recurrence-relations time-complexity asymptotic-analysis loops graph dynamic-programming asymptotic-notation greedy substitution-method a-star np-completeness nested-loops vertex-coloring mdp log analysis probability stochastic heuristic master-theorem markov-model n-puzzle csp graph-coloring mvcs small-oh exponent proof viterbi bayes-rule hmm tree-search grid-world admissible n-queens conflict ai clique exam coins reduction dfs prime-numbers sqrt count easy sorted-lists logn example recursive gcd independent-set unsolvable pcp counter-example not-master-theorem modulus algebra most-likely-estimate reinforcement-learning direct-evaluation meu articulation-point hotel-room small-omega limit-method mle graph-search while-loop greedy-suboptimal job-assignment maximize-value gold constraint-satisfaction-problem 8-puzzle task-environments min-max peak randomized satisfiability random-graph-generation proxy network sudoku branchandbound d&c degree-constrained spanning-tree vertex-cover branch subtree series pmi bound contradiction math backtracking tree minimize floors