming/viterbi.r

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+# ==============================================================
+# 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")